id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-14300 | Sum of Bernoulli Numbers by Power of Two and Binomial Coefficient | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \dbinom {2 n + 1} {2 k} 2^{2 k} B_{2 k}
| r = \binom {2 n + 1} 2 2^2 B_2 + \binom {2 n + 1} 4 2^4 B_4 + \binom {2 n + 1} 6 2^6 B_6 + \cdots
| c =
}}
{{eqn | r = 2 n
| c =
}}
{{end-eqn}}
where... | Let $B_k$ denote the $k$th Bernoulli number
Let $\map {B_k} x$ denote the $k$th Bernoulli polynomial
By Value of Odd Bernoulli Polynomial at One Half:
{{begin-eqn}}
{{eqn | l = \map {B_{2 n + 1} } {\frac 1 2}
| r = 0
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 0}^{2 n + 1} \binom {2 n + 1} k \map {B_k} ... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \dbinom {2 n + 1} {2 k} 2^{2 k} B_{2 k}
| r = \binom {2 n + 1} 2 2^2 B_2 + \binom {2 n + 1} 4 2^4 B_4 + \binom {2 n + 1} 6 2^6 B_6 + \cdots
| c =
}}
{{eqn |... | Let $B_k$ denote the $k$th [[Definition:Bernoulli Numbers|Bernoulli number]]
Let $\map {B_k} x$ denote the $k$th [[Definition: Bernoulli Polynomial|Bernoulli polynomial]]
By [[Value of Odd Bernoulli Polynomial at One Half]]:
{{begin-eqn}}
{{eqn | l = \map {B_{2 n + 1} } {\frac 1 2}
| r = 0
}}
{{eqn | ll= \lea... | Sum of Bernoulli Numbers by Power of Two and Binomial Coefficient/Proof 1 | https://proofwiki.org/wiki/Sum_of_Bernoulli_Numbers_by_Power_of_Two_and_Binomial_Coefficient | https://proofwiki.org/wiki/Sum_of_Bernoulli_Numbers_by_Power_of_Two_and_Binomial_Coefficient/Proof_1 | [
"Sum of Bernoulli Numbers by Power of Two and Binomial Coefficient",
"Bernoulli Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Bernoulli Numbers"
] | [
"Definition:Bernoulli Numbers",
"Definition: Bernoulli Polynomial",
"Value of Odd Bernoulli Polynomial at One Half",
"Odd Bernoulli Numbers Vanish"
] |
proofwiki-14301 | Sum of Bernoulli Numbers by Power of Two and Binomial Coefficient | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \dbinom {2 n + 1} {2 k} 2^{2 k} B_{2 k}
| r = \binom {2 n + 1} 2 2^2 B_2 + \binom {2 n + 1} 4 2^4 B_4 + \binom {2 n + 1} 6 2^6 B_6 + \cdots
| c =
}}
{{eqn | r = 2 n
| c =
}}
{{end-eqn}}
where... | The proof proceeds by induction.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$\ds \sum_{k \mathop = 1}^n \dbinom {2 n + 1} {2 k} 2^{2 k} B_{2 k} = 2 n$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = \binom {2 \times 1 + 1} 2 2^2 B_2
| r = \frac {2 \times 3} 2 \... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \dbinom {2 n + 1} {2 k} 2^{2 k} B_{2 k}
| r = \binom {2 n + 1} 2 2^2 B_2 + \binom {2 n + 1} 4 2^4 B_4 + \binom {2 n + 1} 6 2^6 B_6 + \cdots
| c =
}}
{{eqn |... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{k \mathop = 1}^n \dbinom {2 n + 1} {2 k} 2^{2 k} B_{2 k} = 2 n$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | ... | Sum of Bernoulli Numbers by Power of Two and Binomial Coefficient/Proof 2 | https://proofwiki.org/wiki/Sum_of_Bernoulli_Numbers_by_Power_of_Two_and_Binomial_Coefficient | https://proofwiki.org/wiki/Sum_of_Bernoulli_Numbers_by_Power_of_Two_and_Binomial_Coefficient/Proof_2 | [
"Sum of Bernoulli Numbers by Power of Two and Binomial Coefficient",
"Bernoulli Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Bernoulli Numbers"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Binomial Coefficient with Two",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Binomial Coefficient with Self minus One",
"Definition:Bernoulli Numbers/Recurrence Relation",
"Pri... |
proofwiki-14302 | Moment in terms of Moment Generating Function | Let $X$ be a random variable.
Let $M_X$ be the moment generating function of $X$.
Then:
:$\expect {X^n} = \map { {M_X}^{\paren n} } 0$
where:
:$n$ is a non-negative integer
:${M_X}^{\paren n}$ denotes the $n$th derivative of $M_X$
:$\expect {X^n}$ denotes the expectation of $X^n$. | {{begin-eqn}}
{{eqn | l = \map { {M_X}^{\paren n} } t
| r = \frac {\d^n} {\d t^n} \expect {e^{t X} }
| c = {{Defof|Moment Generating Function}}
}}
{{eqn | r = \frac {\d^n} {\d t^n} \expect {\sum_{m \mathop = 0}^\infty \frac {t^m X^m} {m!} }
| c = Power Series Expansion for Exponential Function
}}
{{eq... | Let $X$ be a [[Definition:Random Variable|random variable]].
Let $M_X$ be the [[Definition:Moment Generating Function|moment generating function]] of $X$.
Then:
:$\expect {X^n} = \map { {M_X}^{\paren n} } 0$
where:
:$n$ is a non-negative [[Definition:Integer|integer]]
:${M_X}^{\paren n}$ denotes the [[Definition:... | {{begin-eqn}}
{{eqn | l = \map { {M_X}^{\paren n} } t
| r = \frac {\d^n} {\d t^n} \expect {e^{t X} }
| c = {{Defof|Moment Generating Function}}
}}
{{eqn | r = \frac {\d^n} {\d t^n} \expect {\sum_{m \mathop = 0}^\infty \frac {t^m X^m} {m!} }
| c = [[Power Series Expansion for Exponential Function]]
}}
... | Moment in terms of Moment Generating Function | https://proofwiki.org/wiki/Moment_in_terms_of_Moment_Generating_Function | https://proofwiki.org/wiki/Moment_in_terms_of_Moment_Generating_Function | [
"Moment Generating Functions",
"Moments (Probability Theory)"
] | [
"Definition:Random Variable",
"Definition:Moment Generating Function",
"Definition:Integer",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Definition:Expectation"
] | [
"Power Series Expansion for Exponential Function",
"Expectation is Linear",
"Expectation is Linear",
"Power Series is Termwise Differentiable within Radius of Convergence",
"Nth Derivative of Mth Power",
"Falling Factorial as Quotient of Factorials"
] |
proofwiki-14303 | Moment Generating Function of Continuous Uniform Distribution | Let $X \sim \ContinuousUniform a b$ for some $a, b \in \R$ denote the continuous uniform distribution on the interval $\closedint a b$.
Then the moment generating function of $X$ is given by:
:$\map {M_X} t = \begin {cases} \dfrac {e^{t b} - e^{t a} } {t \paren {b - a} } & t \ne 0 \\ 1 & t = 0 \end{cases}$ | From the definition of the continuous uniform distribution, $X$ has probability density function:
:$\map {f_X} x = \begin{cases} \dfrac 1 {b - a} & a \le x \le b \\ 0 & \text{otherwise} \end{cases}$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect {e^{t X} } = \int_{-\infty}^\infty e^{... | Let $X \sim \ContinuousUniform a b$ for some $a, b \in \R$ denote the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]] on the [[Definition:Closed Real Interval|interval]] $\closedint a b$.
Then the [[Definition:Moment Generating Function|moment generating function]] of $X$ is given by:
... | From the definition of the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \begin{cases} \dfrac 1 {b - a} & a \le x \le b \\ 0 & \text{otherwise} \end{cases}$
From the definition of a [... | Moment Generating Function of Continuous Uniform Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Continuous_Uniform_Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Continuous_Uniform_Distribution | [
"Moment Generating Functions",
"Continuous Uniform Distribution"
] | [
"Definition:Uniform Distribution/Continuous",
"Definition:Real Interval/Closed",
"Definition:Moment Generating Function"
] | [
"Definition:Uniform Distribution/Continuous",
"Definition:Probability Density Function",
"Definition:Moment Generating Function",
"Definition:Expectation",
"Primitive of Exponential of a x",
"Fundamental Theorem of Calculus"
] |
proofwiki-14304 | Moment Generating Function of Normal Distribution | Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
Then the moment generating function $M_X$ of $X$ is given by:
:$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$ | From the definition of the normal distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt{2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect { e^{t X} } = \int_{-\infty}^\infty e^{t x} \ma... | Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the [[Definition:Normal Distribution|normal distribution]].
Then the [[Definition:Moment Generating Function|moment generating function]] $M_X$ of $X$ is given by:
:$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^... | From the definition of the [[Definition:Normal Distribution|normal distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt{2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
From the definition of a [[Definition:Moment Gen... | Moment Generating Function of Normal Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Normal_Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Normal_Distribution | [
"Moment Generating Function of Normal Distribution",
"Moment Generating Functions",
"Normal Distribution"
] | [
"Definition:Normal Distribution",
"Definition:Moment Generating Function"
] | [
"Definition:Normal Distribution",
"Definition:Probability Density Function",
"Definition:Moment Generating Function",
"Integration by Substitution",
"Integration by Substitution",
"Gaussian Integral"
] |
proofwiki-14305 | Bernoulli Number in terms of Euler Numbers | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
{{begin-eqn}}
{{eqn | l = B_{2 n}
| r = \frac {2 n} {2^{2 n} \paren {2^{2 n} - 1} } \paren {\sum_{k \mathop = 0}^{n - 1} \dbinom {2 n - 2} {2 k} E_{2 k} E_{2 n - 2 k - 2} }
| c =
}}
{{eqn | r = \frac {2 n} {2^{2 n} \paren {2^{2 n} - 1} } \paren {\... | {{begin-eqn}}
{{eqn | l = \tan x
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c = Power Series Expansion for Tangent Function
}}
{{eqn | ll = \leadsto
| l = \map {\dfrac \d {\d x} } {\tan x}
| r = \sum_{n \ma... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = B_{2 n}
| r = \frac {2 n} {2^{2 n} \paren {2^{2 n} - 1} } \paren {\sum_{k \mathop = 0}^{n - 1} \dbinom {2 n - 2} {2 k} E_{2 k} E_{2 n - 2 k - 2} }
| c =
}}
{{eqn | r = \frac {2 n}... | {{begin-eqn}}
{{eqn | l = \tan x
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c = [[Power Series Expansion for Tangent Function]]
}}
{{eqn | ll = \leadsto
| l = \map {\dfrac \d {\d x} } {\tan x}
| r = \sum_{n... | Bernoulli Number in terms of Euler Numbers | https://proofwiki.org/wiki/Bernoulli_Number_in_terms_of_Euler_Numbers | https://proofwiki.org/wiki/Bernoulli_Number_in_terms_of_Euler_Numbers | [
"Bernoulli Number in terms of Euler Numbers",
"Euler Numbers",
"Bernoulli Numbers",
"Examples of Equating Coefficients"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Bernoulli Numbers",
"Definition:Euler Numbers"
] | [
"Power Series Expansion for Tangent Function",
"Power Rule for Derivatives",
"Definition:Coefficient",
"Derivative of Tangent Function",
"Power Series Expansion for Secant Function",
"Definition:Coefficient",
"Definition:Coefficient"
] |
proofwiki-14306 | Taylor Series of Logarithm of Gamma Function | Let $\gamma$ denote the Euler-Mascheroni constant.
Let $\map \zeta s$ denote the Riemann zeta function.
Let $\map \Gamma z$ denote the gamma function.
Let $\Log$ denote the natural logarithm.
Then $\map \Log {\map \Gamma z}$ has the power series expansion:
{{begin-eqn}}
{{eqn | l = \map \Log {\map \Gamma z}
| r =... | {{begin-eqn}}
{{eqn | l = \map \Gamma {x + 1}
| r = x \map \Gamma x
| c = Gamma Difference Equation
}}
{{eqn | r = \paren {x } \paren {x - 1 } \map \Gamma {x - 1 }
| c =
}}
{{eqn | r = \paren {x } \paren {x - 1 } \paren {x - 2 } \map \Gamma {x - 2 }
| c =
}}
{{eqn | r = \paren {x } \paren {x -... | Let $\gamma$ denote the [[Definition:Euler-Mascheroni Constant|Euler-Mascheroni constant]].
Let $\map \zeta s$ denote the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Let $\map \Gamma z$ denote the [[Definition:Gamma Function|gamma function]].
Let $\Log$ denote the [[Definition:Natural Logarithm|natur... | {{begin-eqn}}
{{eqn | l = \map \Gamma {x + 1}
| r = x \map \Gamma x
| c = [[Gamma Difference Equation]]
}}
{{eqn | r = \paren {x } \paren {x - 1 } \map \Gamma {x - 1 }
| c =
}}
{{eqn | r = \paren {x } \paren {x - 1 } \paren {x - 2 } \map \Gamma {x - 2 }
| c =
}}
{{eqn | r = \paren {x } \paren ... | Taylor Series of Logarithm of Gamma Function | https://proofwiki.org/wiki/Taylor_Series_of_Logarithm_of_Gamma_Function | https://proofwiki.org/wiki/Taylor_Series_of_Logarithm_of_Gamma_Function | [
"Gamma Function",
"Natural Logarithms"
] | [
"Definition:Euler-Mascheroni Constant",
"Definition:Riemann Zeta Function",
"Definition:Gamma Function",
"Definition:Natural Logarithm",
"Definition:Power Series"
] | [
"Gamma Difference Equation",
"Sum of Logarithms",
"Sum Rule for Derivatives",
"Derivative of Natural Logarithm Function",
"Sum Rule for Derivatives",
"Nth Derivative of Reciprocal of Mth Power",
"Stirling's Formula for Gamma Function",
"Sum of Logarithms",
"Logarithm of Power/Natural Logarithm",
"... |
proofwiki-14307 | Equality of Vector Quantities | Two vector quantities are equal {{iff}} they have the same magnitude and direction.
That is:
:$\mathbf a = \mathbf b \iff \paren {\size {\mathbf a} = \size {\mathbf b} \land \hat {\mathbf a} = \hat {\mathbf b} }$
where:
:$\hat {\mathbf a}$ denotes the unit vector in the direction of $\mathbf a$
:$\size {\mathbf a}$ den... | Let $\mathbf a$ and $\mathbf b$ be expressed in component form:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = a_1 \mathbf e_1 + a_2 \mathbf e_2 + \cdots + a_n \mathbf e_n
| c =
}}
{{eqn | l = \mathbf b
| r = b_1 \mathbf e_1 + b_2 \mathbf e_2 + \cdots + b_n \mathbf e_n
| c =
}}
{{end-eqn}}
where $\... | Two [[Definition:Vector Quantity|vector quantities]] are [[Definition:Equality|equal]] {{iff}} they have the same [[Definition:Magnitude|magnitude]] and [[Definition:Direction|direction]].
That is:
:$\mathbf a = \mathbf b \iff \paren {\size {\mathbf a} = \size {\mathbf b} \land \hat {\mathbf a} = \hat {\mathbf b} }$
w... | Let $\mathbf a$ and $\mathbf b$ be expressed in [[Definition:Component of Vector|component form]]:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = a_1 \mathbf e_1 + a_2 \mathbf e_2 + \cdots + a_n \mathbf e_n
| c =
}}
{{eqn | l = \mathbf b
| r = b_1 \mathbf e_1 + b_2 \mathbf e_2 + \cdots + b_n \mathbf e_n
... | Equality of Vector Quantities | https://proofwiki.org/wiki/Equality_of_Vector_Quantities | https://proofwiki.org/wiki/Equality_of_Vector_Quantities | [
"Vectors",
"Equality"
] | [
"Definition:Vector Quantity",
"Definition:Equals",
"Definition:Magnitude",
"Definition:Direction",
"Definition:Unit Vector",
"Definition:Direction",
"Definition:Magnitude"
] | [
"Definition:Vector Quantity/Component",
"Definition:Unit Vector",
"Definition:Axis/Positive Direction",
"Definition:Axis/Coordinate Axes",
"Definition:Cartesian Coordinate System",
"Definition:Vector Length/Real Vector Space",
"Vector Quantity as Scalar Product of Unit Vector Quantity",
"Vector Quanti... |
proofwiki-14308 | Vector Addition is Commutative | Let $\mathbf a, \mathbf b$ be vector quantities.
Then:
:$\mathbf a + \mathbf b = \mathbf b + \mathbf a$ | From the Parallelogram Law:
:350px
{{finish}} | Let $\mathbf a, \mathbf b$ be [[Definition:Vector Quantity|vector quantities]].
Then:
:$\mathbf a + \mathbf b = \mathbf b + \mathbf a$ | From the [[Parallelogram Law]]:
:[[File:ParallelogramLaw.png|350px]]
{{finish}} | Vector Addition is Commutative | https://proofwiki.org/wiki/Vector_Addition_is_Commutative | https://proofwiki.org/wiki/Vector_Addition_is_Commutative | [
"Vectors",
"Vector Algebra"
] | [
"Definition:Vector Quantity"
] | [
"Parallelogram Law",
"File:ParallelogramLaw.png"
] |
proofwiki-14309 | Vector Addition is Associative | Let $\mathbf a, \mathbf b, \mathbf c$ be vectors.
Then:
:$\mathbf a + \paren {\mathbf b + \mathbf c} = \paren {\mathbf a + \mathbf b} + \mathbf c$
where $+$ denotes vector addition. | :420px
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be positioned in space so they are end to end as in the above diagram.
Let $\mathbf v$ be a vector representing the closing side of the polygon whose other $3$ sides are represented by $\mathbf a$, $\mathbf b$ and $\mathbf c$.
By the Parallelogram Law we can add any p... | Let $\mathbf a, \mathbf b, \mathbf c$ be [[Definition:Vector Quantity|vectors]].
Then:
:$\mathbf a + \paren {\mathbf b + \mathbf c} = \paren {\mathbf a + \mathbf b} + \mathbf c$
where $+$ denotes [[Definition:Vector Sum|vector addition]]. | :[[File:Vector-Addition-is-Associative.png|420px]]
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be positioned in space so they are end to end as in the above diagram.
Let $\mathbf v$ be a [[Definition:Vector Quantity|vector]] representing the closing [[Definition:Side of Polygon|side]] of the [[Definition:Polygon|pol... | Vector Addition is Associative/Proof 1 | https://proofwiki.org/wiki/Vector_Addition_is_Associative | https://proofwiki.org/wiki/Vector_Addition_is_Associative/Proof_1 | [
"Vector Addition is Associative",
"Vectors",
"Vector Addition"
] | [
"Definition:Vector Quantity",
"Definition:Vector Sum"
] | [
"File:Vector-Addition-is-Associative.png",
"Definition:Vector Quantity",
"Definition:Polygon/Side",
"Definition:Polygon",
"Definition:Polygon/Side",
"Parallelogram Law",
"Definition:Doubleton",
"Definition:Vector Quantity",
"Definition:Vector Quantity",
"Definition:Vector Sum"
] |
proofwiki-14310 | Vector Addition is Associative | Let $\mathbf a, \mathbf b, \mathbf c$ be vectors.
Then:
:$\mathbf a + \paren {\mathbf b + \mathbf c} = \paren {\mathbf a + \mathbf b} + \mathbf c$
where $+$ denotes vector addition. | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be expressed in component form:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = a_1 \mathbf e_1 + a_2 \mathbf e_2 + \dotsb + a_n \mathbf e_n
}}
{{eqn | l = \mathbf b
| r = b_1 \mathbf e_1 + b_2 \mathbf e_2 + \dotsb + b_n \mathbf e_n
}}
{{eqn | l = \mathbf c
| r =... | Let $\mathbf a, \mathbf b, \mathbf c$ be [[Definition:Vector Quantity|vectors]].
Then:
:$\mathbf a + \paren {\mathbf b + \mathbf c} = \paren {\mathbf a + \mathbf b} + \mathbf c$
where $+$ denotes [[Definition:Vector Sum|vector addition]]. | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be expressed in [[Definition:Component of Vector|component form]]:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = a_1 \mathbf e_1 + a_2 \mathbf e_2 + \dotsb + a_n \mathbf e_n
}}
{{eqn | l = \mathbf b
| r = b_1 \mathbf e_1 + b_2 \mathbf e_2 + \dotsb + b_n \mathbf e_n
... | Vector Addition is Associative/Proof 2 | https://proofwiki.org/wiki/Vector_Addition_is_Associative | https://proofwiki.org/wiki/Vector_Addition_is_Associative/Proof_2 | [
"Vector Addition is Associative",
"Vectors",
"Vector Addition"
] | [
"Definition:Vector Quantity",
"Definition:Vector Sum"
] | [
"Definition:Vector Quantity/Component",
"Scalar Multiplication of Vectors is Distributive over Vector Addition",
"Scalar Multiplication of Vectors is Distributive over Vector Addition",
"Associative Law of Addition",
"Scalar Multiplication of Vectors is Distributive over Vector Addition",
"Scalar Multipli... |
proofwiki-14311 | Scalar Multiplication of Vectors is Associative | Let $\mathbf a$ be a vector quantity.
Let $m, n$ be scalar quantities.
Then:
:$m \paren {n \mathbf a} = \paren {m n} \mathbf a = n \paren {m \mathbf a}$ | {{ProofWanted|Need to consider which definition you start from}} | Let $\mathbf a$ be a [[Definition:Vector Quantity|vector quantity]].
Let $m, n$ be [[Definition:Scalar Quantity|scalar quantities]].
Then:
:$m \paren {n \mathbf a} = \paren {m n} \mathbf a = n \paren {m \mathbf a}$ | {{ProofWanted|Need to consider which definition you start from}} | Scalar Multiplication of Vectors is Associative | https://proofwiki.org/wiki/Scalar_Multiplication_of_Vectors_is_Associative | https://proofwiki.org/wiki/Scalar_Multiplication_of_Vectors_is_Associative | [
"Vectors",
"Vector Algebra",
"Scalar Multiplication"
] | [
"Definition:Vector Quantity",
"Definition:Scalar Quantity"
] | [] |
proofwiki-14312 | Scalar Multiplication of Vectors is Distributive over Scalar Addition | Let $\mathbf a$ be a vector quantity.
Let $m, n$ be scalar quantities.
Then:
:$\paren {m + n} \mathbf a = m \mathbf a + n \mathbf a$ | {{ProofWanted|Need to consider which definition you start from}} | Let $\mathbf a$ be a [[Definition:Vector Quantity|vector quantity]].
Let $m, n$ be [[Definition:Scalar Quantity|scalar quantities]].
