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-8900 | Absolute Value Function is Convex | Let $f: \R \to \R$ be the absolute value function on the real numbers.
Then $f$ is convex. | Let $x_1, x_2, x_3 \in \R$ such that $x_1 < x_2 < x_3$.
Consider the expressions:
:$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1}$
:$\dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$
The following cases are investigated:
:$(1): \quad x_1, x_2, x_3 < 0$:
Then:
{{begin-eqn}}
{{eqn | l = \frac {\map f {x_2} - \map f ... | Let $f: \R \to \R$ be the [[Definition:Absolute Value|absolute value function]] on the [[Definition:Real Number|real numbers]].
Then $f$ is [[Definition:Convex Real Function|convex]]. | Let $x_1, x_2, x_3 \in \R$ such that $x_1 < x_2 < x_3$.
Consider the expressions:
:$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1}$
:$\dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$
The following cases are investigated:
:$(1): \quad x_1, x_2, x_3 < 0$:
Then:
{{begin-eqn}}
{{eqn | l = \frac {\map f {x_2} - \... | Absolute Value Function is Convex/Proof 1 | https://proofwiki.org/wiki/Absolute_Value_Function_is_Convex | https://proofwiki.org/wiki/Absolute_Value_Function_is_Convex/Proof_1 | [
"Absolute Value Function is Convex",
"Absolute Value Function",
"Convex Real Functions"
] | [
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Convex Real Function"
] | [
"Definition:Convex Real Function"
] |
proofwiki-8901 | Absolute Value Function is Convex | Let $f: \R \to \R$ be the absolute value function on the real numbers.
Then $f$ is convex. | Let $x, y \in \R$.
Let $\alpha, \beta \in \R_{\ge 0}$ where $\alpha + \beta = 1$.
{{begin-eqn}}
{{eqn | l = \map f {\alpha x + \beta y}
| r = \size {\alpha x + \beta y}
| c = Definition of $f$
}}
{{eqn | o = \le
| r = \size {\alpha x} + \size {\beta y}
| c = Triangle Inequality for Real Numbers
... | Let $f: \R \to \R$ be the [[Definition:Absolute Value|absolute value function]] on the [[Definition:Real Number|real numbers]].
Then $f$ is [[Definition:Convex Real Function|convex]]. | Let $x, y \in \R$.
Let $\alpha, \beta \in \R_{\ge 0}$ where $\alpha + \beta = 1$.
{{begin-eqn}}
{{eqn | l = \map f {\alpha x + \beta y}
| r = \size {\alpha x + \beta y}
| c = Definition of $f$
}}
{{eqn | o = \le
| r = \size {\alpha x} + \size {\beta y}
| c = [[Triangle Inequality for Real Numb... | Absolute Value Function is Convex/Proof 2 | https://proofwiki.org/wiki/Absolute_Value_Function_is_Convex | https://proofwiki.org/wiki/Absolute_Value_Function_is_Convex/Proof_2 | [
"Absolute Value Function is Convex",
"Absolute Value Function",
"Convex Real Functions"
] | [
"Definition:Absolute Value",
"Definition:Real Number",
"Definition:Convex Real Function"
] | [
"Triangle Inequality/Real Numbers",
"Absolute Value Function is Completely Multiplicative",
"Definition:Convex Real Function/Definition 1"
] |
proofwiki-8902 | Real Function is Concave iff Derivative is Decreasing | Let $f$ be a real function which is differentiable on the open interval $\openint a b$.
Then $f$ is concave on $\openint a b$ {{iff}} its derivative $f'$ is decreasing on $\openint a b$.
Thus the intuitive result that a concave function "gets less steep". | === Necessary Condition ===
Let $f$ be concave on $\openint a b$.
Let $x_1, x_2, x_3, x_4 \in \openint a b$ such that:
:$x_1 < x_2 < x_3 < x_4$
By the definition of concave:
:$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \dfrac {\map f {x_3} - \map f {x_2} } {x_3 - x_2} \ge \dfrac {\map f {x_4} - \map f {x_3} ... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then $f$ is [[Definition:Concave Real Function|concave]] on $\openint a b$ {{iff}} its [[Definition:Derivative on Interval|de... | === Necessary Condition ===
Let $f$ be [[Definition:Concave Real Function|concave]] on $\openint a b$.