Then:
:$\paren {m + n} \mathbf a = m \mathbf a + n \mathbf a$ | {{ProofWanted|Need to consider which definition you start from}} | Scalar Multiplication of Vectors is Distributive over Scalar Addition | https://proofwiki.org/wiki/Scalar_Multiplication_of_Vectors_is_Distributive_over_Scalar_Addition | https://proofwiki.org/wiki/Scalar_Multiplication_of_Vectors_is_Distributive_over_Scalar_Addition | [
"Vectors",
"Vector Algebra",
"Scalar Multiplication",
"Vector Addition"
] | [
"Definition:Vector Quantity",
"Definition:Scalar Quantity"
] | [] |
proofwiki-14313 | Scalar Multiplication of Vectors is Distributive over Vector Addition | Let $\mathbf a, \mathbf b$ be a vector quantities.
Let $m$ be a scalar quantity.
Then:
:$m \paren {\mathbf a + \mathbf b} = m \mathbf a + m \mathbf b$ | :400px
Let $\mathbf a = \vec {OP}$ and $\mathbf b = \vec {PQ}$.
Then:
:$\vec {OQ} = \mathbf a + \mathbf b$
Let $P'$ and $Q'$ be points on $OP$ and $OQ$ respectively so that:
:$OP' : OP = OQ' : OQ = m$
Then $P'Q'$ is parallel to $PQ$ and $m$ times it in length.
Thus:
:$\vec {P'Q'} = m \mathbf b$
which shows that:
{{begi... | Let $\mathbf a, \mathbf b$ be a [[Definition:Vector Quantity|vector quantities]].
Let $m$ be a [[Definition:Scalar Quantity|scalar quantity]].
Then:
:$m \paren {\mathbf a + \mathbf b} = m \mathbf a + m \mathbf b$ | :[[File:Scalar-product-distributes-over-vector-addition.png|400px]]
Let $\mathbf a = \vec {OP}$ and $\mathbf b = \vec {PQ}$.
Then:
:$\vec {OQ} = \mathbf a + \mathbf b$
Let $P'$ and $Q'$ be [[Definition:Point|points]] on $OP$ and $OQ$ respectively so that:
:$OP' : OP = OQ' : OQ = m$
Then $P'Q'$ is [[Definition:Paral... | Scalar Multiplication of Vectors is Distributive over Vector Addition | https://proofwiki.org/wiki/Scalar_Multiplication_of_Vectors_is_Distributive_over_Vector_Addition | https://proofwiki.org/wiki/Scalar_Multiplication_of_Vectors_is_Distributive_over_Vector_Addition | [
"Vectors",
"Vector Algebra",
"Scalar Multiplication",
"Vector Addition"
] | [
"Definition:Vector Quantity",
"Definition:Scalar Quantity"
] | [
"File:Scalar-product-distributes-over-vector-addition.png",
"Definition:Point",
"Definition:Parallel (Geometry)/Lines",
"Definition:Linear Measure/Length"
] |
proofwiki-14314 | Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors | Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space of $3$ dimensions:
Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$.
Then $\norm {\mathbf a \times \mathbf b}$ equals the area of the parallelogram two of whose sides are $\mathbf a$ and $\mathbf b$. | By definition of vector cross product:
:$\mathbf a \times \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$
where:
:$\norm {\mathbf a}$ denotes the length of $\mathbf a$
:$\theta$ denotes the angle from $\mathbf a$ to $\mathbf b$, measured in the positive direction
:$\mathbf {\hat n}$ is ... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector (Linear Algebra)|vectors]] in a [[Definition:Vector Space|vector space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]]:
Let $\mathbf a \times \mathbf b$ denote the [[Definition:Vector Cross Product|vector cross product]] of $\mathbf a$ with $\mathbf b... | By definition of [[Definition:Vector Cross Product/Definition 2|vector cross product]]:
:$\mathbf a \times \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$
where:
:$\norm {\mathbf a}$ denotes the [[Definition:Vector Length|length]] of $\mathbf a$
:$\theta$ denotes the [[Definition:Angle|... | Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors | https://proofwiki.org/wiki/Magnitude_of_Vector_Cross_Product_equals_Area_of_Parallelogram_Contained_by_Vectors | https://proofwiki.org/wiki/Magnitude_of_Vector_Cross_Product_equals_Area_of_Parallelogram_Contained_by_Vectors | [
"Vector Cross Product"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Vector Cross Product",
"Definition:Area",
"Definition:Quadrilateral/Parallelogram",
"Definition:Polygon/Side"
] | [
"Definition:Vector Cross Product/Definition 2",
"Definition:Vector Length",
"Definition:Angle",
"Definition:Axis/Positive Direction",
"Definition:Unit Vector",
"Definition:Right Angle/Perpendicular",
"Definition:Right-Hand Rule/Cross Product",
"Definition:Unit Vector",
"Area of Parallelogram",
"De... |
proofwiki-14315 | Differential Entropy of Normal Distribution | Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
Then the differential entropy $\map h X$ of $X$ is given by:
:$\map h X = \map \ln {\sigma \sqrt {2 \pi} } + \dfrac 1 2$ | From the definition of the normal distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \, \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
From the definition of differential entropy:
:$\ds \map h X = -\int_{-\infty}^\infty \map {f_X} x \ln \map {f_X} x \rd x$
So... | Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the [[Definition:Normal Distribution|normal distribution]].
Then the [[Definition:Differential Entropy|differential entropy]] $\map h X$ of $X$ is given by:
:$\map h X = \map \ln {\sigma \sqrt {2 \pi} } + \dfrac 1 2$ | From the definition of the [[Definition:Normal Distribution|normal distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \, \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
From the definition of [[Definition:Differen... | Differential Entropy of Normal Distribution | https://proofwiki.org/wiki/Differential_Entropy_of_Normal_Distribution | https://proofwiki.org/wiki/Differential_Entropy_of_Normal_Distribution | [
"Normal Distribution",
"Differential Entropy"
] | [
"Definition:Normal Distribution",
"Definition:Differential Entropy"
] | [
"Definition:Normal Distribution",
"Definition:Probability Density Function",
"Definition:Differential Entropy",
"Logarithm of Reciprocal",
"Integration by Substitution",
"Sum of Logarithms",
"Gaussian Integral",
"Integration by Parts",
"Fundamental Theorem of Calculus",
"Exponential Tends to Zero ... |
proofwiki-14316 | Magnitude of Scalar Triple Product equals Volume of Parallelepiped Contained by Vectors | Let $\mathbf a, \mathbf b, \mathbf c$ be vectors in a vector space of $3$ dimensions:
Let $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$ denote the scalar triple product of $\mathbf a, \mathbf b, \mathbf c$.
Then $\size {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} }$ equals the volume of the parallelepip... | Let us construct the parallelepiped $P$ contained by $\mathbf a, \mathbf b, \mathbf c$.
:500px
We have by Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors that:
:$\mathbf b \times \mathbf c$ is a vector area equal to and normal to the area of the bottom face $S$ of $P$.
The dot produc... | Let $\mathbf a, \mathbf b, \mathbf c$ be [[Definition:Vector (Linear Algebra)|vectors]] in a [[Definition:Vector Space|vector space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]]:
Let $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$ denote the [[Definition:Scalar Triple Product|scalar triple product... | Let us construct the [[Definition:Parallelepiped|parallelepiped]] $P$ contained by $\mathbf a, \mathbf b, \mathbf c$.
:[[File:Scalar-triple-product-parallelepiped.png|500px]]
We have by [[Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors]] that:
:$\mathbf b \times \mathbf c$ is a [[D... | Magnitude of Scalar Triple Product equals Volume of Parallelepiped Contained by Vectors | https://proofwiki.org/wiki/Magnitude_of_Scalar_Triple_Product_equals_Volume_of_Parallelepiped_Contained_by_Vectors | https://proofwiki.org/wiki/Magnitude_of_Scalar_Triple_Product_equals_Volume_of_Parallelepiped_Contained_by_Vectors | [
"Scalar Triple Product",
"Parallelepipeds"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Scalar Triple Product",
"Definition:Volume",
"Definition:Parallelepiped"
] | [
"Definition:Parallelepiped",
"File:Scalar-triple-product-parallelepiped.png",
"Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors",
"Definition:Vector Area",
"Definition:Normal Vector",
"Definition:Area",
"Definition:Polyhedron/Face",
"Definition:Dot Product",
"Defin... |
proofwiki-14317 | Lagrange's Formula/Corollary | :$\paren {\mathbf a \times \mathbf b} \times \mathbf c = \paren {\mathbf a \cdot \mathbf c} \mathbf b - \paren {\mathbf b \cdot \mathbf c} \mathbf a$ | {{begin-eqn}}
{{eqn | l = \mathbf c \times \paren {\mathbf a \times \mathbf b}
| r = \paren {\mathbf c \cdot \mathbf b} \mathbf a - \paren {\mathbf c \cdot \mathbf a} \mathbf b
| c = Lagrange's Formula
}}
{{eqn | ll= \leadsto
| l = \paren {\mathbf a \times \mathbf b} \times \mathbf c
| r = -\par... | :$\paren {\mathbf a \times \mathbf b} \times \mathbf c = \paren {\mathbf a \cdot \mathbf c} \mathbf b - \paren {\mathbf b \cdot \mathbf c} \mathbf a$ | {{begin-eqn}}
{{eqn | l = \mathbf c \times \paren {\mathbf a \times \mathbf b}
| r = \paren {\mathbf c \cdot \mathbf b} \mathbf a - \paren {\mathbf c \cdot \mathbf a} \mathbf b
| c = [[Lagrange's Formula]]
}}
{{eqn | ll= \leadsto
| l = \paren {\mathbf a \times \mathbf b} \times \mathbf c
| r = -... | Lagrange's Formula/Corollary | https://proofwiki.org/wiki/Lagrange's_Formula/Corollary | https://proofwiki.org/wiki/Lagrange's_Formula/Corollary | [
"Vector Algebra",
"Dot Product",
"Vector Cross Product"
] | [] | [
"Lagrange's Formula",
"Vector Cross Product is Anticommutative",
"Dot Product Operator is Commutative",
"Real Addition is Commutative"
] |
proofwiki-14318 | Differential Entropy of Continuous Uniform Distribution | Let $X \sim \ContinuousUniform a b$ for some $a, b \in \R$, $a \ne b$, where $\operatorname U$ is the continuous uniform distribution.
Then the differential entropy of $X$, $\map h X$, is given by:
:$\map h X = \map \ln {b - a}$ | From the definition of the continuous uniform distribution, $X$ has probability density function:
:$\map {f_X} x = \begin{cases} \dfrac 1 {b - a} & : a \le x \le b \\ 0 & : \text{otherwise} \end{cases}$
From the definition of differential entropy:
:$\ds \map h X = - \int_{-\infty}^\infty \map {f_X} x \map \ln {\map {... | Let $X \sim \ContinuousUniform a b$ for some $a, b \in \R$, $a \ne b$, where $\operatorname U$ is the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]].
Then the [[Definition:Differential Entropy|differential entropy]] of $X$, $\map h X$, is given by:
:$\map h X = \map \ln {b - a}$ | From the definition of the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \begin{cases} \dfrac 1 {b - a} & : a \le x \le b \\ 0 & : \text{otherwise} \end{cases}$
From the definition of ... | Differential Entropy of Continuous Uniform Distribution | https://proofwiki.org/wiki/Differential_Entropy_of_Continuous_Uniform_Distribution | https://proofwiki.org/wiki/Differential_Entropy_of_Continuous_Uniform_Distribution | [
"Continuous Uniform Distribution",
"Differential Entropy"
] | [
"Definition:Uniform Distribution/Continuous",
"Definition:Differential Entropy"
] | [
"Definition:Uniform Distribution/Continuous",
"Definition:Probability Density Function",
"Definition:Differential Entropy",
"Logarithm of Reciprocal",
"Primitive of Constant",
"Fundamental Theorem of Calculus",
"Category:Continuous Uniform Distribution",
"Category:Differential Entropy"
] |
proofwiki-14319 | Dot Product of Vector Cross Products | Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be vectors in a vector space $\mathbf V$ of $3$ dimensions.
Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$.
Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ with $\mathbf b$.
Then:
:$\paren {\mathbf a... | {{begin-eqn}}
{{eqn | l = \paren {\mathbf a \times \mathbf b} \cdot \paren {\mathbf c \times \mathbf d}
| r = \sqbrk {\mathbf a, \mathbf b, \mathbf c \times \mathbf d}
| c = {{Defof|Scalar Triple Product}}
}}
{{eqn | r = \sqbrk {\mathbf b, \mathbf c \times \mathbf d, \mathbf a}
| c = Equivalent Expres... | Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be [[Definition:Vector (Linear Algebra)|vectors]] in a [[Definition:Vector Space|vector space]] $\mathbf V$ of [[Definition:Dimension of Vector Space|$3$ dimensions]].
Let $\mathbf a \times \mathbf b$ denote the [[Definition:Vector Cross Product|vector cross product]] o... | {{begin-eqn}}
{{eqn | l = \paren {\mathbf a \times \mathbf b} \cdot \paren {\mathbf c \times \mathbf d}
| r = \sqbrk {\mathbf a, \mathbf b, \mathbf c \times \mathbf d}
| c = {{Defof|Scalar Triple Product}}
}}
{{eqn | r = \sqbrk {\mathbf b, \mathbf c \times \mathbf d, \mathbf a}
| c = [[Equivalent Expr... | Dot Product of Vector Cross Products/Proof 1 | https://proofwiki.org/wiki/Dot_Product_of_Vector_Cross_Products | https://proofwiki.org/wiki/Dot_Product_of_Vector_Cross_Products/Proof_1 | [
"Dot Product of Vector Cross Products",
"Vector Cross Product",
"Dot Product"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Vector Cross Product",
"Definition:Dot Product"
] | [
"Equivalent Expressions for Scalar Triple Product",
"Lagrange's Formula",
"Dot Product Distributes over Addition",
"Dot Product Operator is Commutative"
] |
proofwiki-14320 | Vector Cross Product of Vector Cross Products | Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be vectors in a vector space $\mathbf V$ of $3$ dimensions:
Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$.
Let $\sqbrk {\mathbf a, \mathbf b, \mathbf c}$ denote the scalar triple product of $\mathbf a$, $\mathbf b$ and $... | {{begin-eqn}}
{{eqn | l = \paren {\mathbf a \times \mathbf b} \times \paren {\mathbf c \times \mathbf d}
| r = \paren {\paren {\mathbf a \times \mathbf b} \cdot \mathbf d} \mathbf c - \paren {\paren {\mathbf a \times \mathbf b} \cdot \mathbf c} \mathbf d
| c = Lagrange's Formula
}}
{{eqn | r = \sqbrk {\math... | Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be [[Definition:Vector (Linear Algebra)|vectors]] in a [[Definition:Vector Space|vector space]] $\mathbf V$ of [[Definition:Dimension of Vector Space|$3$ dimensions]]:
Let $\mathbf a \times \mathbf b$ denote the [[Definition:Vector Cross Product|vector cross product]] o... | {{begin-eqn}}
{{eqn | l = \paren {\mathbf a \times \mathbf b} \times \paren {\mathbf c \times \mathbf d}
| r = \paren {\paren {\mathbf a \times \mathbf b} \cdot \mathbf d} \mathbf c - \paren {\paren {\mathbf a \times \mathbf b} \cdot \mathbf c} \mathbf d
| c = [[Lagrange's Formula]]
}}
{{eqn | r = \sqbrk {\... | Vector Cross Product of Vector Cross Products | https://proofwiki.org/wiki/Vector_Cross_Product_of_Vector_Cross_Products | https://proofwiki.org/wiki/Vector_Cross_Product_of_Vector_Cross_Products | [
"Vector Cross Product of Vector Cross Products",
"Vector Cross Product"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Vector Cross Product",
"Definition:Scalar Triple Product"
] | [
"Lagrange's Formula",
"Lagrange's Formula/Corollary"
] |
proofwiki-14321 | Derivative of Scalar Triple Product of Vector-Valued Functions | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be differentiable vector-valued functions in Cartesian $3$-space.
The derivative of their scalar triple product is given by:
:$\map {\dfrac \d {\d x} } {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} } = \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathb... | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} }
| r = \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathbf c} + \mathbf a \cdot \map {\dfrac \d {\d x} } {\mathbf b \times \mathbf c}
| c = Derivative of Dot Product of Vector-Valued Func... | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be [[Definition:Differentiable Vector-Valued Function|differentiable]] [[Definition:Vector-Valued Function|vector-valued functions]] in [[Definition:Cartesian 3-Space|Cartesian $3$-space]].
The [[Definition:Derivative of Vector-Valued Function|derivative]] of their [[Defin... | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} }
| r = \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathbf c} + \mathbf a \cdot \map {\dfrac \d {\d x} } {\mathbf b \times \mathbf c}
| c = [[Derivative of Dot Product of Vector-Valued Fu... | Derivative of Scalar Triple Product of Vector-Valued Functions | https://proofwiki.org/wiki/Derivative_of_Scalar_Triple_Product_of_Vector-Valued_Functions | https://proofwiki.org/wiki/Derivative_of_Scalar_Triple_Product_of_Vector-Valued_Functions | [
"Differential Calculus",
"Vector Calculus",
"Scalar Triple Product"
] | [
"Definition:Differentiable Mapping/Vector-Valued Function",
"Definition:Vector-Valued Function",
"Definition:Cartesian 3-Space",
"Definition:Derivative/Vector-Valued Function",
"Definition:Scalar Triple Product"
] | [
"Derivative of Dot Product of Vector-Valued Functions",
"Derivative of Vector Cross Product of Vector-Valued Functions",
"Dot Product Distributes over Addition"
] |
proofwiki-14322 | Dot Product of Vector-Valued Function with its Derivative | Let:
:$\map {\mathbf f} x = \ds \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$
be a differentiable vector-valued function.
The dot product of $\mathbf f$ with its derivative is given by:
:$\map {\mathbf f} x \cdot \dfrac {\map {\d \mathbf f} x} {\d x} = \size {\map {\mathbf f} x} \dfrac {\d \size {\map {\mathbf f} x}... | {{begin-eqn}}
{{eqn | l = \map {\mathbf f} x \cdot \dfrac {\map {\d \mathbf f} x} {\d x}
| r = \map {\mathbf f} x \cdot \sum_{k \mathop = 0}^n \dfrac {\map {\d f_k} x} {\d x} \mathbf e_k
| c = {{Defof|Derivative of Vector-Valued Function}}
}}
{{eqn | r = \sum_{k \mathop = 0}^n \map {f_k} x \dfrac {\map {\d ... | Let:
:$\map {\mathbf f} x = \ds \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$
be a [[Definition:Differentiable Vector-Valued Function|differentiable]] [[Definition:Vector-Valued Function|vector-valued function]].
The [[Definition:Dot Product|dot product]] of $\mathbf f$ with its [[Definition:Derivative of Vector-... | {{begin-eqn}}
{{eqn | l = \map {\mathbf f} x \cdot \dfrac {\map {\d \mathbf f} x} {\d x}
| r = \map {\mathbf f} x \cdot \sum_{k \mathop = 0}^n \dfrac {\map {\d f_k} x} {\d x} \mathbf e_k
| c = {{Defof|Derivative of Vector-Valued Function}}
}}
{{eqn | r = \sum_{k \mathop = 0}^n \map {f_k} x \dfrac {\map {\d ... | Dot Product of Vector-Valued Function with its Derivative | https://proofwiki.org/wiki/Dot_Product_of_Vector-Valued_Function_with_its_Derivative | https://proofwiki.org/wiki/Dot_Product_of_Vector-Valued_Function_with_its_Derivative | [
"Differential Calculus",
"Vector Calculus",
"Dot Product"
] | [
"Definition:Differentiable Mapping/Vector-Valued Function",
"Definition:Vector-Valued Function",
"Definition:Dot Product",
"Definition:Derivative/Vector-Valued Function"
] | [
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Sum Rule for Derivatives/General Result",
"Derivative of Composite Function",
"Power Rule for Derivatives"
] |
proofwiki-14323 | Expectation of Beta Distribution | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the beta distribution.
Then:
:$\expect X = \dfrac \alpha {\alpha + \beta}$ | From the definition of the beta distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$
From the definition of the expected value of a continuous random variable:
:$\ds \expect X = \int_0^1 x \map {f_X} x \rd x$
So:
{{begin... | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the [[Definition:Beta Distribution|beta distribution]].
Then:
:$\expect X = \dfrac \alpha {\alpha + \beta}$ | From the definition of the [[Definition:Beta Distribution|beta distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$
From the definition of the [[Definition:Expectation of Co... | Expectation of Beta Distribution/Proof 1 | https://proofwiki.org/wiki/Expectation_of_Beta_Distribution | https://proofwiki.org/wiki/Expectation_of_Beta_Distribution/Proof_1 | [
"Expectation of Beta Distribution",
"Expectation",
"Beta Distribution"
] | [
"Definition:Beta Distribution"
] | [
"Definition:Beta Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Gamma Difference Equation"
] |
proofwiki-14324 | Expectation of Beta Distribution | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the beta distribution.
Then:
:$\expect X = \dfrac \alpha {\alpha + \beta}$ | {{begin-eqn}}
{{eqn | l = \expect X
| r = \prod_{r \mathop = 0}^0 \frac {\alpha + r} {\alpha + \beta + r}
| c = Raw Moment of Beta Distribution
}}
{{eqn | r = \frac {\alpha + 0} {\alpha + \beta + 0}
}}
{{eqn | r = \frac \alpha {\alpha + \beta}
}}
{{end-eqn}}
{{qed}} | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the [[Definition:Beta Distribution|beta distribution]].
Then:
:$\expect X = \dfrac \alpha {\alpha + \beta}$ | {{begin-eqn}}
{{eqn | l = \expect X
| r = \prod_{r \mathop = 0}^0 \frac {\alpha + r} {\alpha + \beta + r}
| c = [[Raw Moment of Beta Distribution]]
}}
{{eqn | r = \frac {\alpha + 0} {\alpha + \beta + 0}
}}
{{eqn | r = \frac \alpha {\alpha + \beta}
}}
{{end-eqn}}
{{qed}} | Expectation of Beta Distribution/Proof 2 | https://proofwiki.org/wiki/Expectation_of_Beta_Distribution | https://proofwiki.org/wiki/Expectation_of_Beta_Distribution/Proof_2 | [
"Expectation of Beta Distribution",
"Expectation",
"Beta Distribution"
] | [
"Definition:Beta Distribution"
] | [
"Raw Moment of Beta Distribution"
] |
proofwiki-14325 | Variance of Beta Distribution | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the beta distribution.
Then:
:$\var X = \dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} }$ | From the definition of the Beta distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$
From Variance as Expectation of Square minus Square of Expectation:
:$\ds \var X = \int_0^1 x^2 \map {f_X} X \rd x - \paren {\expect X}... | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the [[Definition:Beta Distribution|beta distribution]].
Then:
:$\var X = \dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} }$ | From the definition of the [[Definition:Beta Distribution|Beta distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$
From [[Variance as Expectation of Square minus Square of ... | Variance of Beta Distribution/Proof 1 | https://proofwiki.org/wiki/Variance_of_Beta_Distribution | https://proofwiki.org/wiki/Variance_of_Beta_Distribution/Proof_1 | [
"Variance of Beta Distribution",
"Variance",
"Beta Distribution"
] | [
"Definition:Beta Distribution"
] | [
"Definition:Beta Distribution",
"Definition:Probability Density Function",
"Variance as Expectation of Square minus Square of Expectation/Continuous",
"Expectation of Beta Distribution",
"Gamma Difference Equation"
] |
proofwiki-14326 | Variance of Beta Distribution | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the beta distribution.
Then:
:$\var X = \dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} }$ | From the definition of Variance as Expectation of Square minus Square of Expectation:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
From Expectation of Beta Distribution:
:$\expect X = \dfrac \alpha {\alpha + \beta}$
From Raw Moment of Beta Distribution:
:$\ds \expect {X^n} = \prod_{r \mathop = 0}^{n - 1} \frac {\al... | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ denotes the [[Definition:Beta Distribution|beta distribution]].
Then:
:$\var X = \dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} }$ | From the definition of [[Variance as Expectation of Square minus Square of Expectation]]:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
From [[Expectation of Beta Distribution]]:
:$\expect X = \dfrac \alpha {\alpha + \beta}$
From [[Raw Moment of Beta Distribution]]:
:$\ds \expect {X^n} = \prod_{r \mathop = 0}^{n ... | Variance of Beta Distribution/Proof 2 | https://proofwiki.org/wiki/Variance_of_Beta_Distribution | https://proofwiki.org/wiki/Variance_of_Beta_Distribution/Proof_2 | [
"Variance of Beta Distribution",
"Variance",
"Beta Distribution"
] | [
"Definition:Beta Distribution"
] | [
"Variance as Expectation of Square minus Square of Expectation",
"Expectation of Beta Distribution",
"Raw Moment of Beta Distribution",
"Raw Moment of Beta Distribution"
] |
proofwiki-14327 | Moment Generating Function of Gamma Distribution | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
Then the moment generating function of $X$ is given by:
:$\map {M_X} t = \begin {cases} \paren {1 - \dfrac t \beta}^{-\alpha} & t < \beta \\ \text {does not exist} & t \ge \beta \end {cases}$ | From the definition of the Gamma distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\map \Gamma \alpha}$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect {e^{t X} } = \int_0^\infty e^{t x} \map {f_X} x \rd x$
First ta... | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the [[Definition:Gamma Distribution|Gamma distribution]].
Then the [[Definition:Moment Generating Function|moment generating function]] of $X$ is given by:
:$\map {M_X} t = \begin {cases} \paren {1 - \dfrac t \beta}^{-\alpha} & ... | From the definition of the [[Definition:Gamma Distribution|Gamma distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {\beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\map \Gamma \alpha}$
From the definition of a [[Definition:Moment Generating Function|... | Moment Generating Function of Gamma Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Gamma_Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Gamma_Distribution | [
"Moment Generating Function of Gamma Distribution",
"Moment Generating Functions",
"Gamma Distribution"
] | [
"Definition:Gamma Distribution",
"Definition:Moment Generating Function"
] | [
"Definition:Gamma Distribution",
"Definition:Probability Density Function",
"Definition:Moment Generating Function",
"Integration by Substitution",
"Primitive of Power",
"Fundamental Theorem of Calculus",
"Definition:Positive/Real Number",
"Exponential Dominates Polynomial"
] |
proofwiki-14328 | Gradient Operator Distributes over Addition | Let $\mathbf V$ be a vector space of $n$ dimensions.
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis of $\mathbf V$.
Let $\map f {x_1, x_2, \ldots, x_n}, \map g {x_1, x_2, \ldots, x_n}: \mathbf V \to \R$ be differentiable real-valued functions on $\mathbf V$.
Let $\nabla f$ de... | {{begin-eqn}}
{{eqn | l = \nabla \paren {f + g}
| r = \sum_{k \mathop = 1}^n \frac {\partial \paren {f + g} } {\partial x_k} \mathbf e_k
| c = {{Defof|Gradient Operator}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {\frac {\partial f} {\partial x_k} \mathbf e_k + \frac {\partial g} {\partial x_k} \mathbf e... | Let $\mathbf V$ be a [[Definition:Vector Space|vector space]] of [[Definition:Dimension of Vector Space|$n$ dimensions]].
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the [[Definition:Standard Ordered Basis on Vector Space|standard ordered basis of $\mathbf V$]].
Let $\map f {x_1, x_2, \ldots, x_n}... | {{begin-eqn}}
{{eqn | l = \nabla \paren {f + g}
| r = \sum_{k \mathop = 1}^n \frac {\partial \paren {f + g} } {\partial x_k} \mathbf e_k
| c = {{Defof|Gradient Operator}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {\frac {\partial f} {\partial x_k} \mathbf e_k + \frac {\partial g} {\partial x_k} \mathbf e... | Gradient Operator Distributes over Addition | https://proofwiki.org/wiki/Gradient_Operator_Distributes_over_Addition | https://proofwiki.org/wiki/Gradient_Operator_Distributes_over_Addition | [
"Gradient Operator"
] | [
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Differentiable Mapping/Real-Valued Function",
"Definition:Gradient Operator"
] | [
"Linear Combination of Partial Derivatives",
"Sum of Summations equals Summation of Sum"
] |
proofwiki-14329 | Divergence Operator Distributes over Addition | Let $\map {\mathbf V} {x_1, x_2, \ldots, x_n}$ be a vector space of $n$ dimensions.
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis of $\mathbf V$.
Let $\mathbf f$ and $\mathbf g: \mathbf V \to \mathbf V$ be vector-valued functions on $\mathbf V$:
:$\mathbf f := \tuple {\map {... | {{begin-eqn}}
{{eqn | l = \nabla \cdot \paren {\mathbf f + \mathbf g}
| r = \sum_{k \mathop = 1}^n \frac {\map \partial {f_k + g_k} } {\partial x_k}
| c = {{Defof|Divergence Operator}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {\frac {\partial f_k} {\partial x_k} + \frac {\partial g_k} {\partial x_k} }
... | Let $\map {\mathbf V} {x_1, x_2, \ldots, x_n}$ be a [[Definition:Vector Space|vector space]] of [[Definition:Dimension of Vector Space|$n$ dimensions]].
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the [[Definition:Standard Ordered Basis on Vector Space|standard ordered basis of $\mathbf V$]].
Let... | {{begin-eqn}}
{{eqn | l = \nabla \cdot \paren {\mathbf f + \mathbf g}
| r = \sum_{k \mathop = 1}^n \frac {\map \partial {f_k + g_k} } {\partial x_k}
| c = {{Defof|Divergence Operator}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {\frac {\partial f_k} {\partial x_k} + \frac {\partial g_k} {\partial x_k} }
... | Divergence Operator Distributes over Addition | https://proofwiki.org/wiki/Divergence_Operator_Distributes_over_Addition | https://proofwiki.org/wiki/Divergence_Operator_Distributes_over_Addition | [
"Divergence Operator"
] | [
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Vector-Valued Function",
"Definition:Divergence Operator"
] | [
"Linear Combination of Partial Derivatives"
] |
proofwiki-14330 | Curl Operator Distributes over Addition | Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions..
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Let $\mathbf f$ and $\mathbf g: \R^3 \to \R^3$ be vector-valued functions on $\R^3$:
:$\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf ... | {{begin-eqn}}
{{eqn | l = \nabla \times \paren {\mathbf f + \mathbf g}
| r = \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ \dfrac \partial {\partial x} & \dfrac \partial {\partial y} & \dfrac \partial {\partial z} \\ f_x + g_x & f_y + g_y & f_z + g_x \end {vmatrix}
| c = {{Defof|Curl Operator}}
}}
{... | Let $\map {\R^3} {x, y, z}$ denote the [[Definition:Real Cartesian Space|real Cartesian space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]]..
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the [[Definition:Standard Ordered Basis on Vector Space|standard ordered basis on $\R^3$]].
Let $\mathbf f$ and... | {{begin-eqn}}
{{eqn | l = \nabla \times \paren {\mathbf f + \mathbf g}
| r = \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ \dfrac \partial {\partial x} & \dfrac \partial {\partial y} & \dfrac \partial {\partial z} \\ f_x + g_x & f_y + g_y & f_z + g_x \end {vmatrix}
| c = {{Defof|Curl Operator}}
}}
{... | Curl Operator Distributes over Addition | https://proofwiki.org/wiki/Curl_Operator_Distributes_over_Addition | https://proofwiki.org/wiki/Curl_Operator_Distributes_over_Addition | [
"Curl Operator"
] | [
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Dimension of Vector Space",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Vector-Valued Function",
"Definition:Curl Operator"
] | [
"Determinant as Sum of Determinants"
] |
proofwiki-14331 | Product Rule for Divergence | Let $\map {\mathbf V} {x_1, x_2, \ldots, x_n}$ be a vector space of $n$ dimensions.
Let $\mathbf A$ be a vector field over $\mathbf V$.
Let $U$ be a scalar field over $\mathbf V$.
Then:
:$\map {\operatorname {div} } {U \mathbf A} = \map U {\operatorname {div} \mathbf A} + \mathbf A \cdot \grad U$
where
:$\operatorname ... | From Divergence Operator on Vector Space is Dot Product of Del Operator and definition of the gradient operator:
{{begin-eqn}}
{{eqn | l = \operatorname {div} \mathbf V
| r = \nabla \cdot \mathbf V
}}
{{eqn | l = \grad \mathbf U
| r = \nabla U
}}
{{end-eqn}}
where $\nabla$ denotes the del operator.
Hence we... | Let $\map {\mathbf V} {x_1, x_2, \ldots, x_n}$ be a [[Definition:Vector Space|vector space]] of [[Definition:Dimension of Vector Space|$n$ dimensions]].
Let $\mathbf A$ be a [[Definition:Vector Field|vector field]] over $\mathbf V$.
Let $U$ be a [[Definition:Scalar Field|scalar field]] over $\mathbf V$.
Then:
:$\ma... | From [[Divergence Operator on Vector Space is Dot Product of Del Operator]] and definition of the [[Definition:Gradient Operator|gradient operator]]:
{{begin-eqn}}
{{eqn | l = \operatorname {div} \mathbf V
| r = \nabla \cdot \mathbf V
}}
{{eqn | l = \grad \mathbf U
| r = \nabla U
}}
{{end-eqn}}
where $\na... | Product Rule for Divergence | https://proofwiki.org/wiki/Product_Rule_for_Divergence | https://proofwiki.org/wiki/Product_Rule_for_Divergence | [
"Divergence Operator"
] | [
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Vector Field",
"Definition:Scalar Field",
"Definition:Divergence Operator",
"Definition:Gradient Operator",
"Definition:Dot Product"
] | [
"Divergence Operator on Vector Space is Dot Product of Del Operator",
"Definition:Gradient Operator",
"Definition:Del Operator",
"Definition:Vector-Valued Function",
"Definition:Element",
"Definition:Standard Ordered Basis/Vector Space",
"Product Rule for Derivatives"
] |
proofwiki-14332 | Product Rule for Curl | Let $R$ be a region of space embedded in Cartesian $3$ space $\R^3$.
Let $\mathbf A$ be a vector field over $\mathbf V$.
Let $U$ be a scalar field over $\mathbf V$.
Then:
{{begin-eqn}}
{{eqn | l = \map \curl {U \mathbf A}
| r = U \curl \mathbf A + \grad U \times \mathbf A
}}
{{eqn | r = U \curl \mathbf A - \mathb... | From Curl Operator on Vector Space is Cross Product of Del Operator and definition of the gradient operator:
{{begin-eqn}}
{{eqn | l = \curl \mathbf A
| r = \nabla \times \mathbf A
}}
{{eqn | l = \grad \mathbf U
| r = \nabla U
}}
{{end-eqn}}
where $\nabla$ denotes the del operator.
Hence we are to demonstra... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]] embedded in [[Definition:Cartesian 3-Space|Cartesian $3$ space $\R^3$]].
Let $\mathbf A$ be a [[Definition:Vector Field|vector field]] over $\mathbf V$.
Let $U$ be a [[Definition:Scalar Field (Physics)|scalar field]] over $\mathbf V$.
... | From [[Curl Operator on Vector Space is Cross Product of Del Operator]] and definition of the [[Definition:Gradient Operator|gradient operator]]:
{{begin-eqn}}
{{eqn | l = \curl \mathbf A
| r = \nabla \times \mathbf A
}}
{{eqn | l = \grad \mathbf U
| r = \nabla U
}}
{{end-eqn}}
where $\nabla$ denotes the ... | Product Rule for Curl | https://proofwiki.org/wiki/Product_Rule_for_Curl | https://proofwiki.org/wiki/Product_Rule_for_Curl | [
"Curl Operator"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Cartesian 3-Space",
"Definition:Vector Field",
"Definition:Scalar Field (Physics)",
"Definition:Curl Operator",
"Definition:Gradient Operator",
"Definition:Vector Cross Product"
] | [
"Curl Operator on Vector Space is Cross Product of Del Operator",
"Definition:Gradient Operator",
"Definition:Del Operator",
"Definition:Vector-Valued Function",
"Definition:Position Vector",
"Definition:Point",
"Definition:Standard Ordered Basis/Vector Space",
"Product Rule for Derivatives"
] |
proofwiki-14333 | Moment Generating Function of Beta Distribution | Let $X \sim \BetaDist \alpha \beta$ denote the Beta distribution fior some $\alpha, \beta > 0$.
Then the moment generating function $M_X$ of $X$ is given by:
:$\ds \map {M_X} t = 1 + \sum_{k \mathop = 1}^\infty \paren {\prod_{r \mathop = 0}^{k - 1} \frac {\alpha + r} {\alpha + \beta + r} } \frac {t^k} {k!}$ | From the definition of the Beta distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect {e^{t X} } = \int_0^1 e^{t x} \map {f_X} x \rd x$
So:
... | Let $X \sim \BetaDist \alpha \beta$ denote the [[Definition:Beta Distribution|Beta distribution]] fior some $\alpha, \beta > 0$.
Then the [[Definition:Moment Generating Function|moment generating function]] $M_X$ of $X$ is given by:
:$\ds \map {M_X} t = 1 + \sum_{k \mathop = 1}^\infty \paren {\prod_{r \mathop = 0}^{... | From the definition of the [[Definition:Beta Distribution|Beta distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$
From the definition of a [[Definition:Moment Generating F... | Moment Generating Function of Beta Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Beta_Distribution | https://proofwiki.org/wiki/Moment_Generating_Function_of_Beta_Distribution | [
"Moment Generating Functions",
"Beta Distribution"
] | [
"Definition:Beta Distribution",
"Definition:Moment Generating Function"
] | [
"Definition:Beta Distribution",
"Definition:Probability Density Function",
"Definition:Moment Generating Function",
"Power Series Expansion for Exponential Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Gamma Difference Equation",
"Product of Products",
"Category:Momen... |
proofwiki-14334 | Curl of Gradient is Zero | Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions..
Let $\map U {x, y, z}$ be a scalar field on $\R^3$.
Then:
:$\map \curl {\grad U} = \mathbf 0$
where:
:$\curl$ denotes the curl operator
:$\grad$ denotes the gradient operator. | From Curl Operator on Vector Space is Cross Product of Del Operator and definition of the gradient operator:
{{begin-eqn}}
{{eqn | l = \grad \mathbf U
| r = \nabla U
}}
{{eqn | l = \map \curl {\grad U}
| r = \nabla \times \paren {\nabla U}
}}
{{end-eqn}}
where $\nabla$ denotes the del operator.
Hence we are... | Let $\map {\R^3} {x, y, z}$ denote the [[Definition:Real Cartesian Space|real Cartesian space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]]..
Let $\map U {x, y, z}$ be a [[Definition:Scalar Field (Physics)|scalar field]] on $\R^3$.
Then:
:$\map \curl {\grad U} = \mathbf 0$
where:
:$\curl$ denotes the... | From [[Curl Operator on Vector Space is Cross Product of Del Operator]] and definition of the [[Definition:Gradient Operator|gradient operator]]:
{{begin-eqn}}
{{eqn | l = \grad \mathbf U
| r = \nabla U
}}
{{eqn | l = \map \curl {\grad U}
| r = \nabla \times \paren {\nabla U}
}}
{{end-eqn}}
where $\nabla$... | Curl of Gradient is Zero | https://proofwiki.org/wiki/Curl_of_Gradient_is_Zero | https://proofwiki.org/wiki/Curl_of_Gradient_is_Zero | [
"Curl of Gradient is Zero",
"Gradient Operator",
"Curl Operator"
] | [
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Dimension of Vector Space",
"Definition:Scalar Field (Physics)",
"Definition:Curl Operator",
"Definition:Gradient Operator"
] | [
"Curl Operator on Vector Space is Cross Product of Del Operator",
"Definition:Gradient Operator",
"Definition:Del Operator",
"Definition:Standard Ordered Basis/Vector Space",
"Clairaut's Theorem",
"Definition:Partial Derivative"
] |
proofwiki-14335 | Curl of Curl is Gradient of Divergence minus Laplacian | Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.
Let $\mathbf V$ be a vector field on $\R^3$.
Then:
:$\curl \curl \mathbf V = \grad \operatorname {div} \mathbf V - \nabla^2 \mathbf V$
where:
:$\curl$ denotes the curl operator
:$\operatorname {div}$ denotes the divergence operator
:$\grad$ ... | From Curl Operator on Vector Space is Cross Product of Del Operator, and Divergence Operator on Vector Space is Dot Product of Del Operator and the definition of the gradient operator:
{{begin-eqn}}
{{eqn | l = \curl \mathbf V
| r = \nabla \times \mathbf V
}}
{{eqn | l = \operatorname {div} \mathbf V
| r = ... | Let $\map {\R^3} {x, y, z}$ denote the [[Definition:Real Cartesian Space|real Cartesian space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]].
Let $\mathbf V$ be a [[Definition:Vector Field|vector field]] on $\R^3$.
Then:
:$\curl \curl \mathbf V = \grad \operatorname {div} \mathbf V - \nabla^2 \mathbf V... | From [[Curl Operator on Vector Space is Cross Product of Del Operator]], and [[Divergence Operator on Vector Space is Dot Product of Del Operator]] and the definition of the [[Definition:Gradient Operator|gradient operator]]:
{{begin-eqn}}
{{eqn | l = \curl \mathbf V
| r = \nabla \times \mathbf V
}}
{{eqn | l = ... | Curl of Curl is Gradient of Divergence minus Laplacian | https://proofwiki.org/wiki/Curl_of_Curl_is_Gradient_of_Divergence_minus_Laplacian | https://proofwiki.org/wiki/Curl_of_Curl_is_Gradient_of_Divergence_minus_Laplacian | [
"Gradient Operator",
"Divergence Operator",
"Curl Operator",
"Laplacian"
] | [
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Dimension of Vector Space",
"Definition:Vector Field",
"Definition:Curl Operator",
"Definition:Divergence Operator",
"Definition:Gradient Operator",
"Definition:Laplacian/Vector Field/Cartesian 3-Space"
] | [
"Curl Operator on Vector Space is Cross Product of Del Operator",
"Divergence Operator on Vector Space is Dot Product of Del Operator",
"Definition:Gradient Operator",
"Definition:Del Operator",
"Definition:Vector-Valued Function",
"Definition:Position Vector",
"Definition:Point",
"Definition:Standard... |
proofwiki-14336 | Raw Moment of Beta Distribution | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ is the Beta distribution.
Then:
:$\ds \expect {X^n} = \prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r}$
for positive integer $n$. | By Moment Generating Function of Beta Distribution, the moment generating function, $M_X$, of $X$ is given by:
:$\ds \map {M_X} t = \expect {e^{t X} } = 1 + \sum_{n \mathop = 1}^\infty \paren {\prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r} } \frac {t^n} {n!}$
We also have:
{{begin-eqn}}
{{eqn |... | Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ is the [[Definition:Beta Distribution|Beta distribution]].
Then:
:$\ds \expect {X^n} = \prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r}$
for [[Definition:Positive Integer|positive integer]] $n$. | By [[Moment Generating Function of Beta Distribution]], the [[Definition:Moment Generating Function|moment generating function]], $M_X$, of $X$ is given by:
:$\ds \map {M_X} t = \expect {e^{t X} } = 1 + \sum_{n \mathop = 1}^\infty \paren {\prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r} } \frac {... | Raw Moment of Beta Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Beta_Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Beta_Distribution | [
"Beta Distribution",
"Raw Moments"
] | [
"Definition:Beta Distribution",
"Definition:Positive/Integer"
] | [
"Moment Generating Function of Beta Distribution",
"Definition:Moment Generating Function",
"Power Series Expansion for Exponential Function",
"Expectation is Linear",
"Expectation is Linear",
"Expectation of Constant",
"Category:Beta Distribution",
"Category:Raw Moments"
] |
proofwiki-14337 | Expectation is Linear/Discrete | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$.
Let $\expect X$ denote the expectation of $X$.
Then:
:$\forall \alpha, \beta \in \R: \expect {\alpha X + \beta Y} = \alpha \, \expect X + \beta \, \expect Y$ | Follows directly from Expectation of Function of Joint Probability Mass Distribution, thus:
{{begin-eqn}}
{{eqn | l = \expect {\alpha X + \beta Y}
| r = \sum_x \sum_y \paren {\alpha x + \beta y} \, \map \Pr {X = x, Y = y}
| c = Expectation of Function of Joint Probability Mass Distribution
}}
{{eqn | r = \a... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Discrete Random Variable|random variables]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $\expect X$ denote the [[Definition:Expectation|expectation]] of $X$.
Then:
:$\forall \alpha, \beta \in \R:... | Follows directly from [[Expectation of Function of Joint Probability Mass Distribution]], thus:
{{begin-eqn}}
{{eqn | l = \expect {\alpha X + \beta Y}
| r = \sum_x \sum_y \paren {\alpha x + \beta y} \, \map \Pr {X = x, Y = y}
| c = [[Expectation of Function of Joint Probability Mass Distribution]]
}}
{{eq... | Expectation is Linear/Discrete | https://proofwiki.org/wiki/Expectation_is_Linear/Discrete | https://proofwiki.org/wiki/Expectation_is_Linear/Discrete | [
"Expectation is Linear"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Discrete",
"Definition:Expectation"
] | [
"Expectation of Function of Joint Probability Mass Distribution",
"Expectation of Function of Joint Probability Mass Distribution"
] |
proofwiki-14338 | Expectation is Linear/Continuous | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$.
Let $E$ denote the expectation function.
Then:
:$\forall \alpha, \beta \in \R: \expect {\alpha X + \beta Y} = \alpha \expect X + \beta \expect Y$ | Let $\map \supp X$ and $\map \supp Y$ be the supports of $X$ and $Y$ respectively.
Let $f_{X, Y} : \map \supp X \times \map \supp Y \to \R$ be the joint probability density function of $X$ and $Y$.
Let $f_X$ and $f_Y$ be the marginal probability density functions of $X$ and $Y$.
Then:
{{begin-eqn}}
{{eqn | l = \expe... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Continuous Random Variable|random variables]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $E$ denote the [[Definition:Expectation|expectation]] function.
Then:
:$\forall \alpha, \beta \in \R: \ex... | Let $\map \supp X$ and $\map \supp Y$ be the [[Definition:Support of Random Variable|supports]] of $X$ and $Y$ respectively.
Let $f_{X, Y} : \map \supp X \times \map \supp Y \to \R$ be the [[Definition:Joint Probability Density Function|joint probability density function]] of $X$ and $Y$.
Let $f_X$ and $f_Y$ be the... | Expectation is Linear/Continuous | https://proofwiki.org/wiki/Expectation_is_Linear/Continuous | https://proofwiki.org/wiki/Expectation_is_Linear/Continuous | [
"Expectation is Linear"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Continuous",
"Definition:Expectation"
] | [
"Definition:Support of Random Variable",
"Definition:Joint Probability Density Function",
"Definition:Marginal Probability Density Function",
"Linear Combination of Integrals/Definite"
] |
proofwiki-14339 | Derivative of Constant Multiple/Real | Let $f$ be a real function which is differentiable on $\R$.
Let $c \in \R$ be a constant.
Then:
:$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {c \map f x}
| r = c \map {\dfrac \d {\d x} } {\map f x} + \map f x \map {\dfrac \d {\d x} } c
| c = Product Rule for Derivatives
}}
{{eqn | r = c \map {\dfrac \d {\d x} } {\map f x} + 0
| c = Derivative of Constant
}}
{{eqn | r = c \map {\dfrac \d {\... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Differentiable on Interval|differentiable]] on $\R$.
Let $c \in \R$ be a [[Definition:Constant|constant]].
Then:
:$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {c \map f x}
| r = c \map {\dfrac \d {\d x} } {\map f x} + \map f x \map {\dfrac \d {\d x} } c
| c = [[Product Rule for Derivatives]]
}}
{{eqn | r = c \map {\dfrac \d {\d x} } {\map f x} + 0
| c = [[Derivative of Constant]]
}}
{{eqn | r = c \map {\dfr... | Derivative of Constant Multiple/Real | https://proofwiki.org/wiki/Derivative_of_Constant_Multiple/Real | https://proofwiki.org/wiki/Derivative_of_Constant_Multiple/Real | [
"Derivative of Constant Multiple"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Constant"
] | [
"Product Rule for Derivatives",
"Derivative of Constant"
] |
proofwiki-14340 | Derivative of Constant Multiple/Complex | Let $D$ be an open subset of the set of complex numbers $\C$.
Let $f: D \to \C$ be a complex-differentiable function on $D$.
Let $c \in \C$ be a constant.
Then:
:$\forall z \in D : \map {D_z} {c \map f z} = c \map {D_z} {\map f z}$ | {{begin-eqn}}
{{eqn | l = \map {D_z} {c \map f z}
| r = c \map {D_z} {\map f z} + \map f z \map {D_z} c
| c = Product Rule for Complex Derivatives
}}
{{eqn | r = c \map {D_z} {\map f z} + 0
| c = Complex Derivative of Constant
}}
{{eqn | r = c \map {D_z} {\map f z}
}}
{{end-eqn}}
{{qed}}
Category:Deri... | Let $D$ be an [[Definition:Open Set (Complex Analysis)|open]] [[Definition:Subset|subset]] of the [[Definition:Set|set]] of [[Definition:Complex Number|complex numbers]] $\C$.
Let $f: D \to \C$ be a [[Definition:Complex-Differentiable Function|complex-differentiable function]] on $D$.
Let $c \in \C$ be a [[Definition... | {{begin-eqn}}
{{eqn | l = \map {D_z} {c \map f z}
| r = c \map {D_z} {\map f z} + \map f z \map {D_z} c
| c = [[Product Rule for Complex Derivatives]]
}}
{{eqn | r = c \map {D_z} {\map f z} + 0
| c = [[Complex Derivative of Constant]]
}}
{{eqn | r = c \map {D_z} {\map f z}
}}
{{end-eqn}}
{{qed}}
[[Ca... | Derivative of Constant Multiple/Complex | https://proofwiki.org/wiki/Derivative_of_Constant_Multiple/Complex | https://proofwiki.org/wiki/Derivative_of_Constant_Multiple/Complex | [
"Derivative of Constant Multiple",
"Complex Analysis"
] | [
"Definition:Open Set/Complex Analysis",
"Definition:Subset",
"Definition:Set",
"Definition:Complex Number",
"Definition:Differentiable Mapping/Complex Function",
"Definition:Constant"
] | [
"Combination Theorem for Complex Derivatives/Product Rule",
"Derivative of Constant/Complex",
"Category:Derivative of Constant Multiple",
"Category:Complex Analysis"
] |
proofwiki-14341 | Bound on Complex Values of Gamma Function | Let $\map \Gamma z$ denote the Gamma function.
Then for any complex number $z = s + i t$, we have for $\size b \le \size t$:
:$\size {\map \Gamma {s + i t} } \le \dfrac {\size {s + i b} } {\size {s + i t} } \size {\map \Gamma {s + i b} }$ | From the Euler Form of the Gamma Function:
{{begin-eqn}}
{{eqn | l = \size {\map \Gamma {s + i t} }
| r = \lim_{M \mathop \to \infty} \size {\dfrac 1 {s + i t} \prod_{n \mathop = 1}^M \dfrac {\paren {1 + \frac 1 n}^{s + i t} } {1 + \frac {s + i t} n} }
}}
{{eqn | r = \lim_{M \mathop \to \infty} \dfrac 1 {\size {s... | Let $\map \Gamma z$ denote the [[Definition:Gamma Function|Gamma function]].
Then for any [[Definition:Complex Number|complex number]] $z = s + i t$, we have for $\size b \le \size t$:
:$\size {\map \Gamma {s + i t} } \le \dfrac {\size {s + i b} } {\size {s + i t} } \size {\map \Gamma {s + i b} }$ | From the [[Definition:Euler Form of Gamma Function|Euler Form of the Gamma Function]]:
{{begin-eqn}}
{{eqn | l = \size {\map \Gamma {s + i t} }
| r = \lim_{M \mathop \to \infty} \size {\dfrac 1 {s + i t} \prod_{n \mathop = 1}^M \dfrac {\paren {1 + \frac 1 n}^{s + i t} } {1 + \frac {s + i t} n} }
}}
{{eqn | r = \... | Bound on Complex Values of Gamma Function | https://proofwiki.org/wiki/Bound_on_Complex_Values_of_Gamma_Function | https://proofwiki.org/wiki/Bound_on_Complex_Values_of_Gamma_Function | [
"Gamma Function"
] | [
"Definition:Gamma Function",
"Definition:Complex Number"
] | [
"Definition:Gamma Function/Euler Form",
"Modulus of Exponential of Imaginary Number is One",
"Category:Gamma Function"
] |
proofwiki-14342 | Contour Integral of Gamma Function | Let $\Gamma$ denote the gamma function.
Let $y$ be a (strictly) positive real number.
Then for any (strictly) positive real number $c$:
:$\ds \frac 1 {2 \pi i} \int_{c - i \infty}^{c + i \infty} \map \Gamma t y^{-t} \rd t = e^{-y}$ | Let $L$ be the rectangular contour with the vertices $c \pm i R$, $- N - \dfrac 1 2 \pm i R$.
We will take the Contour Integral of $\map \Gamma t y^{-t}$ about the rectangular contour $L$.
Note from Poles of Gamma Function, that the poles of this function are located at the non-positive integers.
Thus, by Cauchy's Resi... | Let $\Gamma$ denote the [[Definition:Gamma Function|gamma function]].
Let $y$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Then for any [[Definition:Strictly Positive Real Number|(strictly) positive real number]] $c$:
:$\ds \frac 1 {2 \pi i} \int_{c - i \infty}^{c + i \infty} \... | Let $L$ be the rectangular [[Definition:Contour (Complex Plane)|contour]] with the vertices $c \pm i R$, $- N - \dfrac 1 2 \pm i R$.
We will take the [[Definition:Complex Contour Integral|Contour Integral]] of $\map \Gamma t y^{-t}$ about the rectangular [[Definition:Contour (Complex Plane)|contour]] $L$.
Note from [... | Contour Integral of Gamma Function | https://proofwiki.org/wiki/Contour_Integral_of_Gamma_Function | https://proofwiki.org/wiki/Contour_Integral_of_Gamma_Function | [
"Gamma Function",
"Complex Analysis"
] | [
"Definition:Gamma Function",
"Definition:Strictly Positive/Real Number",
"Definition:Strictly Positive/Real Number"
] | [
"Definition:Contour/Complex Plane",
"Definition:Contour Integral/Complex",
"Definition:Contour/Complex Plane",
"Poles of Gamma Function",
"Definition:Isolated Singularity/Pole",
"Definition:Negative/Integer",
"Cauchy's Residue Theorem",
"Residues of Gamma Function",
"Power Series Expansion for Expon... |
proofwiki-14343 | Set of Vectors defined by Directed Line Segments in Space forms Vector Space | Let $\R^3$ be a real cartesian space of $3$ dimensions.
Consider the set $S$ of directed line segments in $\R^3$.
Let the equivalence relation $\sim$ be applied to $\R^3$ such that:
:$\forall L_1, L_2 \in \R^3: L_1 \sim L_2$ {{iff}} there exists a translation $T$ such that $\map T {L_1} = L_2$
Let $\mathbb V$ denote th... | {{ProofWanted|This needs to be made considerably less clumsy}} | Let $\R^3$ be a [[Definition:Real Cartesian Space|real cartesian space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]].
Consider the [[Definition:Set|set]] $S$ of [[Definition:Directed Line Segment|directed line segments]] in $\R^3$.
Let the [[Definition:Equivalence Relation|equivalence relation]] $\sim$... | {{ProofWanted|This needs to be made considerably less clumsy}} | Set of Vectors defined by Directed Line Segments in Space forms Vector Space | https://proofwiki.org/wiki/Set_of_Vectors_defined_by_Directed_Line_Segments_in_Space_forms_Vector_Space | https://proofwiki.org/wiki/Set_of_Vectors_defined_by_Directed_Line_Segments_in_Space_forms_Vector_Space | [
"Vector Spaces"
] | [
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Dimension of Vector Space",
"Definition:Set",
"Definition:Directed Line Segment",
"Definition:Equivalence Relation",
"Definition:Translation Mapping/Euclidean Space",
"Definition:Set",
"Definition:Equivalence Class",
"D... | [] |
proofwiki-14344 | Matrix is Nonsingular iff Determinant has Multiplicative Inverse/Sufficient Condition | Let $R$ be a commutative ring with unity.
Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$.
Let the determinant of $\mathbf A$ be invertible in $R$.
Then $\mathbf A$ is a nonsingular matrix. | Let $\map \det {\mathbf A}$ be invertible in $R$.
From Matrix Product with Adjugate Matrix:
{{begin-eqn}}
{{eqn | l = \mathbf A \cdot \adj {\mathbf A}
| r = \map \det {\mathbf A} \cdot \mathbf I_n
}}
{{eqn | l = \adj {\mathbf A} \cdot \mathbf A
| r = \map \det {\mathbf A} \cdot \mathbf I_n
}}
{{end-eqn}}
Th... | Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $\mathbf A \in R^{n \times n}$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order]] $n$.
Let the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$ be [[Definition:Unit of... | Let $\map \det {\mathbf A}$ be [[Definition:Unit of Ring|invertible]] in $R$.
From [[Matrix Product with Adjugate Matrix]]:
{{begin-eqn}}
{{eqn | l = \mathbf A \cdot \adj {\mathbf A}
| r = \map \det {\mathbf A} \cdot \mathbf I_n
}}
{{eqn | l = \adj {\mathbf A} \cdot \mathbf A
| r = \map \det {\mathbf A} \... | Matrix is Nonsingular iff Determinant has Multiplicative Inverse/Sufficient Condition | https://proofwiki.org/wiki/Matrix_is_Nonsingular_iff_Determinant_has_Multiplicative_Inverse/Sufficient_Condition | https://proofwiki.org/wiki/Matrix_is_Nonsingular_iff_Determinant_has_Multiplicative_Inverse/Sufficient_Condition | [
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Unit of Ring",
"Definition:Nonsingular Matrix"
] | [
"Definition:Unit of Ring",
"Matrix Product with Adjugate Matrix",
"Definition:Nonsingular Matrix"
] |
proofwiki-14345 | Bienaymé-Chebyshev Inequality | Let $X$ be a random variable.
Let $\expect X = \mu$ for some $\mu \in \R$.
Let $\var X = \sigma^2$ for some $\sigma^2 \in \R_{> 0}$.
Then, for all $k > 0$:
:$\map \Pr {\size {X - \mu} \ge k \sigma} \le \dfrac 1 {k^2}$ | Let $f$ be the function:
:$\map f x = \begin{cases} k^2 \sigma^2 & : \size {x - \mu} \ge k \sigma \\ 0 & : \text{otherwise} \end{cases}$
By construction:
:$\forall x \in \Dom f: \map f x \le \size {x - \mu}^2 = \paren {x - \mu}^2$
Hence from Expectation Preserves Inequality:
:$\expect {\map f X} \le \expect {\paren {X ... | Let $X$ be a [[Definition:Random Variable|random variable]].
Let $\expect X = \mu$ for some $\mu \in \R$.
Let $\var X = \sigma^2$ for some $\sigma^2 \in \R_{> 0}$.
Then, for all $k > 0$:
:$\map \Pr {\size {X - \mu} \ge k \sigma} \le \dfrac 1 {k^2}$ | Let $f$ be the function:
:$\map f x = \begin{cases} k^2 \sigma^2 & : \size {x - \mu} \ge k \sigma \\ 0 & : \text{otherwise} \end{cases}$
By construction:
:$\forall x \in \Dom f: \map f x \le \size {x - \mu}^2 = \paren {x - \mu}^2$
Hence from [[Expectation Preserves Inequality]]:
:$\expect {\map f X} \le \expect {\p... | Bienaymé-Chebyshev Inequality/Proof 1 | https://proofwiki.org/wiki/Bienaymé-Chebyshev_Inequality | https://proofwiki.org/wiki/Bienaymé-Chebyshev_Inequality/Proof_1 | [
"Bienaymé-Chebyshev Inequality",
"Probability Theory",
"Inequalities"
] | [
"Definition:Random Variable"
] | [
"Expectation Preserves Inequality",
"Definition:Variance",
"Definition:Expectation/Discrete"
] |
proofwiki-14346 | Bienaymé-Chebyshev Inequality | Let $X$ be a random variable.
Let $\expect X = \mu$ for some $\mu \in \R$.
Let $\var X = \sigma^2$ for some $\sigma^2 \in \R_{> 0}$.
Then, for all $k > 0$:
:$\map \Pr {\size {X - \mu} \ge k \sigma} \le \dfrac 1 {k^2}$ | Note that as $k > 0$ and $\sigma > 0$, we have $k \sigma > 0$.
We therefore have:
{{begin-eqn}}
{{eqn | l = \map \Pr {\size {X - \mu} \ge k \sigma}
| r = \map \Pr {\paren {X - \mu}^2 \ge \paren {k \sigma}^2}
}}
{{eqn | o = \le
| r = \frac {\expect {\paren {X - \mu}^2} } {\paren {k \sigma}^2}
| c = as ... | Let $X$ be a [[Definition:Random Variable|random variable]].
Let $\expect X = \mu$ for some $\mu \in \R$.
Let $\var X = \sigma^2$ for some $\sigma^2 \in \R_{> 0}$.
Then, for all $k > 0$:
:$\map \Pr {\size {X - \mu} \ge k \sigma} \le \dfrac 1 {k^2}$ | Note that as $k > 0$ and $\sigma > 0$, we have $k \sigma > 0$.
We therefore have:
{{begin-eqn}}
{{eqn | l = \map \Pr {\size {X - \mu} \ge k \sigma}
| r = \map \Pr {\paren {X - \mu}^2 \ge \paren {k \sigma}^2}
}}
{{eqn | o = \le
| r = \frac {\expect {\paren {X - \mu}^2} } {\paren {k \sigma}^2}
| c = a... | Bienaymé-Chebyshev Inequality/Proof 2 | https://proofwiki.org/wiki/Bienaymé-Chebyshev_Inequality | https://proofwiki.org/wiki/Bienaymé-Chebyshev_Inequality/Proof_2 | [
"Bienaymé-Chebyshev Inequality",
"Probability Theory",
"Inequalities"
] | [
"Definition:Random Variable"
] | [] |
proofwiki-14347 | Sign of Composition of Permutations | Let $n \in \N$ be a natural number.
Let $N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$.
Let $S_n$ denote the set of permutations on $N_n$.
Let $\map \sgn \pi$ denote the sign of $\pi$ of a permutation $\pi$ of $N_n$.
Let $\pi_1, \pi_2 \in S_n$.
Then:
:$\map \sgn {\pi_1} \map \sgn {\pi_2} = \map \sgn {... | From Sign of Permutation on n Letters is Well-Defined, it is established that the sign each of $\pi_1$, $\pi_2$ and $\pi_1 \circ \pi_2$ is either $+1$ and $-1$.
By Existence and Uniqueness of Cycle Decomposition, each of $\pi_1$ and $\pi_2$ has a unique cycle decomposition.
Thus each of $\pi_1$ and $\pi_2$ can be expre... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $N_n$ denote the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\set {1, 2, \ldots, n}$.
Let $S_n$ denote the [[Definition:Set of Permutations on n Letters|set of permutations]] on $N_n$.
Let $\map \sgn \pi$ denote the [... | From [[Sign of Permutation on n Letters is Well-Defined]], it is established that the [[Definition:Sign of Permutation on n Letters|sign]] each of $\pi_1$, $\pi_2$ and $\pi_1 \circ \pi_2$ is either $+1$ and $-1$.
By [[Existence and Uniqueness of Cycle Decomposition]], each of $\pi_1$ and $\pi_2$ has a unique [[Definit... | Sign of Composition of Permutations | https://proofwiki.org/wiki/Sign_of_Composition_of_Permutations | https://proofwiki.org/wiki/Sign_of_Composition_of_Permutations | [
"Sign of Permutation"
] | [
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Permutation on n Letters/Set of Permutations",
"Definition:Sign of Permutation on n Letters",
"Definition:Permutation on n Letters",
"Definition:Composition of Mappings"
] | [
"Sign of Permutation on n Letters is Well-Defined",
"Definition:Sign of Permutation on n Letters",
"Existence and Uniqueness of Cycle Decomposition",
"Definition:Cycle Decomposition",
"Definition:Composition of Mappings",
"Definition:Transposition",
"Definition:Composition of Mappings",
"Definition:Tr... |
proofwiki-14348 | Sign of Permutation on n Letters is Well-Defined | Let $n \in \N$ be a natural number.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\rho \in S_n$ be a permutation in $S_n$.
Let $\map \sgn \rho$ denote the sign of $\rho$.
Then $\map \sgn \rho$ is well-defined, in that it is either $1$ or $-1$. | We need to prove that for any permutation $\rho \in S_n$, $\rho$ cannot be expressed as the composite of both an even number and an odd number of transpositions.
Consider the permutation formed by composing $\rho$ with an arbitrary transposition $\tau$.
Let $\rho$ be expressed as the composite of disjoint cycles whose ... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $\rho \in S_n$ be a [[Definition:Permutation on n Letters|permutation in $S_n$]].
Let $\map \sgn \rho$ denote the [[Definition:Sign of Permutation on ... | We need to prove that for any [[Definition:Permutation on n Letters|permutation]] $\rho \in S_n$, $\rho$ cannot be expressed as the [[Definition:Composition of Mappings|composite]] of both an [[Definition:Even Integer|even number]] and an [[Definition:Odd Integer|odd number]] of [[Definition:Transposition|transposition... | Sign of Permutation on n Letters is Well-Defined | https://proofwiki.org/wiki/Sign_of_Permutation_on_n_Letters_is_Well-Defined | https://proofwiki.org/wiki/Sign_of_Permutation_on_n_Letters_is_Well-Defined | [
"Sign of Permutation"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group/n Letters",
"Definition:Permutation on n Letters",
"Definition:Sign of Permutation on n Letters",
"Definition:Well-Defined/Mapping"
] | [
"Definition:Permutation on n Letters",
"Definition:Composition of Mappings",
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Transposition",
"Definition:Permutation on n Letters",
"Definition:Composition of Mappings",
"Definition:Transposition",
"Definition:Composition of Mappings",... |
proofwiki-14349 | Odd and Even Permutations of Set are Equivalent | Let $n \in \N_{> 0}$ be a natural number greater than $0$.
Let $S$ be a set of cardinality $n$.
Let $S_n$ denote the symmetric group on $S$ of order $n$.
Let $R_e$ and $R_o$ denote the subsets of $S_n$ consisting of even permutations and odd permutations respectively.
Then $R_e$ and $R_o$ are equivalent. | From Symmetric Group on n Letters is Isomorphic to Symmetric Group, it is sufficient to investigate the symmetric group on n letters.
Let $\tau$ be a transposition.
By definition of sign:
:$\map \sgn \rho = -1$
By definition of odd permutation:
:$\tau \in R_o$
Moreover, also by definition of sign, for any $\rho \in R_e... | Let $n \in \N_{> 0}$ be a [[Definition:Natural Number|natural number]] greater than $0$.
Let $S$ be a [[Definition:Set|set]] of [[Definition:Cardinality|cardinality]] $n$.
Let $S_n$ denote the [[Definition:Symmetric Group|symmetric group]] on $S$ of [[Definition:Order of Structure|order $n$]].
Let $R_e$ and $R_o$ de... | From [[Symmetric Group on n Letters is Isomorphic to Symmetric Group]], it is sufficient to investigate the [[Definition:Symmetric Group on n Letters|symmetric group on n letters]].
Let $\tau$ be a [[Definition:Transposition|transposition]].
By definition of [[Definition:Sign of Permutation on n Letters|sign]]:
:$\m... | Odd and Even Permutations of Set are Equivalent | https://proofwiki.org/wiki/Odd_and_Even_Permutations_of_Set_are_Equivalent | https://proofwiki.org/wiki/Odd_and_Even_Permutations_of_Set_are_Equivalent | [
"Even Permutations",
"Odd Permutations",
"Set Equivalence"
] | [
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Cardinality",
"Definition:Symmetric Group",
"Definition:Order of Structure",
"Definition:Subset",
"Definition:Even Permutation",
"Definition:Odd Permutation",
"Definition:Set Equivalence"
] | [
"Symmetric Group on n Letters is Isomorphic to Symmetric Group",
"Definition:Symmetric Group/n Letters",
"Definition:Transposition",
"Definition:Sign of Permutation on n Letters",
"Definition:Odd Permutation",
"Definition:Sign of Permutation on n Letters",
"Definition:Mapping",
"Definition:Injective",... |
proofwiki-14350 | Cardinality of Even and Odd Permutations on Finite Set | Let $n \in \N_{> 0}$ be a natural number greater than $0$.
Let $S$ be a set of cardinality $n$.
Let $S_n$ denote the symmetric group on $S$ of order $n$.
Let $R_e$ and $R_o$ denote the subsets of $S_n$ consisting of even permutations and odd permutations respectively.
Then the cardinality of both $R_e$ and $R_o$ is $\d... | From Order of Symmetric Group:
:$\order {S_n} = n!$
where:
:$\order {S_n}$ denotes the order of $S_n$
:$n!$ denotes the factorial of $n$.
By definition:
:$\card {R_e} + \card {R_o} = \order {S_n}$
From Odd and Even Permutations of Set are Equivalent:
:$\card {R_e} = \card {R_o}$
The result follows.
{{qed}} | Let $n \in \N_{> 0}$ be a [[Definition:Natural Number|natural number]] greater than $0$.
Let $S$ be a [[Definition:Set|set]] of [[Definition:Cardinality|cardinality]] $n$.
Let $S_n$ denote the [[Definition:Symmetric Group|symmetric group]] on $S$ of [[Definition:Order of Structure|order $n$]].
Let $R_e$ and $R_o$ de... | From [[Order of Symmetric Group]]:
:$\order {S_n} = n!$
where:
:$\order {S_n}$ denotes the [[Definition:Order of Structure|order]] of $S_n$
:$n!$ denotes the [[Definition:Factorial|factorial]] of $n$.
By definition:
:$\card {R_e} + \card {R_o} = \order {S_n}$
From [[Odd and Even Permutations of Set are Equivalent]]:... | Cardinality of Even and Odd Permutations on Finite Set | https://proofwiki.org/wiki/Cardinality_of_Even_and_Odd_Permutations_on_Finite_Set | https://proofwiki.org/wiki/Cardinality_of_Even_and_Odd_Permutations_on_Finite_Set | [
"Odd Permutations",
"Even Permutations"
] | [
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Cardinality",
"Definition:Symmetric Group",
"Definition:Order of Structure",
"Definition:Subset",
"Definition:Even Permutation",
"Definition:Odd Permutation",
"Definition:Cardinality"
] | [
"Order of Symmetric Group",
"Definition:Order of Structure",
"Definition:Factorial",
"Odd and Even Permutations of Set are Equivalent"
] |
proofwiki-14351 | Congruence Modulo Integer is Equivalence Relation | For all $z \in \Z$, congruence modulo $z$ is an equivalence relation. | Checking in turn each of the criteria for equivalence: | For all $z \in \Z$, [[Definition:Congruence Modulo Integer|congruence modulo $z$]] is an [[Definition:Equivalence Relation|equivalence relation]]. | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Congruence Modulo Integer is Equivalence Relation | https://proofwiki.org/wiki/Congruence_Modulo_Integer_is_Equivalence_Relation | https://proofwiki.org/wiki/Congruence_Modulo_Integer_is_Equivalence_Relation | [
"Congruence (Number Theory)",
"Modulo Arithmetic",
"Examples of Equivalence Relations"
] | [
"Definition:Congruence (Number Theory)/Integers",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-14352 | Integer Combination of Coprime Integers/General Result | Let $a_1, a_2, \ldots, a_n$ be integers.
Then $\gcd \set {a_1, a_2, \ldots, a_n} = 1$ {{iff}} there exists an integer combination of them equal to $1$:
:$\exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$ | First let $\exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$.
Let $\gcd \set {a_1, a_2, \ldots, a_n} = d$.
Then $\ds \sum_{k \mathop = 1}^n m_k a_k$ has $d$ as a divisor.
That means $d$ is a divisor of $1$.
Thus $\gcd \set {a_1, a_2, \ldots, a_n} = 1$.
{{qed|lemma}}
It remains to be shown th... | Let $a_1, a_2, \ldots, a_n$ be [[Definition:Integer|integers]].
Then $\gcd \set {a_1, a_2, \ldots, a_n} = 1$ {{iff}} there exists an [[Definition:Integer Combination|integer combination]] of them equal to $1$:
:$\exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$ | First let $\exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$.
Let $\gcd \set {a_1, a_2, \ldots, a_n} = d$.
Then $\ds \sum_{k \mathop = 1}^n m_k a_k$ has $d$ as a [[Definition:Divisor of Integer|divisor]].
That means $d$ is a [[Definition:Divisor of Integer|divisor]] of $1$.
Thus $\gcd \s... | Integer Combination of Coprime Integers/General Result | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/General_Result | https://proofwiki.org/wiki/Integer_Combination_of_Coprime_Integers/General_Result | [
"Integer Combination of Coprime Integers"
] | [
"Definition:Integer",
"Definition:Integer Combination"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-14353 | Beta Function of x with y+m+1 | Let $\map \Beta {x, y}$ denote the Beta function.
Then:
:$\map \Beta {x, y} = \dfrac {\map {\Gamma_m} y m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}$
where $\Gamma_m$ is the partial Gamma function:
{{begin-eqn}}
{{eqn | l = \map {\Gamma_m} y
| r = \dfrac {m^y m!} {y \paren {y + 1} \paren {y + 2} \c... | {{begin-eqn}}
{{eqn | l = \map \Beta {x, y}
| r = \dfrac {x + y} y \map \Beta {x, y + 1}
| c = Beta Function of x with y+1 by x+y over y
}}
{{eqn | r = \paren {\dfrac {x + y} y } \paren {\dfrac {x + y + 1} {y + 1} } \map \Beta {x, y + 1 + 1}
| c = applying Beta Function of x with y+1 by x+y over ... | Let $\map \Beta {x, y}$ denote the [[Definition:Beta Function|Beta function]].
Then:
:$\map \Beta {x, y} = \dfrac {\map {\Gamma_m} y m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}$
where $\Gamma_m$ is the [[Definition:Partial Gamma Function|partial Gamma function]]:
{{begin-eqn}}
{{eqn | l = \map {\Gamma_... | {{begin-eqn}}
{{eqn | l = \map \Beta {x, y}
| r = \dfrac {x + y} y \map \Beta {x, y + 1}
| c = [[Beta Function of x with y+1 by x+y over y]]
}}
{{eqn | r = \paren {\dfrac {x + y} y } \paren {\dfrac {x + y + 1} {y + 1} } \map \Beta {x, y + 1 + 1}
| c = applying [[Beta Function of x with y+1 by x+y... | Beta Function of x with y+m+1/Proof 1 | https://proofwiki.org/wiki/Beta_Function_of_x_with_y+m+1 | https://proofwiki.org/wiki/Beta_Function_of_x_with_y+m+1/Proof_1 | [
"Beta Function of x with y+m+1",
"Beta Function"
] | [
"Definition:Beta Function",
"Definition:Gamma Function/Partial"
] | [
"Beta Function of x with y+1 by x+y over y",
"Beta Function of x with y+1 by x+y over y",
"Definition:Rising Factorial",
"Beta Function of x with y+1 by x+y over y",
"Beta Function of x with y+1 by x+y over y"
] |
proofwiki-14354 | Beta Function of x with y+m+1 | Let $\map \Beta {x, y}$ denote the Beta function.
Then:
:$\map \Beta {x, y} = \dfrac {\map {\Gamma_m} y m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}$
where $\Gamma_m$ is the partial Gamma function:
{{begin-eqn}}
{{eqn | l = \map {\Gamma_m} y
| r = \dfrac {m^y m!} {y \paren {y + 1} \paren {y + 2} \c... | {{begin-eqn}}
{{eqn | l = \map \Beta {x, y}
| r = \dfrac {x + y} y \map \Beta {x, y + 1}
| c = Beta Function of x with y+1 by x+y over y
}}
{{eqn | r = \paren {\dfrac {x + y} y } \paren {\dfrac {x + y + 1} {y + 1} } \map \Beta {x, y + 1 + 1}
| c = applying Beta Function of x with y+1 by x+y over ... | Let $\map \Beta {x, y}$ denote the [[Definition:Beta Function|Beta function]].
Then:
:$\map \Beta {x, y} = \dfrac {\map {\Gamma_m} y m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}$
where $\Gamma_m$ is the [[Definition:Partial Gamma Function|partial Gamma function]]:
{{begin-eqn}}
{{eqn | l = \map {\Gamma_... | {{begin-eqn}}
{{eqn | l = \map \Beta {x, y}
| r = \dfrac {x + y} y \map \Beta {x, y + 1}
| c = [[Beta Function of x with y+1 by x+y over y]]
}}
{{eqn | r = \paren {\dfrac {x + y} y } \paren {\dfrac {x + y + 1} {y + 1} } \map \Beta {x, y + 1 + 1}
| c = applying [[Beta Function of x with y+1 by x+y... | Beta Function of x with y+m+1/Proof 2 | https://proofwiki.org/wiki/Beta_Function_of_x_with_y+m+1 | https://proofwiki.org/wiki/Beta_Function_of_x_with_y+m+1/Proof_2 | [
"Beta Function of x with y+m+1",
"Beta Function"
] | [
"Definition:Beta Function",
"Definition:Gamma Function/Partial"
] | [
"Beta Function of x with y+1 by x+y over y",
"Beta Function of x with y+1 by x+y over y",
"Definition:Rising Factorial"
] |
proofwiki-14355 | Beta Function expressed using Gamma Functions | Let $\map \Beta {x, y}$ denote the Beta function.
Then:
:$\map \Beta {x, y} = \dfrac {\map \Gamma x \, \map \Gamma y} {\map \Gamma {x + y} }$
where $\Gamma$ is the Gamma function: | From Beta Function of x with y+m+1:
:$\map \Beta {x, y} = \dfrac {\map {\Gamma_m} y \, m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}$
where $\Gamma_m$ is the partial Gamma function:
:$\map {\Gamma_m} y := \dfrac {m^y m!} {y \paren {y + 1} \paren {y + 2} \dotsm \paren {y + m} }$
From Partial Gamma Function e... | Let $\map \Beta {x, y}$ denote the [[Definition:Beta Function|Beta function]].
Then:
:$\map \Beta {x, y} = \dfrac {\map \Gamma x \, \map \Gamma y} {\map \Gamma {x + y} }$
where $\Gamma$ is the [[Definition:Gamma Function|Gamma function]]: | From [[Beta Function of x with y+m+1]]:
:$\map \Beta {x, y} = \dfrac {\map {\Gamma_m} y \, m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}$
where $\Gamma_m$ is the [[Definition:Partial Gamma Function|partial Gamma function]]:
:$\map {\Gamma_m} y := \dfrac {m^y m!} {y \paren {y + 1} \paren {y + 2} \dotsm \pa... | Beta Function expressed using Gamma Functions | https://proofwiki.org/wiki/Beta_Function_expressed_using_Gamma_Functions | https://proofwiki.org/wiki/Beta_Function_expressed_using_Gamma_Functions | [
"Beta Function",
"Gamma Function"
] | [
"Definition:Beta Function",
"Definition:Gamma Function"
] | [
"Beta Function of x with y+m+1",
"Definition:Gamma Function/Partial",
"Partial Gamma Function expressed as Integral",
"Definition:Monotone (Order Theory)/Real Function",
"Definition:Integer",
"Definition:Real Number"
] |
proofwiki-14356 | Binomial Coefficient expressed using Beta Function | Let $\dbinom r k$ denote a binomial coefficient.
Then:
:$\dbinom r k = \dfrac 1 {\paren {r + 1} \map B {k + 1, r - k + 1} }$ | {{begin-eqn}}
{{eqn | l = \dbinom r k
| r = \dfrac {r!} {k! \, \paren {r - k}!}
| c = {{Defof|Binomial Coefficient/Integers|Binomial Coefficient|index = 1}}
}}
{{eqn | r = \dfrac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} }
| c = Gamma Function Extends Factorial
}}
{{eqn | r =... | Let $\dbinom r k$ denote a [[Definition:Binomial Coefficient|binomial coefficient]].
Then:
:$\dbinom r k = \dfrac 1 {\paren {r + 1} \map B {k + 1, r - k + 1} }$ | {{begin-eqn}}
{{eqn | l = \dbinom r k
| r = \dfrac {r!} {k! \, \paren {r - k}!}
| c = {{Defof|Binomial Coefficient/Integers|Binomial Coefficient|index = 1}}
}}
{{eqn | r = \dfrac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} }
| c = [[Gamma Function Extends Factorial]]
}}
{{eqn |... | Binomial Coefficient expressed using Beta Function | https://proofwiki.org/wiki/Binomial_Coefficient_expressed_using_Beta_Function | https://proofwiki.org/wiki/Binomial_Coefficient_expressed_using_Beta_Function | [
"Beta Function",
"Binomial Coefficients"
] | [
"Definition:Binomial Coefficient"
] | [
"Gamma Function Extends Factorial",
"Gamma Difference Equation"
] |
proofwiki-14357 | Symmetry Rule for Binomial Coefficients/Complex Numbers | For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer:
:$\dbinom z w = \dbinom z {z - w}$ | From the definition of the binomial coefficient:
:$\dbinom z w := \ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} }$
where $\Gamma$ denotes the Gamma function.
{{begin-eqn}}
{{eqn | l = \dbinom z w
| r = \lim_{... | For all $z, w \in \C$ such that it is not the case that $z$ is a [[Definition:Negative Integer|negative integer]] and $w$ an [[Definition:Integer|integer]]:
:$\dbinom z w = \dbinom z {z - w}$ | From the definition of the [[Definition:Binomial Coefficient/Complex Numbers|binomial coefficient]]:
:$\dbinom z w := \ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} }$
where $\Gamma$ denotes the [[Definition:Gamma ... | Symmetry Rule for Binomial Coefficients/Complex Numbers | https://proofwiki.org/wiki/Symmetry_Rule_for_Binomial_Coefficients/Complex_Numbers | https://proofwiki.org/wiki/Symmetry_Rule_for_Binomial_Coefficients/Complex_Numbers | [
"Symmetry Rule for Binomial Coefficients"
] | [
"Definition:Negative/Integer",
"Definition:Integer"
] | [
"Definition:Binomial Coefficient/Complex Numbers",
"Definition:Gamma Function"
] |
proofwiki-14358 | Factors of Binomial Coefficient/Complex Numbers | For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer:
:$\dbinom z w = \dfrac z w \dbinom {z - 1} {w - 1}$
where $\dbinom z w$ is a binomial coefficient. | {{begin-eqn}}
{{eqn | l = \dbinom z w
| r = \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}
| c = {{Defof|Binomial Coefficient/Complex Numbers|Binomial Coefficient}}
}}
{{eqn | r = \li... | For all $z, w \in \C$ such that it is not the case that $z$ is a [[Definition:Negative Integer|negative integer]] and $w$ an [[Definition:Integer|integer]]:
:$\dbinom z w = \dfrac z w \dbinom {z - 1} {w - 1}$
where $\dbinom z w$ is a [[Definition:Binomial Coefficient/Complex Numbers|binomial coefficient]]. | {{begin-eqn}}
{{eqn | l = \dbinom z w
| r = \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}
| c = {{Defof|Binomial Coefficient/Complex Numbers|Binomial Coefficient}}
}}
{{eqn | r = \li... | Factors of Binomial Coefficient/Complex Numbers | https://proofwiki.org/wiki/Factors_of_Binomial_Coefficient/Complex_Numbers | https://proofwiki.org/wiki/Factors_of_Binomial_Coefficient/Complex_Numbers | [
"Binomial Coefficients"
] | [
"Definition:Negative/Integer",
"Definition:Integer",
"Definition:Binomial Coefficient/Complex Numbers"
] | [
"Gamma Difference Equation"
] |
proofwiki-14359 | Pascal's Rule/Complex Numbers | For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer:
:$\dbinom z {w - 1} + \dbinom z w = \dbinom {z + 1} w$
where $\dbinom z w$ is a binomial coefficient. | {{begin-eqn}}
{{eqn | l = \binom z {w - 1} + \binom z w
| r = \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma \omega \map \Gamma {\zeta - \omega + 2} } + \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\om... | For all $z, w \in \C$ such that it is not the case that $z$ is a [[Definition:Negative Integer|negative integer]] and $w$ an [[Definition:Integer|integer]]:
:$\dbinom z {w - 1} + \dbinom z w = \dbinom {z + 1} w$
where $\dbinom z w$ is a [[Definition:Binomial Coefficient/Complex Numbers|binomial coefficient]]. | {{begin-eqn}}
{{eqn | l = \binom z {w - 1} + \binom z w
| r = \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma \omega \map \Gamma {\zeta - \omega + 2} } + \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\om... | Pascal's Rule/Complex Numbers | https://proofwiki.org/wiki/Pascal's_Rule/Complex_Numbers | https://proofwiki.org/wiki/Pascal's_Rule/Complex_Numbers | [
"Pascal's Rule"
] | [
"Definition:Negative/Integer",
"Definition:Integer",
"Definition:Binomial Coefficient/Complex Numbers"
] | [
"Combination Theorem for Limits of Functions/Complex/Sum Rule",
"Gamma Difference Equation",
"Definition:Fraction/Denominator",
"Gamma Difference Equation",
"Gamma Difference Equation"
] |
proofwiki-14360 | Product of r Choose m with m Choose k/Complex Numbers | For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $t, w$ integers:
:$\dbinom z t \dbinom t w = \dbinom z w \dbinom {z - w} {t - w}$
where $\dbinom z w$ is a binomial coefficient. | {{begin-eqn}}
{{eqn | l = \dbinom z t \dbinom t w
| r = \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to t} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\tau + 1} \map \Gamma {\zeta - \tau + 1} } \lim_{\tau \mathop \to t} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\tau + 1} } {\map \Gamma {\omega + 1} \... | For all $z, w \in \C$ such that it is not the case that $z$ is a [[Definition:Negative Integer|negative integer]] and $t, w$ [[Definition:Integer|integers]]:
:$\dbinom z t \dbinom t w = \dbinom z w \dbinom {z - w} {t - w}$
where $\dbinom z w$ is a [[Definition:Binomial Coefficient/Complex Numbers|binomial coefficient]]... | {{begin-eqn}}
{{eqn | l = \dbinom z t \dbinom t w
| r = \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to t} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\tau + 1} \map \Gamma {\zeta - \tau + 1} } \lim_{\tau \mathop \to t} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\tau + 1} } {\map \Gamma {\omega + 1} \... | Product of r Choose m with m Choose k/Complex Numbers | https://proofwiki.org/wiki/Product_of_r_Choose_m_with_m_Choose_k/Complex_Numbers | https://proofwiki.org/wiki/Product_of_r_Choose_m_with_m_Choose_k/Complex_Numbers | [
"Product of r Choose m with m Choose k"
] | [
"Definition:Negative/Integer",
"Definition:Integer",
"Definition:Binomial Coefficient/Complex Numbers"
] | [
"Combination Theorem for Limits of Functions/Complex/Product Rule"
] |
proofwiki-14361 | Negated Upper Index of Binomial Coefficient/Complex Numbers | For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $t, w$ integers:
:$\dbinom z w = \dfrac {\map \sin {\pi \paren {w - z - 1} } } {\map \sin {\pi z} } \dbinom {w - z - 1} w$
where $\dbinom z w$ is a binomial coefficient. | By definition of Binomial Coefficient:
:$\dbinom z w = \ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} }$
Euler's Reflection Formula gives:
:$\forall z \notin \Z: \map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \s... | For all $z, w \in \C$ such that it is not the case that $z$ is a [[Definition:Negative Integer|negative integer]] and $t, w$ [[Definition:Integer|integers]]:
:$\dbinom z w = \dfrac {\map \sin {\pi \paren {w - z - 1} } } {\map \sin {\pi z} } \dbinom {w - z - 1} w$
where $\dbinom z w$ is a [[Definition:Binomial Coefficie... | By definition of [[Definition:Binomial Coefficient/Complex Numbers|Binomial Coefficient]]:
:$\dbinom z w = \ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} }$
[[Euler's Reflection Formula]] gives:
:$\forall z \notin \... | Negated Upper Index of Binomial Coefficient/Complex Numbers | https://proofwiki.org/wiki/Negated_Upper_Index_of_Binomial_Coefficient/Complex_Numbers | https://proofwiki.org/wiki/Negated_Upper_Index_of_Binomial_Coefficient/Complex_Numbers | [
"Negated Upper Index of Binomial Coefficient"
] | [
"Definition:Negative/Integer",
"Definition:Integer",
"Definition:Binomial Coefficient/Complex Numbers"
] | [
"Definition:Binomial Coefficient/Complex Numbers",
"Euler's Reflection Formula",
"Euler's Reflection Formula",
"Euler's Reflection Formula",
"Combination Theorem for Limits of Functions/Complex",
"Combination Theorem for Limits of Functions/Complex",
"Sine Function is Odd",
"Sine of Angle plus Straigh... |
proofwiki-14362 | Chu-Vandermonde Identity/Extended | Let $r, s, \alpha, \beta \in \C$ be complex numbers.
Then:
:$\ds \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} \dbinom s {\beta - k} = \dbinom {r + s} {\alpha + \beta}$
where $\dbinom r {\alpha + k}$ denotes a binomial coefficient. | From the Chu-Vandermonde Identity, we have:
:$\ds \sum_{k \mathop \in \Z} \binom r k \binom s {n - k} = \binom {r + s} n$
Let $n = \alpha + \beta$
Let $k = \alpha + k$
Then:
:$\ds \sum_{k \mathop \in \Z} \binom r {\alpha + k} \binom s {\alpha + \beta - \paren {\alpha + k} } = \binom {r + s} {\alpha + \beta}$
:$\ds \sum... | Let $r, s, \alpha, \beta \in \C$ be [[Definition:Complex Number|complex numbers]].
Then:
:$\ds \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} \dbinom s {\beta - k} = \dbinom {r + s} {\alpha + \beta}$
where $\dbinom r {\alpha + k}$ denotes a [[Definition:Binomial Coefficient/Complex Numbers|binomial coefficient]]. | From the [[Chu-Vandermonde Identity]], we have:
:$\ds \sum_{k \mathop \in \Z} \binom r k \binom s {n - k} = \binom {r + s} n$
Let $n = \alpha + \beta$
Let $k = \alpha + k$
Then:
:$\ds \sum_{k \mathop \in \Z} \binom r {\alpha + k} \binom s {\alpha + \beta - \paren {\alpha + k} } = \binom {r + s} {\alpha + \beta}$
:... | Chu-Vandermonde Identity/Extended | https://proofwiki.org/wiki/Chu-Vandermonde_Identity/Extended | https://proofwiki.org/wiki/Chu-Vandermonde_Identity/Extended | [
"Chu-Vandermonde Identity"
] | [
"Definition:Complex Number",
"Definition:Binomial Coefficient/Complex Numbers"
] | [
"Chu-Vandermonde Identity"
] |
proofwiki-14363 | Beta Function of Half with Half | :$\map \Beta {\dfrac 1 2, \dfrac 1 2} = \pi$
where $\Beta$ denotes the Beta function. | By definition of the Beta function:
:$\ds \map \Beta {x, y} := 2 \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 1} \paren {\cos \theta}^{2 y - 1} \rd \theta$
Thus:
{{begin-eqn}}
{{eqn | l = \map \Beta {\dfrac 1 2, \dfrac 1 2}
| r = 2 \int_0^{\pi / 2} \paren {\sin \theta}^{2 \times \frac 1 2 - 1} \paren {\cos \theta... | :$\map \Beta {\dfrac 1 2, \dfrac 1 2} = \pi$
where $\Beta$ denotes the [[Definition:Beta Function|Beta function]]. | By definition of the [[Definition:Beta Function/Definition 2|Beta function]]:
:$\ds \map \Beta {x, y} := 2 \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 1} \paren {\cos \theta}^{2 y - 1} \rd \theta$
Thus:
{{begin-eqn}}
{{eqn | l = \map \Beta {\dfrac 1 2, \dfrac 1 2}
| r = 2 \int_0^{\pi / 2} \paren {\sin \theta}... | Beta Function of Half with Half/Proof 1 | https://proofwiki.org/wiki/Beta_Function_of_Half_with_Half | https://proofwiki.org/wiki/Beta_Function_of_Half_with_Half/Proof_1 | [
"Beta Function",
"Beta Function of Half with Half"
] | [
"Definition:Beta Function"
] | [
"Definition:Beta Function/Definition 2"
] |
proofwiki-14364 | Beta Function of Half with Half | :$\map \Beta {\dfrac 1 2, \dfrac 1 2} = \pi$
where $\Beta$ denotes the Beta function. | By definition of the Beta function:
:$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$
Thus:
{{begin-eqn}}
{{eqn | l = \map \Beta {\dfrac 1 2, \dfrac 1 2}
| r = \int_{\mathop \to 0}^{\mathop \to 1} t^{\frac 1 2 - 1} \paren {1 - t}^{\frac 1 2 - 1} \rd t
| ... | :$\map \Beta {\dfrac 1 2, \dfrac 1 2} = \pi$
where $\Beta$ denotes the [[Definition:Beta Function|Beta function]]. | By definition of the [[Definition:Beta Function/Definition 2|Beta function]]:
:$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$
Thus:
{{begin-eqn}}
{{eqn | l = \map \Beta {\dfrac 1 2, \dfrac 1 2}
| r = \int_{\mathop \to 0}^{\mathop \to 1} t^{\frac 1 2 - 1} ... | Beta Function of Half with Half/Proof 2 | https://proofwiki.org/wiki/Beta_Function_of_Half_with_Half | https://proofwiki.org/wiki/Beta_Function_of_Half_with_Half/Proof_2 | [
"Beta Function",
"Beta Function of Half with Half"
] | [
"Definition:Beta Function"
] | [
"Definition:Beta Function/Definition 2",
"Integration by Substitution",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-14365 | Beta Function of Half with Half | :$\map \Beta {\dfrac 1 2, \dfrac 1 2} = \pi$
where $\Beta$ denotes the Beta function. | By definition of the Beta function:
:$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$
Thus:
{{begin-eqn}}
{{eqn | l = \map \Beta {\dfrac 1 2, \dfrac 1 2}
| r = \int_{\mathop \to 0}^{\mathop \to 1} t^{\frac 1 2 - 1} \paren {1 - t}^{\frac 1 2 - 1} \rd t
| ... | :$\map \Beta {\dfrac 1 2, \dfrac 1 2} = \pi$
where $\Beta$ denotes the [[Definition:Beta Function|Beta function]]. | By definition of the [[Definition:Beta Function/Definition 2|Beta function]]:
:$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$
Thus:
{{begin-eqn}}
{{eqn | l = \map \Beta {\dfrac 1 2, \dfrac 1 2}
| r = \int_{\mathop \to 0}^{\mathop \to 1} t^{\frac 1 2 - 1} ... | Beta Function of Half with Half/Proof 3 | https://proofwiki.org/wiki/Beta_Function_of_Half_with_Half | https://proofwiki.org/wiki/Beta_Function_of_Half_with_Half/Proof_3 | [
"Beta Function",
"Beta Function of Half with Half"
] | [
"Definition:Beta Function"
] | [
"Definition:Beta Function/Definition 2",
"Integration by Substitution",
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form"
] |
proofwiki-14366 | Binomial Coefficient of Real Number with Half | :$\dbinom r {1 / 2} = \dfrac {2^{2 r + 1} } {\dbinom {2 r} r \pi}$
where $\dbinom r {1 / 2}$ denotes a binomial coefficient. | {{begin-eqn}}
{{eqn | l = \dbinom r {1 / 2}
| r = \lim_{\rho \mathop \to r} \dfrac {\map \Gamma {\rho + 1} } {\map \Gamma {\frac 1 2 + 1} \map \Gamma {\rho - \frac 1 2 + 1} }
| c = {{Defof|Binomial Coefficient/Complex Numbers|Binomial Coefficient}}
}}
{{eqn | r = \lim_{\rho \mathop \to r} \dfrac {\map \Gamm... | :$\dbinom r {1 / 2} = \dfrac {2^{2 r + 1} } {\dbinom {2 r} r \pi}$
where $\dbinom r {1 / 2}$ denotes a [[Definition:Binomial Coefficient/Complex Numbers|binomial coefficient]]. | {{begin-eqn}}
{{eqn | l = \dbinom r {1 / 2}
| r = \lim_{\rho \mathop \to r} \dfrac {\map \Gamma {\rho + 1} } {\map \Gamma {\frac 1 2 + 1} \map \Gamma {\rho - \frac 1 2 + 1} }
| c = {{Defof|Binomial Coefficient/Complex Numbers|Binomial Coefficient}}
}}
{{eqn | r = \lim_{\rho \mathop \to r} \dfrac {\map \Gamm... | Binomial Coefficient of Real Number with Half | https://proofwiki.org/wiki/Binomial_Coefficient_of_Real_Number_with_Half | https://proofwiki.org/wiki/Binomial_Coefficient_of_Real_Number_with_Half | [
"Examples of Binomial Coefficients"
] | [
"Definition:Binomial Coefficient/Complex Numbers"
] | [
"Gamma Difference Equation",
"Gamma Function of One Half",
"Legendre's Duplication Formula",
"Gamma Difference Equation"
] |
proofwiki-14367 | Approximation to x+y Choose y | :$\ds \lim_{x, y \mathop \to \infty} \dbinom {x + y} y = \sqrt {\dfrac 1 {2 \pi} \paren {\frac 1 x + \frac 1 y} } \paren {1 + \dfrac y x}^x \paren {1 + \dfrac x y}^y$ | It can be assumed that both $x$ and $y$ are integers.
{{begin-eqn}}
{{eqn | l = \dbinom {x + y} y
| r = \dfrac {\paren {x + y}!} {x! \, y!}
| c =
}}
{{eqn | ll= \leadsto
| l = \ds \lim_{x, y \mathop \to \infty} \dbinom {x + y} y
| r = \dfrac {\sqrt {2 \pi \paren {x + y} } \paren {\dfrac {x + y}... | :$\ds \lim_{x, y \mathop \to \infty} \dbinom {x + y} y = \sqrt {\dfrac 1 {2 \pi} \paren {\frac 1 x + \frac 1 y} } \paren {1 + \dfrac y x}^x \paren {1 + \dfrac x y}^y$ | It can be assumed that both $x$ and $y$ are [[Definition:Integer|integers]].
{{begin-eqn}}
{{eqn | l = \dbinom {x + y} y
| r = \dfrac {\paren {x + y}!} {x! \, y!}
| c =
}}
{{eqn | ll= \leadsto
| l = \ds \lim_{x, y \mathop \to \infty} \dbinom {x + y} y
| r = \dfrac {\sqrt {2 \pi \paren {x + y}... | Approximation to x+y Choose y | https://proofwiki.org/wiki/Approximation_to_x+y_Choose_y | https://proofwiki.org/wiki/Approximation_to_x+y_Choose_y | [
"Binomial Coefficients",
"Stirling's Formula"
] | [] | [
"Definition:Integer",
"Stirling's Formula"
] |
proofwiki-14368 | Approximation to 2n Choose n | :$\ds \lim_{n \mathop \to \infty} \dbinom {2 n} n = \dfrac {4^n} {\sqrt {n \pi} }$ | From Approximation to $\dbinom {x + y} y$:
:$\ds \lim_{x, y \mathop \to \infty} \dbinom {x + y} y = \sqrt {\dfrac 1 {2 \pi} \paren {\frac 1 x + \frac 1 y} } \paren {1 + \dfrac y x}^x \paren {1 + \dfrac x y}^y$
Thus:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \dbinom {2 n} n
| r = \sqrt {\dfrac 1 {2 \pi... | :$\ds \lim_{n \mathop \to \infty} \dbinom {2 n} n = \dfrac {4^n} {\sqrt {n \pi} }$ | From [[Approximation to x+y Choose y|Approximation to $\dbinom {x + y} y$]]:
:$\ds \lim_{x, y \mathop \to \infty} \dbinom {x + y} y = \sqrt {\dfrac 1 {2 \pi} \paren {\frac 1 x + \frac 1 y} } \paren {1 + \dfrac y x}^x \paren {1 + \dfrac x y}^y$
Thus:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \dbinom {2 n... | Approximation to 2n Choose n | https://proofwiki.org/wiki/Approximation_to_2n_Choose_n | https://proofwiki.org/wiki/Approximation_to_2n_Choose_n | [
"Binomial Coefficients"
] | [] | [
"Approximation to x+y Choose y"
] |
proofwiki-14369 | Product of r Choose k with r Minus Half Choose k/Formulation 1 | Let $k \in \Z$, $r \in \R$.
:$\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} k \dbinom {2 r - k} k} {4^k}$
where $\dbinom r k$ denotes a binomial coefficient. | First we establish the following:
{{begin-eqn}}
{{eqn | l = \paren {r - \frac 1 2}^{\underline k}
| r = \paren {r - \frac 1 2} \paren {r - \frac 3 2} \paren {r - \frac 5 2} \dotsm \paren {r - \frac 1 2 - k + 1}
| c = {{Defof|Falling Factorial}}
}}
{{eqn | r = \dfrac {2^k \paren {r - \frac 1 2} \paren {r - \... | Let $k \in \Z$, $r \in \R$.
:$\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} k \dbinom {2 r - k} k} {4^k}$
where $\dbinom r k$ denotes a [[Definition:Binomial Coefficient|binomial coefficient]]. | First we establish the following:
{{begin-eqn}}
{{eqn | l = \paren {r - \frac 1 2}^{\underline k}
| r = \paren {r - \frac 1 2} \paren {r - \frac 3 2} \paren {r - \frac 5 2} \dotsm \paren {r - \frac 1 2 - k + 1}
| c = {{Defof|Falling Factorial}}
}}
{{eqn | r = \dfrac {2^k \paren {r - \frac 1 2} \paren {r - ... | Product of r Choose k with r Minus Half Choose k/Formulation 1/Proof 1 | https://proofwiki.org/wiki/Product_of_r_Choose_k_with_r_Minus_Half_Choose_k/Formulation_1 | https://proofwiki.org/wiki/Product_of_r_Choose_k_with_r_Minus_Half_Choose_k/Formulation_1/Proof_1 | [
"Product of r Choose k with r Minus Half Choose k"
] | [
"Definition:Binomial Coefficient"
] | [
"Falling Factorial of Sum of Integers"
] |
proofwiki-14370 | Product of r Choose k with r Minus Half Choose k/Formulation 1 | Let $k \in \Z$, $r \in \R$.
:$\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} k \dbinom {2 r - k} k} {4^k}$
where $\dbinom r k$ denotes a binomial coefficient. | From Binomial Coefficient expressed using Beta Function:
:$(1): \quad \dbinom r k \dbinom {r - \frac 1 2} k = \dfrac 1 {\paren {r + 1} \map \Beta {k + 1, r - k + 1} \paren {r + \frac 1 2} \map \Beta {k + 1, r - k + \frac 1 2} }$
Then:
{{begin-eqn}}
{{eqn | l = \dbinom r {k + 1} \dbinom {r - \frac 1 2} {k + 1}
| r... | Let $k \in \Z$, $r \in \R$.
:$\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} k \dbinom {2 r - k} k} {4^k}$
where $\dbinom r k$ denotes a [[Definition:Binomial Coefficient|binomial coefficient]]. | From [[Binomial Coefficient expressed using Beta Function]]:
:$(1): \quad \dbinom r k \dbinom {r - \frac 1 2} k = \dfrac 1 {\paren {r + 1} \map \Beta {k + 1, r - k + 1} \paren {r + \frac 1 2} \map \Beta {k + 1, r - k + \frac 1 2} }$
Then:
{{begin-eqn}}
{{eqn | l = \dbinom r {k + 1} \dbinom {r - \frac 1 2} {k + 1}
... | Product of r Choose k with r Minus Half Choose k/Formulation 1/Proof 2 | https://proofwiki.org/wiki/Product_of_r_Choose_k_with_r_Minus_Half_Choose_k/Formulation_1 | https://proofwiki.org/wiki/Product_of_r_Choose_k_with_r_Minus_Half_Choose_k/Formulation_1/Proof_2 | [
"Product of r Choose k with r Minus Half Choose k"
] | [
"Definition:Binomial Coefficient"
] | [
"Binomial Coefficient expressed using Beta Function",
"Beta Function of x with y+1 by x+y over y",
"Beta Function of x with y+1 by x+y over y",
"Binomial Coefficient expressed using Beta Function",
"Beta Function of x with y+1 by x+y over y",
"Beta Function of x with y+1 by x+y over y",
"Beta Function o... |
proofwiki-14371 | Provable Consequence of Theorems is Theorem | Let $\PP$ be a proof system for a formal language $\LL$.
Let $\FF$ be a collection of theorems of $\PP$.
Denote with $\map {\mathscr P} \FF$ the proof system obtained from $\mathscr P$ by adding all the WFFs from $\FF$ as axioms.
Let $\phi$ be a provable consequence of $\FF$:
:$\vdash_{\mathscr P} \FF$
:$\FF \vdash_{\m... | We have that $\phi$ is a provable consequence of $\FF$.
Hence it is a theorem of $\map {\mathscr P} \FF$, the proof system obtained from $\mathscr P$ by adding all of $\FF$ as axioms.
Now in the formal proof of $\phi$ in $\map {\mathscr P} \FF$, both axioms and rules of inference are used.
Each rule of inference of $\m... | Let $\PP$ be a [[Definition:Proof System|proof system]] for a [[Definition:Formal Language|formal language]] $\LL$.
Let $\FF$ be a collection of [[Definition:Theorem (Formal Systems)|theorems]] of $\PP$.
Denote with $\map {\mathscr P} \FF$ the [[Definition:Proof System|proof system]] obtained from $\mathscr P$ by add... | We have that $\phi$ is a [[Definition:Provable Consequence|provable consequence]] of $\FF$.
Hence it is a [[Definition:Theorem (Formal Systems)|theorem]] of $\map {\mathscr P} \FF$, the [[Definition:Proof System|proof system]] obtained from $\mathscr P$ by adding all of $\FF$ as [[Definition:Axiom (Formal Systems)|axi... | Provable Consequence of Theorems is Theorem | https://proofwiki.org/wiki/Provable_Consequence_of_Theorems_is_Theorem | https://proofwiki.org/wiki/Provable_Consequence_of_Theorems_is_Theorem | [
"Proof Systems"
] | [
"Definition:Proof System",
"Definition:Formal Language",
"Definition:Theorem/Formal System",
"Definition:Proof System",
"Definition:Well-Formed Formula",
"Definition:Axiom/Formal Systems",
"Definition:Provable Consequence",
"Definition:Theorem/Formal System"
] | [
"Definition:Provable Consequence",
"Definition:Theorem/Formal System",
"Definition:Proof System",
"Definition:Axiom/Formal Systems",
"Definition:Proof System/Formal Proof",
"Definition:Axiom/Formal Systems",
"Definition:Rule of Inference",
"Definition:Rule of Inference",
"Definition:Rule of Inferenc... |
proofwiki-14372 | Binomial Coefficient of Minus Half | Let $k \in \Z$.
:$\dbinom {-\frac 1 2} k = \dfrac {\paren {-1}^k} {4^k} \dbinom {2 k} k$
where $\dbinom {-\frac 1 2} k$ denotes a binomial coefficient. | From Product of r Choose k with r Minus Half Choose k:
:$\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} k \dbinom {2 r - k} k} {4^k}$
Setting $r = -\dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \dbinom {-\frac 1 2} k \dbinom {-1} k
| r = \frac 1 {4^k} \dbinom {-1} k \dbinom {-1 - k} k
| c =
}}
{{eq... | Let $k \in \Z$.
:$\dbinom {-\frac 1 2} k = \dfrac {\paren {-1}^k} {4^k} \dbinom {2 k} k$
where $\dbinom {-\frac 1 2} k$ denotes a [[Definition:Binomial Coefficient|binomial coefficient]]. | From [[Product of r Choose k with r Minus Half Choose k]]:
:$\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} k \dbinom {2 r - k} k} {4^k}$
Setting $r = -\dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \dbinom {-\frac 1 2} k \dbinom {-1} k
| r = \frac 1 {4^k} \dbinom {-1} k \dbinom {-1 - k} k
| c = ... | Binomial Coefficient of Minus Half | https://proofwiki.org/wiki/Binomial_Coefficient_of_Minus_Half | https://proofwiki.org/wiki/Binomial_Coefficient_of_Minus_Half | [
"Examples of Binomial Coefficients"
] | [
"Definition:Binomial Coefficient"
] | [
"Product of r Choose k with r Minus Half Choose k",
"Negated Upper Index of Binomial Coefficient"
] |
proofwiki-14373 | Falling Factorial of Sum of Integers | Let $r \in \R$ be a real number.
Let $a, b \in \Z$ be (positive) integers.
Then:
:$r^{\underline {a + b} } = r^{\underline a} \paren {r - a}^{\underline b}$
where $r^{\underline a}$ denotes the $a$th falling factorial of $r$. | {{begin-eqn}}
{{eqn | l = r^{\underline {a + b} }
| r = \prod_{j \mathop = 0}^{a + b - 1} \paren {r - j}
| c = {{Defof|Falling Factorial}}
}}
{{eqn | r = \paren {\prod_{j \mathop = 0}^{a - 1} \paren {r - j} } \paren {\prod_{j \mathop = a}^{a + b - 1} \paren {r - j} }
| c =
}}
{{eqn | r = \paren {\pro... | Let $r \in \R$ be a [[Definition:Real Number|real number]].
Let $a, b \in \Z$ be [[Definition:Positive Integer|(positive) integers]].
Then:
:$r^{\underline {a + b} } = r^{\underline a} \paren {r - a}^{\underline b}$
where $r^{\underline a}$ denotes the $a$th [[Definition:Falling Factorial|falling factorial]] of $r$. | {{begin-eqn}}
{{eqn | l = r^{\underline {a + b} }
| r = \prod_{j \mathop = 0}^{a + b - 1} \paren {r - j}
| c = {{Defof|Falling Factorial}}
}}
{{eqn | r = \paren {\prod_{j \mathop = 0}^{a - 1} \paren {r - j} } \paren {\prod_{j \mathop = a}^{a + b - 1} \paren {r - j} }
| c =
}}
{{eqn | r = \paren {\pro... | Falling Factorial of Sum of Integers | https://proofwiki.org/wiki/Falling_Factorial_of_Sum_of_Integers | https://proofwiki.org/wiki/Falling_Factorial_of_Sum_of_Integers | [
"Falling Factorials"
] | [
"Definition:Real Number",
"Definition:Positive/Integer",
"Definition:Falling Factorial"
] | [
"Translation of Index Variable of Product",
"Category:Falling Factorials"
] |
proofwiki-14374 | Approximate Size of Sum of Harmonic Series | Let $H_n$ denote the sum of the harmonic series:
:$H_n = \ds \sum_{k \mathop = 1}^n \frac 1 k$
Then $H_n$ can be approximated as follows:
:$H_n \approx \ln n + \gamma + \dfrac 1 {2 n} - \dfrac 1 {12 n^2} + \dfrac 1 {120 n^4} - \epsilon$
where:
:$\gamma$ denotes the Euler-Mascheroni constant: $\gamma \approx 0 \cdotp 57... | {{tidy}}
{{begin-eqn}}
{{eqn | l = H_n
| r = \sum_{i=1}^n\frac{1}{k}
| c = Definition of harmonic numbers
}}
{{eqn | r = \int_1^n \frac 1 x \rd x + \sum_{k \mathop = 1}^\infty \frac {B_k} {k!} \paren {\map {f^{\paren {k - 1} } } n - \paren {-1}^k \map {f^{\paren {k - 1} } } 1}
| c = Euler-Maclaurin Su... | Let $H_n$ denote the [[Definition:Summation|sum]] of the [[Definition:Harmonic Series|harmonic series]]:
:$H_n = \ds \sum_{k \mathop = 1}^n \frac 1 k$
Then $H_n$ can be approximated as follows:
:$H_n \approx \ln n + \gamma + \dfrac 1 {2 n} - \dfrac 1 {12 n^2} + \dfrac 1 {120 n^4} - \epsilon$
where:
:$\gamma$ denotes t... | {{tidy}}
{{begin-eqn}}
{{eqn | l = H_n
| r = \sum_{i=1}^n\frac{1}{k}
| c = Definition of [[Definition:Harmonic Numbers|harmonic numbers]]
}}
{{eqn | r = \int_1^n \frac 1 x \rd x + \sum_{k \mathop = 1}^\infty \frac {B_k} {k!} \paren {\map {f^{\paren {k - 1} } } n - \paren {-1}^k \map {f^{\paren {k - 1} } } ... | Approximate Size of Sum of Harmonic Series | https://proofwiki.org/wiki/Approximate_Size_of_Sum_of_Harmonic_Series | https://proofwiki.org/wiki/Approximate_Size_of_Sum_of_Harmonic_Series | [
"Harmonic Numbers",
"Harmonic Series"
] | [
"Definition:Summation",
"Definition:Harmonic Series",
"Definition:Euler-Mascheroni Constant"
] | [
"Definition:Harmonic Numbers",
"Euler-Maclaurin Summation Formula",
"Definition:Natural Logarithm/Positive Real",
"Natural Logarithm of 1 is 0",
"Definition:Bernoulli Numbers/Sequence",
"Odd Bernoulli Numbers Vanish",
"Power Rule for Derivatives",
"Definition:Harmonic Numbers",
"Definition:Natural L... |
proofwiki-14375 | Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function | Let $z \in \C$ with a strictly positive real part and $\size {\arg z} < \dfrac \pi 2$.
Then:
:$\ds \map \Ln {\map \Gamma z} = \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \sum_{n \mathop = 1}^d \frac {B_{2 n} } {2 n \paren {2 n - 1} z^{2 n - 1} } + \OO \paren {\dfrac 1 {z^{2 d + 1} } }$
where:
:$\G... | From Binet's Formula for Logarithm of Gamma Function Formulation 2, we have:
:$\ds \Ln \map \Gamma z = \paren {z - \dfrac 1 2} \Ln z - z + \dfrac 1 2 \ln 2 \pi + 2 \int_0^\infty \dfrac {\map \arctan {t / z} } {e^{2 \pi t} - 1} \rd t$
:Let $A = \paren {z - \dfrac 1 2} \Ln z - z + \dfrac 1 2 \ln 2 \pi$
Then:
{{begin-eqn}... | Let $z \in \C$ with a [[Definition:Strictly Positive|strictly positive]] [[Definition:Real Part|real part]] and $\size {\arg z} < \dfrac \pi 2$.
Then:
:$\ds \map \Ln {\map \Gamma z} = \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \sum_{n \mathop = 1}^d \frac {B_{2 n} } {2 n \paren {2 n - 1} z^{2 n ... | From [[Binet's Formula for Logarithm of Gamma Function/Formulation 2|Binet's Formula for Logarithm of Gamma Function Formulation 2]], we have:
:$\ds \Ln \map \Gamma z = \paren {z - \dfrac 1 2} \Ln z - z + \dfrac 1 2 \ln 2 \pi + 2 \int_0^\infty \dfrac {\map \arctan {t / z} } {e^{2 \pi t} - 1} \rd t$
:Let $A = \paren {z... | Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function | https://proofwiki.org/wiki/Logarithmic_Approximation_of_Error_Term_of_Stirling's_Formula_for_Gamma_Function | https://proofwiki.org/wiki/Logarithmic_Approximation_of_Error_Term_of_Stirling's_Formula_for_Gamma_Function | [
"Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function",
"Stirling's Formula",
"Gamma Function",
"Asymptotic Expansions"
] | [
"Definition:Strictly Positive",
"Definition:Complex Number/Real Part",
"Definition:Gamma Function",
"Definition:Natural Logarithm/Complex/Principal Branch",
"Definition:Natural Logarithm/Complex",
"Definition:Bernoulli Numbers",
"Definition:Big-O Notation"
] | [
"Binet's Formula for Logarithm of Gamma Function/Formulation 2",
"Binet's Formula for Logarithm of Gamma Function/Formulation 2",
"Power Series Expansion for Real Arctangent Function",
"Primitive of Power",
"Tonelli's Theorem",
"Exponent Combination Laws/Product of Powers",
"Definition:Fraction/Numerato... |
proofwiki-14376 | Riemann Zeta Function of 8 | The Riemann zeta function of $8$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 8
| r = \dfrac 1 {1^8} + \dfrac 1 {2^8} + \dfrac 1 {3^8} + \dfrac 1 {4^8} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^8} {9450}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 00408 \, 3 \ldots
| c =
}}
{{en... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^{\infty} \frac 1 {n^8}
| r = \map \zeta 8
| c = {{Defof|Riemann Zeta Function}}
}}
{{eqn | r = \paren {-1}^5 \frac {B_8 2^7 \pi^8} {8!}
| c = Riemann Zeta Function at Even Integers
}}
{{eqn | r = \frac 1 {30} \cdot \frac {2^7 \pi^8} {8!}
| c = {{... | The [[Definition:Riemann Zeta Function|Riemann zeta function]] of $8$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 8
| r = \dfrac 1 {1^8} + \dfrac 1 {2^8} + \dfrac 1 {3^8} + \dfrac 1 {4^8} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^8} {9450}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp ... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^{\infty} \frac 1 {n^8}
| r = \map \zeta 8
| c = {{Defof|Riemann Zeta Function}}
}}
{{eqn | r = \paren {-1}^5 \frac {B_8 2^7 \pi^8} {8!}
| c = [[Riemann Zeta Function at Even Integers]]
}}
{{eqn | r = \frac 1 {30} \cdot \frac {2^7 \pi^8} {8!}
| c ... | Riemann Zeta Function of 8 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_8 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_8 | [
"Riemann Zeta Function at Even Integers",
"Formulas for Pi",
"Eighth Powers"
] | [
"Definition:Riemann Zeta Function"
] | [
"Riemann Zeta Function at Even Integers"
] |
proofwiki-14377 | Sum of Sequence of Harmonic Numbers | :$\ds \sum_{k \mathop = 1}^n H_k = \paren {n + 1} H_n - n$
where $H_k$ denotes the $k$th harmonic number. | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n H_k
| r = \sum_{k \mathop = 1}^n \paren {\sum_{j \mathop = 1}^k \frac 1 j}
| c = {{Defof|Harmonic Numbers}}
}}
{{eqn | r = \sum_{j \mathop = 1}^n \paren {\sum_{k \mathop = j}^n \frac 1 j}
| c = Summation of i from 1 to n of Summation of j from 1 to i
}}... | :$\ds \sum_{k \mathop = 1}^n H_k = \paren {n + 1} H_n - n$
where $H_k$ denotes the $k$th [[Definition:Harmonic Number|harmonic number]]. | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n H_k
| r = \sum_{k \mathop = 1}^n \paren {\sum_{j \mathop = 1}^k \frac 1 j}
| c = {{Defof|Harmonic Numbers}}
}}
{{eqn | r = \sum_{j \mathop = 1}^n \paren {\sum_{k \mathop = j}^n \frac 1 j}
| c = [[Summation of i from 1 to n of Summation of j from 1 to i]... | Sum of Sequence of Harmonic Numbers/Proof 1 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Harmonic_Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Harmonic_Numbers/Proof_1 | [
"Harmonic Numbers",
"Sum of Sequence of Harmonic Numbers"
] | [
"Definition:Harmonic Numbers"
] | [
"Summation of i from 1 to n of Summation of j from 1 to i",
"Linear Combination of Convergent Series",
"Linear Combination of Convergent Series"
] |
proofwiki-14378 | Sum of Sequence of Harmonic Numbers | :$\ds \sum_{k \mathop = 1}^n H_k = \paren {n + 1} H_n - n$
where $H_k$ denotes the $k$th harmonic number. | From Sum over k to n of k Choose m by kth Harmonic Number:
:$\ds \sum_{k \mathop = 1}^n \binom k m H_k = \binom {n + 1} {m + 1} \paren {H_{n + 1} - \frac 1 {m + 1} }$
Setting $m = 0$:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \binom k 0 H_k
| r = \binom {n + 1} {0 + 1} \paren {H_{n + 1} - \frac 1 {0 + 1} ... | :$\ds \sum_{k \mathop = 1}^n H_k = \paren {n + 1} H_n - n$
where $H_k$ denotes the $k$th [[Definition:Harmonic Number|harmonic number]]. | From [[Sum over k to n of k Choose m by kth Harmonic Number]]:
:$\ds \sum_{k \mathop = 1}^n \binom k m H_k = \binom {n + 1} {m + 1} \paren {H_{n + 1} - \frac 1 {m + 1} }$
Setting $m = 0$:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \binom k 0 H_k
| r = \binom {n + 1} {0 + 1} \paren {H_{n + 1} - \frac 1 {0... | Sum of Sequence of Harmonic Numbers/Proof 2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Harmonic_Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Harmonic_Numbers/Proof_2 | [
"Harmonic Numbers",
"Sum of Sequence of Harmonic Numbers"
] | [
"Definition:Harmonic Numbers"
] | [
"Sum over k to n of k Choose m by kth Harmonic Number",
"Binomial Coefficient with Zero",
"Binomial Coefficient with One"
] |
proofwiki-14379 | Sum of Sequence of Harmonic Numbers | :$\ds \sum_{k \mathop = 1}^n H_k = \paren {n + 1} H_n - n$
where $H_k$ denotes the $k$th harmonic number. | Let $\sequence {a_n}$ be the sequence defined as:
:$\forall n \in \N_{> 0}: a_n = H_n$
where $H_n$ denotes the $n$th harmonic number.
Let $\map G z$ be the generating function for $\sequence {a_n}$.
From Generating Function for Sequence of Harmonic Numbers:
:$\map G z = \dfrac 1 {1 - z} \map \ln {\dfrac 1 {1 - z} }$
Th... | :$\ds \sum_{k \mathop = 1}^n H_k = \paren {n + 1} H_n - n$
where $H_k$ denotes the $k$th [[Definition:Harmonic Number|harmonic number]]. | Let $\sequence {a_n}$ be the [[Definition:Sequence|sequence]] defined as:
:$\forall n \in \N_{> 0}: a_n = H_n$
where $H_n$ denotes the $n$th [[Definition:Harmonic Number|harmonic number]].
Let $\map G z$ be the [[Definition:Generating Function|generating function]] for $\sequence {a_n}$.
From [[Generating Function f... | Sum of Sequence of Harmonic Numbers/Proof 3 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Harmonic_Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Harmonic_Numbers/Proof_3 | [
"Harmonic Numbers",
"Sum of Sequence of Harmonic Numbers"
] | [
"Definition:Harmonic Numbers"
] | [
"Definition:Sequence",
"Definition:Harmonic Numbers",
"Definition:Generating Function",
"Generating Function for Sequence of Harmonic Numbers",
"Derivative of Generating Function for Sequence of Harmonic Numbers",
"Generating Function for Sequence of Partial Sums of Series",
"Definition:Generating Funct... |
proofwiki-14380 | Sum over k to n of k Choose m by kth Harmonic Number | :$\ds \sum_{k \mathop = 1}^n \binom k m H_k = \binom {n + 1} {m + 1} \paren {H_{n + 1} - \frac 1 {m + 1} }$
where:
:$\dbinom k m$ denotes a binomial coefficient
:$H_k$ denotes the $k$th harmonic number. | First we note that by Pascal's Rule:
:$\dbinom k m = \dbinom {k + 1} {m + 1} - \dbinom k {m + 1}$
Thus:
{{begin-eqn}}
{{eqn | l = \dbinom k m H_k
| r = \dbinom {k + 1} {m + 1} \paren {H_{k + 1} - \dfrac 1 {k + 1} } - \dbinom k {m + 1} H_k
| c =
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^n \bi... | :$\ds \sum_{k \mathop = 1}^n \binom k m H_k = \binom {n + 1} {m + 1} \paren {H_{n + 1} - \frac 1 {m + 1} }$
where:
:$\dbinom k m$ denotes a [[Definition:Binomial Coefficient|binomial coefficient]]
:$H_k$ denotes the $k$th [[Definition:Harmonic Number|harmonic number]]. | First we note that by [[Pascal's Rule]]:
:$\dbinom k m = \dbinom {k + 1} {m + 1} - \dbinom k {m + 1}$
Thus:
{{begin-eqn}}
{{eqn | l = \dbinom k m H_k
| r = \dbinom {k + 1} {m + 1} \paren {H_{k + 1} - \dfrac 1 {k + 1} } - \dbinom k {m + 1} H_k
| c =
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1... | Sum over k to n of k Choose m by kth Harmonic Number | https://proofwiki.org/wiki/Sum_over_k_to_n_of_k_Choose_m_by_kth_Harmonic_Number | https://proofwiki.org/wiki/Sum_over_k_to_n_of_k_Choose_m_by_kth_Harmonic_Number | [
"Harmonic Numbers",
"Binomial Coefficients"
] | [
"Definition:Binomial Coefficient",
"Definition:Harmonic Numbers"
] | [
"Pascal's Rule",
"Sum of Binomial Coefficients over Upper Index"
] |
proofwiki-14381 | Sum over k of n Choose k by x to the k by kth Harmonic Number | Let $x \in \R_{> 0}$ be a real number.
Then:
:$\ds \sum_{k \mathop \in \Z} \binom n k x^k H_k = \paren {x + 1}^n \paren {H_n - \map \ln {1 + \frac 1 x} } + \epsilon$
where:
:$\dbinom n k$ denotes a binomial coefficient
:$H_k$ denotes the $k$th harmonic number
:$0 < \epsilon < \dfrac 1 {x \paren {n + 1} }$ | Let $S_n := \ds \sum_{k \mathop \in \Z} \binom n k x^k H_k$.
Then:
{{begin-eqn}}
{{eqn | l = S_{n + 1}
| r = \sum_{k \mathop \in \Z} \binom {n + 1} k x^k H_k
| c =
}}
{{eqn | r = \sum_{k \mathop \in \Z} \paren {\binom n k + \binom n {k - 1} } x^k H_k
| c = Pascal's Rule
}}
{{eqn | r = \sum_{k \mathop... | Let $x \in \R_{> 0}$ be a [[Definition:Real Number|real number]].
Then:
:$\ds \sum_{k \mathop \in \Z} \binom n k x^k H_k = \paren {x + 1}^n \paren {H_n - \map \ln {1 + \frac 1 x} } + \epsilon$
where:
:$\dbinom n k$ denotes a [[Definition:Binomial Coefficient|binomial coefficient]]
:$H_k$ denotes the $k$th [[Definit... | Let $S_n := \ds \sum_{k \mathop \in \Z} \binom n k x^k H_k$.
Then:
{{begin-eqn}}
{{eqn | l = S_{n + 1}
| r = \sum_{k \mathop \in \Z} \binom {n + 1} k x^k H_k
| c =
}}
{{eqn | r = \sum_{k \mathop \in \Z} \paren {\binom n k + \binom n {k - 1} } x^k H_k
| c = [[Pascal's Rule]]
}}
{{eqn | r = \sum_{k \... | Sum over k of n Choose k by x to the k by kth Harmonic Number | https://proofwiki.org/wiki/Sum_over_k_of_n_Choose_k_by_x_to_the_k_by_kth_Harmonic_Number | https://proofwiki.org/wiki/Sum_over_k_of_n_Choose_k_by_x_to_the_k_by_kth_Harmonic_Number | [
"Harmonic Numbers",
"Binomial Coefficients"
] | [
"Definition:Real Number",
"Definition:Binomial Coefficient",
"Definition:Harmonic Numbers"
] | [
"Pascal's Rule",
"Translation of Index Variable of Summation",
"Factors of Binomial Coefficient",
"Binomial Theorem",
"Binomial Coefficient with Zero",
"Power Series Expansion for Logarithm of 1 - x",
"Logarithm of Reciprocal"
] |
proofwiki-14382 | Upper Bound for Harmonic Number | :$H_{2^m} \le 1 + m$
where $H_{2^m}$ denotes the $2^m$th harmonic number. | :$\ds \sum_{n \mathop = 1}^\infty \frac 1 n = \underbrace 1_{s_0} + \underbrace {\frac 1 2 + \frac 1 3}_{s_1} + \underbrace {\frac 1 4 + \frac 1 5 + \frac 1 6 + \frac 1 7}_{s_2} + \cdots$
where $\ds s_k = \sum_{i \mathop = 2^k}^{2^{k + 1} \mathop - 1} \frac 1 i$
From Ordering of Reciprocals:
:$\forall m, n \in \N_{>0}... | :$H_{2^m} \le 1 + m$
where $H_{2^m}$ denotes the $2^m$th [[Definition:Harmonic Number|harmonic number]]. | :$\ds \sum_{n \mathop = 1}^\infty \frac 1 n = \underbrace 1_{s_0} + \underbrace {\frac 1 2 + \frac 1 3}_{s_1} + \underbrace {\frac 1 4 + \frac 1 5 + \frac 1 6 + \frac 1 7}_{s_2} + \cdots$
where $\ds s_k = \sum_{i \mathop = 2^k}^{2^{k + 1} \mathop - 1} \frac 1 i$
From [[Ordering of Reciprocals]]:
:$\forall m, n \in ... | Upper Bound for Harmonic Number | https://proofwiki.org/wiki/Upper_Bound_for_Harmonic_Number | https://proofwiki.org/wiki/Upper_Bound_for_Harmonic_Number | [
"Harmonic Numbers"
] | [
"Definition:Harmonic Numbers"
] | [
"Ordering of Reciprocals",
"Definition:Summation/Summand",
"Definition:Summation/Summand"
] |
proofwiki-14383 | Power to Real Number by Decimal Expansion is Uniquely Defined | Let $r \in \R_{>1}$ be a real number greater than $1$, expressed by its decimal expansion:
:$r = n \cdotp d_1 d_2 d_3 \ldots$
The power $x^r$ of a (strictly) positive real number $x$ defined as:
:$(1): \quad \ds \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$
where:
{{begin-eqn}}
{{eqn | l = \psi_1
... | If $r$ is rational this has already been established.
{{MissingLinks|Find where.}}
Let $d$ denote the difference between $x^{\psi^1}$ and $x^{\psi^2}$:
{{begin-eqn}}
{{eqn | l = d
| r = x^{\psi^2} - x^{\psi^1}
| c =
}}
{{eqn | r = x^{\psi^1} \paren {x^{\frac 1 {10^k} } - 1}
| c =
}}
{{eqn | r = x^{\... | Let $r \in \R_{>1}$ be a [[Definition:Real Number|real number]] greater than $1$, expressed by its [[Definition:Decimal Expansion|decimal expansion]]:
:$r = n \cdotp d_1 d_2 d_3 \ldots$
The [[Definition:Power (Algebra)/Real Number/Definition 3|power]] $x^r$ of a [[Definition:Strictly Positive Real Number|(strictly) p... | If $r$ is [[Definition:Rational Number|rational]] this has already been established.
{{MissingLinks|Find where.}}
Let $d$ denote the [[Definition:Real Subtraction|difference]] between $x^{\psi^1}$ and $x^{\psi^2}$:
{{begin-eqn}}
{{eqn | l = d
| r = x^{\psi^2} - x^{\psi^1}
| c =
}}
{{eqn | r = x^{\psi^1... | Power to Real Number by Decimal Expansion is Uniquely Defined | https://proofwiki.org/wiki/Power_to_Real_Number_by_Decimal_Expansion_is_Uniquely_Defined | https://proofwiki.org/wiki/Power_to_Real_Number_by_Decimal_Expansion_is_Uniquely_Defined | [
"Powers"
] | [
"Definition:Real Number",
"Definition:Decimal Expansion",
"Definition:Power (Algebra)/Real Number/Definition 3",
"Definition:Strictly Positive/Real Number"
] | [
"Definition:Rational Number",
"Definition:Subtraction/Real Numbers",
"Nth Root of 1 plus x not greater than 1 plus x over n",
"Squeeze Theorem"
] |
proofwiki-14384 | Change of Base of Logarithm/Base 2 to Base 10 | :$\log_{10} x = \left({\lg x}\right) \left({\log_{10} 2}\right) = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots \lg x$ | From Change of Base of Logarithm:
:$\log_a x = \log_a b \ \log_b x$
Substituting $a = 10$ and $b = 2$ gives:
:$\log_{10} x = \left({\lg x}\right) \left({\log_{10} 2}\right)$
The common logarithm of $2$:
:$\log_{10} 2 = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
can be calculated or looked up.
{{qed}} | :$\log_{10} x = \left({\lg x}\right) \left({\log_{10} 2}\right) = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots \lg x$ | From [[Change of Base of Logarithm]]:
:$\log_a x = \log_a b \ \log_b x$
Substituting $a = 10$ and $b = 2$ gives:
:$\log_{10} x = \left({\lg x}\right) \left({\log_{10} 2}\right)$
The [[Common Logarithm of 2|common logarithm of $2$]]:
:$\log_{10} 2 = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
can be ca... | Change of Base of Logarithm/Base 2 to Base 10 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_2_to_Base_10 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_2_to_Base_10 | [
"Examples of Common Logarithms",
"Examples of Binary Logarithms",
"Examples of Change of Base of Logarithm"
] | [] | [
"Change of Base of Logarithm",
"Common Logarithm/Examples/2"
] |
proofwiki-14385 | Smallest Strictly Positive Rational Number does not Exist | There exists no smallest element of the set of strictly positive rational numbers. | {{AimForCont}} $x = \dfrac p q$ is the smallest strictly positive rational number.
By definition of strictly positive:
:$0 < \dfrac p q$
Let us calculate the mediant of $0$ and $\dfrac p q$:
:$\dfrac 0 1 < \dfrac {0 + p} {1 + q} < \dfrac p q$
The inequalities follow from Mediant is Between.
Thus $\dfrac p {1 + q}$ is a... | There exists no [[Definition:Smallest Element|smallest element]] of the [[Definition:Set|set]] of [[Definition:Strictly Positive Rational Number|strictly positive rational numbers]]. | {{AimForCont}} $x = \dfrac p q$ is the [[Definition:Smallest Element|smallest]] [[Definition:Strictly Positive Rational Number|strictly positive rational number]].
By definition of [[Definition:Strictly Positive Rational Number|strictly positive]]:
:$0 < \dfrac p q$
Let us calculate the [[Definition:Mediant|mediant]]... | Smallest Strictly Positive Rational Number does not Exist | https://proofwiki.org/wiki/Smallest_Strictly_Positive_Rational_Number_does_not_Exist | https://proofwiki.org/wiki/Smallest_Strictly_Positive_Rational_Number_does_not_Exist | [
"Rational Numbers"
] | [
"Definition:Smallest Element",
"Definition:Set",
"Definition:Strictly Positive/Rational Number"
] | [
"Definition:Smallest Element",
"Definition:Strictly Positive/Rational Number",
"Definition:Strictly Positive/Rational Number",
"Definition:Mediant",
"Mediant is Between",
"Definition:Strictly Positive/Rational Number",
"Definition:Smallest Element",
"Definition:Strictly Positive/Rational Number",
"P... |
proofwiki-14386 | Necessary Precision for x equal to log base 10 of 2 to determine Decimal expansion of 10 to the x | Let $b = 10$.
Let $x \approx \log_{10} 2$.
Let it be necessary to calculate the decimal expansion of $x$ to determine the first $3$ decimal places of $b^x$.
An infinite number of decimal places of $x$ would in fact be necessary. | This is a trick question:
:''How many decimal places of accuracy of $x$ are needed to determine the first $3$ decimal places of $b^x$?''
We have that $b^x = 10^{\log_{10} 2} = 2$.
Let $x_a < x < x_b$, where $x_a$ and $x_b$ are ever closer approximations to $x$.
Then:
:$x_a$ begins $1 \cdotp 999 \ldots$
:$x_b$ begins $2... | Let $b = 10$.
Let $x \approx \log_{10} 2$.
Let it be necessary to calculate the [[Definition:Decimal Expansion|decimal expansion]] of $x$ to determine the first $3$ decimal places of $b^x$.
An [[Definition:Infinite Set|infinite number]] of decimal places of $x$ would in fact be necessary. | This is a trick question:
:''How many decimal places of accuracy of $x$ are needed to determine the first $3$ decimal places of $b^x$?''
We have that $b^x = 10^{\log_{10} 2} = 2$.
Let $x_a < x < x_b$, where $x_a$ and $x_b$ are ever closer approximations to $x$.
Then:
:$x_a$ begins $1 \cdotp 999 \ldots$
:$x_b$ begi... | Necessary Precision for x equal to log base 10 of 2 to determine Decimal expansion of 10 to the x | https://proofwiki.org/wiki/Necessary_Precision_for_x_equal_to_log_base_10_of_2_to_determine_Decimal_expansion_of_10_to_the_x | https://proofwiki.org/wiki/Necessary_Precision_for_x_equal_to_log_base_10_of_2_to_determine_Decimal_expansion_of_10_to_the_x | [
"Common Logarithms",
"2",
"10"
] | [
"Definition:Decimal Expansion",
"Definition:Infinite Set"
] | [] |
proofwiki-14387 | Change of Base of Logarithm/Base 10 to Base e/Form 1 | :$\ln x = \paren {\ln 10} \paren {\log_{10} x} = 2 \cdotp 30258 \, 50929 \, 94 \ldots \log_{10} x$ | From Change of Base of Logarithm:
:$\log_a x = \log_a b \ \log_b x$
Substituting $a = e$ and $b = 10$ gives:
:$\ln x = \paren {\ln 10} \paren {\log_{10} x}$
The Natural Logarithm of 10:
:$\ln 10 = 2 \cdotp 30258 \, 50929 \, 94 \ldots$
can be calculated or looked up.
{{qed}} | :$\ln x = \paren {\ln 10} \paren {\log_{10} x} = 2 \cdotp 30258 \, 50929 \, 94 \ldots \log_{10} x$ | From [[Change of Base of Logarithm]]:
:$\log_a x = \log_a b \ \log_b x$
Substituting $a = e$ and $b = 10$ gives:
:$\ln x = \paren {\ln 10} \paren {\log_{10} x}$
The [[Natural Logarithm of 10]]:
:$\ln 10 = 2 \cdotp 30258 \, 50929 \, 94 \ldots$
can be calculated or looked up.
{{qed}} | Change of Base of Logarithm/Base 10 to Base e/Form 1 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_10_to_Base_e/Form_1 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_10_to_Base_e/Form_1 | [
"Examples of Change of Base of Logarithm"
] | [] | [
"Change of Base of Logarithm",
"Natural Logarithm/Examples/10"
] |
proofwiki-14388 | Change of Base of Logarithm/Base 10 to Base e/Form 2 | :$\ln x = \dfrac {\log_{10} x} {\log_{10} e} = \dfrac {\log_{10} x} {0 \cdotp 43429 \, 44819 \, 03 \ldots}$ | From Change of Base of Logarithm:
:$\log_a x = \dfrac {\log_b x} {\log_b a}$
Substituting $a = e$ and $b = 10$ gives:
:$\ln x = \dfrac {\log_{10} x} {\log_{10} e}$
The Common Logarithm of e:
:$\log_{10} e = 0 \cdotp 43429 \, 44819 \, 03 \ldots$
can then be calculated or looked up.
{{qed}}
Category:Examples of Change of... | :$\ln x = \dfrac {\log_{10} x} {\log_{10} e} = \dfrac {\log_{10} x} {0 \cdotp 43429 \, 44819 \, 03 \ldots}$ | From [[Change of Base of Logarithm]]:
:$\log_a x = \dfrac {\log_b x} {\log_b a}$
Substituting $a = e$ and $b = 10$ gives:
:$\ln x = \dfrac {\log_{10} x} {\log_{10} e}$
The [[Common Logarithm of e]]:
:$\log_{10} e = 0 \cdotp 43429 \, 44819 \, 03 \ldots$
can then be calculated or looked up.
{{qed}}
[[Category:Example... | Change of Base of Logarithm/Base 10 to Base e/Form 2 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_10_to_Base_e/Form_2 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_10_to_Base_e/Form_2 | [
"Examples of Change of Base of Logarithm"
] | [] | [
"Change of Base of Logarithm",
"Common Logarithm/Examples/e",
"Category:Examples of Change of Base of Logarithm"
] |
proofwiki-14389 | Change of Base of Logarithm/Base e to Base 10/Form 1 | :$\log_{10} x = \paren {\log_{10} e} \paren {\ln x} = 0 \cdotp 43429 \, 44819 \, 03 \ldots \ln x$ | From Change of Base of Logarithm:
:$\log_a x = \log_a b \, \log_b x$
Substituting $a = 10$ and $b = e$ gives:
:$\log_{10} x = \paren {\log_{10} e} \paren {\ln x}$
The Common Logarithm of e:
:$\log_{10} e = 0 \cdotp 43429 \, 44819 \, 03 \ldots$
can be calculated or looked up.
{{qed}} | :$\log_{10} x = \paren {\log_{10} e} \paren {\ln x} = 0 \cdotp 43429 \, 44819 \, 03 \ldots \ln x$ | From [[Change of Base of Logarithm]]:
:$\log_a x = \log_a b \, \log_b x$
Substituting $a = 10$ and $b = e$ gives:
:$\log_{10} x = \paren {\log_{10} e} \paren {\ln x}$
The [[Common Logarithm of e]]:
:$\log_{10} e = 0 \cdotp 43429 \, 44819 \, 03 \ldots$
can be calculated or looked up.
{{qed}} | Change of Base of Logarithm/Base e to Base 10/Form 1 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_e_to_Base_10/Form_1 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_e_to_Base_10/Form_1 | [
"Examples of Change of Base of Logarithm"
] | [] | [
"Change of Base of Logarithm",
"Common Logarithm/Examples/e"
] |
proofwiki-14390 | Change of Base of Logarithm/Base e to Base 10/Form 2 | :$\log_{10} x = \dfrac {\ln x} {\ln 10} = \dfrac {\ln x} {2 \cdotp 30258 \, 50929 \, 94 \ldots}$ | From Change of Base of Logarithm:
:$\log_a x = \dfrac {\log_b x} {\log_b a}$
Substituting $a = 10$ and $b = e$ gives:
:$\log_{10} x = \dfrac {\ln x} {\ln 10}$
as by definition of $\ln x$:
:$\ln x := \log_e x$
The Natural Logarithm of 10:
:$\ln 10 = 2 \cdotp 30258 \, 50929 \, 94 \ldots$
can then be calculated or looked ... | :$\log_{10} x = \dfrac {\ln x} {\ln 10} = \dfrac {\ln x} {2 \cdotp 30258 \, 50929 \, 94 \ldots}$ | From [[Change of Base of Logarithm]]:
:$\log_a x = \dfrac {\log_b x} {\log_b a}$
Substituting $a = 10$ and $b = e$ gives:
:$\log_{10} x = \dfrac {\ln x} {\ln 10}$
as by [[Definition:Natural Logarithm|definition of $\ln x$]]:
:$\ln x := \log_e x$
The [[Natural Logarithm of 10]]:
:$\ln 10 = 2 \cdotp 30258 \, 50929 \,... | Change of Base of Logarithm/Base e to Base 10/Form 2 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_e_to_Base_10/Form_2 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_e_to_Base_10/Form_2 | [
"Examples of Change of Base of Logarithm"
] | [] | [
"Change of Base of Logarithm",
"Definition:Natural Logarithm",
"Natural Logarithm/Examples/10"
] |
proofwiki-14391 | Change of Base of Logarithm/Base 10 to Base 2 | :$\lg x = \dfrac {\log_{10} x} {\log_{10} 2} = \dfrac {\log_{10} x} {0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots}$ | From Change of Base of Logarithm:
:$\log_a x = \dfrac {\log_b x} {\log_b a}$
Substituting $a = e$ and $b = 10$ gives:
:$\log_e x = \dfrac {\log_{10} x} {\log_{10} e}$
The Common Logarithm of 2:
:$\log_{10} 2 = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
can then be calculated or looked up.
{{qed}}
Catego... | :$\lg x = \dfrac {\log_{10} x} {\log_{10} 2} = \dfrac {\log_{10} x} {0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots}$ | From [[Change of Base of Logarithm]]:
:$\log_a x = \dfrac {\log_b x} {\log_b a}$
Substituting $a = e$ and $b = 10$ gives:
:$\log_e x = \dfrac {\log_{10} x} {\log_{10} e}$
The [[Common Logarithm of 2]]:
:$\log_{10} 2 = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$
can then be calculated or looked up.
{{q... | Change of Base of Logarithm/Base 10 to Base 2 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_10_to_Base_2 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_10_to_Base_2 | [
"Examples of Common Logarithms",
"Examples of Binary Logarithms",
"Examples of Change of Base of Logarithm"
] | [] | [
"Change of Base of Logarithm",
"Common Logarithm/Examples/2",
"Category:Examples of Common Logarithms",
"Category:Examples of Binary Logarithms",
"Category:Examples of Change of Base of Logarithm"
] |
proofwiki-14392 | Root of Quotient equals Quotient of Roots | :$\sqrt [n] {\dfrac a b} = \dfrac {\sqrt [n] a} {\sqrt [n] b}$ | {{begin-eqn}}
{{eqn | l = \sqrt [n] {\dfrac a b}
| r = \paren {\dfrac a b}^{1 / n}
| c = {{Defof|Root of Number|$n$th Root}}
}}
{{eqn | r = \paren {a \times \dfrac 1 b}^{1 / n}
| c =
}}
{{eqn | r = a^{1 / n} \times \paren {\dfrac 1 b}^{1 / n}
| c = Power of Product
}}
{{eqn | r = a^{1 / n} \tim... | :$\sqrt [n] {\dfrac a b} = \dfrac {\sqrt [n] a} {\sqrt [n] b}$ | {{begin-eqn}}
{{eqn | l = \sqrt [n] {\dfrac a b}
| r = \paren {\dfrac a b}^{1 / n}
| c = {{Defof|Root of Number|$n$th Root}}
}}
{{eqn | r = \paren {a \times \dfrac 1 b}^{1 / n}
| c =
}}
{{eqn | r = a^{1 / n} \times \paren {\dfrac 1 b}^{1 / n}
| c = [[Power of Product]]
}}
{{eqn | r = a^{1 / n} ... | Root of Quotient equals Quotient of Roots | https://proofwiki.org/wiki/Root_of_Quotient_equals_Quotient_of_Roots | https://proofwiki.org/wiki/Root_of_Quotient_equals_Quotient_of_Roots | [
"Exponent Combination Laws"
] | [] | [
"Exponent Combination Laws/Power of Product",
"Exponent Combination Laws/Negative Power",
"Exponent Combination Laws/Negative Power"
] |
proofwiki-14393 | Logarithm to Own Base equals 1 | Let $b \in \R_{>0}$ be a strictly positive real number such that $b \ne 1$.
Let $\log_b$ denote the logarithm to base $b$.
Then:
:$\log_b b = 1$ | By definition of logarithm:
{{begin-eqn}}
{{eqn | l = y
| r = \log_b b
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = b^y
| r = b
| c = {{Defof|Real General Logarithm}}
}}
{{eqn | ll= \leadstoandfrom
| l = y
| r = 1
| c = {{Defof|Power to Real Number}}
}}
{{end-eqn}}
{{qed}}... | Let $b \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]] such that $b \ne 1$.
Let $\log_b$ denote the [[Definition:Real General Logarithm|logarithm]] to [[Definition:Base of Logarithm|base $b$]].
Then:
:$\log_b b = 1$ | By definition of [[Definition:Real General Logarithm|logarithm]]:
{{begin-eqn}}
{{eqn | l = y
| r = \log_b b
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = b^y
| r = b
| c = {{Defof|Real General Logarithm}}
}}
{{eqn | ll= \leadstoandfrom
| l = y
| r = 1
| c = {{Defof|Power ... | Logarithm to Own Base equals 1 | https://proofwiki.org/wiki/Logarithm_to_Own_Base_equals_1 | https://proofwiki.org/wiki/Logarithm_to_Own_Base_equals_1 | [
"Logarithms"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:General Logarithm/Positive Real",
"Definition:Logarithm/Base"
] | [
"Definition:General Logarithm/Positive Real",
"Category:Logarithms"
] |
proofwiki-14394 | General Logarithm/Examples/Base b of 1 | :$\log_b 1 = 0$ | By definition of logarithm:
{{begin-eqn}}
{{eqn | l = \log_b 1
| r = \frac {\log_e 1} {\log_e b}
| c = Change of Base of Logarithm
}}
{{eqn | r = \frac 0 {\log_e b}
| c = Natural Logarithm of 1 is 0
}}
{{eqn | r = 0
| c = whatever $\log_e b$ happens to be
}}
{{end-eqn}}
{{qed}} | :$\log_b 1 = 0$ | By definition of [[Definition:Real General Logarithm|logarithm]]:
{{begin-eqn}}
{{eqn | l = \log_b 1
| r = \frac {\log_e 1} {\log_e b}
| c = [[Change of Base of Logarithm]]
}}
{{eqn | r = \frac 0 {\log_e b}
| c = [[Natural Logarithm of 1 is 0]]
}}
{{eqn | r = 0
| c = whatever $\log_e b$ happens... | General Logarithm/Examples/Base b of 1 | https://proofwiki.org/wiki/General_Logarithm/Examples/Base_b_of_1 | https://proofwiki.org/wiki/General_Logarithm/Examples/Base_b_of_1 | [
"Examples of General Logarithms"
] | [] | [
"Definition:General Logarithm/Positive Real",
"Change of Base of Logarithm",
"Natural Logarithm of 1 is 0"
] |
proofwiki-14395 | General Logarithm/Examples/Base b of -1 | :$\log_b \left({-1}\right)$ is undefined in the real number line. | {{AimForCont}} $\log_b \left({-1}\right) = y \in \R$.
Then:
:$b^y = -1 < 0$
But from Power of Positive Real Number is Positive:
:$b^y > 0$
The result follows by Proof by Contradiction.
{{qed}} | :$\log_b \left({-1}\right)$ is undefined in the [[Definition:Real Number Line|real number line]]. | {{AimForCont}} $\log_b \left({-1}\right) = y \in \R$.
Then:
:$b^y = -1 < 0$
But from [[Power of Positive Real Number is Positive]]:
:$b^y > 0$
The result follows by [[Proof by Contradiction]].
{{qed}} | General Logarithm/Examples/Base b of -1 | https://proofwiki.org/wiki/General_Logarithm/Examples/Base_b_of_-1 | https://proofwiki.org/wiki/General_Logarithm/Examples/Base_b_of_-1 | [
"Examples of General Logarithms"
] | [
"Definition:Real Number/Real Number Line"
] | [
"Power of Positive Real Number is Positive",
"Proof by Contradiction"
] |
proofwiki-14396 | Change of Base of Logarithm/Base 2 to Base 8 | :$\log_8 x = \dfrac {\lg x} 3$ | From Change of Base of Logarithm:
:$\log_a x = \dfrac {\log_b x} {\log_b a}$
Substituting $a = 8$ and $b = 2$ gives:
:$\log_8 x = \dfrac {\log_2 x} {\log_2 8}$
We have that:
{{begin-eqn}}
{{eqn | l = 2^3
| r = 8
| c =
}}
{{eqn | ll= \leadsto
| l = \lg 8
| r = \log_2 2^3
| c =
}}
{{eqn | ... | :$\log_8 x = \dfrac {\lg x} 3$ | From [[Change of Base of Logarithm]]:
:$\log_a x = \dfrac {\log_b x} {\log_b a}$
Substituting $a = 8$ and $b = 2$ gives:
:$\log_8 x = \dfrac {\log_2 x} {\log_2 8}$
We have that:
{{begin-eqn}}
{{eqn | l = 2^3
| r = 8
| c =
}}
{{eqn | ll= \leadsto
| l = \lg 8
| r = \log_2 2^3
| c =
}}
{... | Change of Base of Logarithm/Base 2 to Base 8 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_2_to_Base_8 | https://proofwiki.org/wiki/Change_of_Base_of_Logarithm/Base_2_to_Base_8 | [
"Examples of Binary Logarithms",
"Examples of Change of Base of Logarithm"
] | [] | [
"Change of Base of Logarithm",
"Logarithm to Own Base equals 1"
] |
proofwiki-14397 | Number of Digits to Represent Integer in Given Number Base | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $b \in \Z$ be an integer such that $b > 1$.
Let $d$ denote the number of digits of $n$ when represented in base $b$.
Then:
:$d = \ceiling {\map {\log_b} {n + 1} }$
where $\ceiling {\, \cdot \,}$ denotes the ceiling function. | Let $n$ have $d$ digits.
Then:
{{begin-eqn}}
{{eqn | l = b^{d - 1}
| o = \le
| m = n
| mo= <
| r = b^d
| c = Basis Representation Theorem
}}
{{eqn | ll= \leadsto
| l = b^{d - 1}
| o = <
| m = n + 1
| mo= \le
| r = b^d
| c =
}}
{{eqn | ll= \leadsto
... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $b \in \Z$ be an [[Definition:Integer|integer]] such that $b > 1$.
Let $d$ denote the number of [[Definition:Digit|digits]] of $n$ when represented in [[Definition:Number Base|base $b$]].
Then:
:$d = \ceiling {\map {\... | Let $n$ have $d$ [[Definition:Digit|digits]].
Then:
{{begin-eqn}}
{{eqn | l = b^{d - 1}
| o = \le
| m = n
| mo= <
| r = b^d
| c = [[Basis Representation Theorem]]
}}
{{eqn | ll= \leadsto
| l = b^{d - 1}
| o = <
| m = n + 1
| mo= \le
| r = b^d
| c =
}}
... | Number of Digits to Represent Integer in Given Number Base | https://proofwiki.org/wiki/Number_of_Digits_to_Represent_Integer_in_Given_Number_Base | https://proofwiki.org/wiki/Number_of_Digits_to_Represent_Integer_in_Given_Number_Base | [
"Number Theory"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Integer",
"Definition:Digit",
"Definition:Number Base",
"Definition:Ceiling Function"
] | [
"Definition:Digit",
"Basis Representation Theorem",
"Integer equals Ceiling iff Number between Integer and One Less",
"Category:Number Theory"
] |
proofwiki-14398 | Number of Bits for Decimal Integer | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $n$ have $m$ digits when expressed in decimal notation.
Then $n$ may require as many as $\ceiling {\dfrac m {\log_{10} 2} }$ bits to represent it. | Let $d$ be the number of bits that may be needed to represent $n$.
Let $n$ have $m$ digits.
Then:
:$n \le 10^m - 1$
and so:
{{begin-eqn}}
{{eqn | l = d
| r = \ceiling {\map {\log_2} {\paren {10^m - 1} + 1} }
| c = Number of Digits to Represent Integer in Given Number Base
}}
{{eqn | r = \ceiling {\map {\log... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $n$ have $m$ [[Definition:Digit|digits]] when expressed in [[Definition:Decimal Notation|decimal notation]].
Then $n$ may require as many as $\ceiling {\dfrac m {\log_{10} 2} }$ [[Definition:Bit|bits]] to represent it. | Let $d$ be the number of [[Definition:Bit|bits]] that may be needed to represent $n$.
Let $n$ have $m$ [[Definition:Digit|digits]].
Then:
:$n \le 10^m - 1$
and so:
{{begin-eqn}}
{{eqn | l = d
| r = \ceiling {\map {\log_2} {\paren {10^m - 1} + 1} }
| c = [[Number of Digits to Represent Integer in Given Nu... | Number of Bits for Decimal Integer | https://proofwiki.org/wiki/Number_of_Bits_for_Decimal_Integer | https://proofwiki.org/wiki/Number_of_Bits_for_Decimal_Integer | [
"Number Theory",
"Number of Bits for Decimal Integer"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Digit",
"Definition:Decimal Notation",
"Definition:Bit"
] | [
"Definition:Bit",
"Definition:Digit",
"Number of Digits to Represent Integer in Given Number Base",
"Reciprocal of Logarithm"
] |
proofwiki-14399 | Argument of x to the n Equals n Times The Argument | Let $z$ be a complex number.
Then:
:$\forall n \in \N_{>0}: \map \arg {z^n} = n \map \arg z$ | For $n = 1$
:$\map \arg {z^1} = 1 \cdot \map \arg z$
Assuming the result is true for $n = k$, we have:
{{begin-eqn}}
{{eqn | l = \map \arg {z^{k + 1} }
| r = \map \arg {z z^k}
}}
{{eqn | r = \map \arg z + \map \arg {z^k}
| c = Argument of Product equals Sum of Arguments
}}
{{eqn | r = \map \arg z + k \map \... | Let $z$ be a [[Definition:Complex Number|complex number]].
Then:
:$\forall n \in \N_{>0}: \map \arg {z^n} = n \map \arg z$ | For $n = 1$
:$\map \arg {z^1} = 1 \cdot \map \arg z$
Assuming the result is true for $n = k$, we have:
{{begin-eqn}}
{{eqn | l = \map \arg {z^{k + 1} }
| r = \map \arg {z z^k}
}}
{{eqn | r = \map \arg z + \map \arg {z^k}
| c = [[Argument of Product equals Sum of Arguments]]
}}
{{eqn | r = \map \arg z + k... | Argument of x to the n Equals n Times The Argument | https://proofwiki.org/wiki/Argument_of_x_to_the_n_Equals_n_Times_The_Argument | https://proofwiki.org/wiki/Argument_of_x_to_the_n_Equals_n_Times_The_Argument | [] | [
"Definition:Complex Number"
] | [
"Argument of Product equals Sum of Arguments"
] |
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