Let $x_1, x_2, x_3, x_4 \in \openint a b$ such that:
:$x_1 < x_2 < x_3 < x_4$
By the definition of [[Definition:Concave Real Function|concave]]:
:$\dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \dfrac {\map... | Real Function is Concave iff Derivative is Decreasing | https://proofwiki.org/wiki/Real_Function_is_Concave_iff_Derivative_is_Decreasing | https://proofwiki.org/wiki/Real_Function_is_Concave_iff_Derivative_is_Decreasing | [
"Differential Calculus",
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Concave Real Function",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Decreasing/Real Function",
"Definition:Concave Real Function"
] | [
"Definition:Concave Real Function",
"Definition:Concave Real Function",
"Definition:Decreasing/Real Function",
"Definition:Decreasing/Real Function",
"Definition:Decreasing/Real Function",
"Definition:Concave Real Function"
] |
proofwiki-8903 | Real Function is Strictly Convex iff Derivative is Strictly Increasing | Let $f$ be a real function which is differentiable on the open interval $\openint a b$.
Then $f$ is strictly convex on $\openint a b$ {{iff}} its derivative $f'$ is strictly increasing on $\openint a b$. | === Necessary Condition ===
Let $f$ be strictly convex on $\openint a b$.
Let $r, s \in \openint a b$ be arbitrarily selected such that $r < s$.
We are to show that:
:$\map {f'} r < \map {f'} s$
Let $x_1, x_2, x_3 \in \openint a b$ be chosen such that:
:$r < x_1 < x_2 < x_3 < s$
By the definition of strictly convex:
:$... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then $f$ is [[Definition:Strictly Convex Real Function|strictly convex]] on $\openint a b$ {{iff}} its [[Definition:Derivativ... | === Necessary Condition ===
Let $f$ be [[Definition:Strictly Convex Real Function|strictly convex]] on $\openint a b$.
Let $r, s \in \openint a b$ be arbitrarily selected such that $r < s$.
We are to show that:
:$\map {f'} r < \map {f'} s$
Let $x_1, x_2, x_3 \in \openint a b$ be chosen such that:
:$r < x_1 < x_2 <... | Real Function is Strictly Convex iff Derivative is Strictly Increasing | https://proofwiki.org/wiki/Real_Function_is_Strictly_Convex_iff_Derivative_is_Strictly_Increasing | https://proofwiki.org/wiki/Real_Function_is_Strictly_Convex_iff_Derivative_is_Strictly_Increasing | [
"Strictly Increasing Real Functions",
"Convex Real Functions",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Strictly Convex Real Function",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Strictly Increasing/Real Function"
] | [
"Definition:Strictly Convex Real Function",
"Definition:Strictly Convex Real Function",
"Inequality Rule for Real Sequences",
"Inequality Rule for Real Sequences",
"Definition:Strictly Decreasing/Real Function",
"Definition:Strictly Convex Real Function"
] |
proofwiki-8904 | Real Function is Strictly Concave iff Derivative is Strictly Decreasing | Let $f$ be a real function which is differentiable on the open interval $\openint a b$.
Then $f$ is strictly concave on $\openint a b$ {{iff}} its derivative $f'$ is strictly decreasing on $\openint a b$. | === Necessary Condition ===
Let $f$ be strictly concave on $\openint a b$.
Let $r, s \in \openint a b$ be arbitrarily selected such that $r < s$.
We are to show that:
:$\map {f'} r > \map {f'} s$
Let $x_1, x_2, x_3 \in \openint a b$ be chosen such that:
:$r < x_1 < x_2 < x_3 < s$
By the definition of strictly concave:
... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$.
Then $f$ is [[Definition:Strictly Concave Real Function|strictly concave]] on $\openint a b$ {{iff}} its [[Definition:Derivat... | === Necessary Condition ===
Let $f$ be [[Definition:Strictly Concave Real Function|strictly concave]] on $\openint a b$.
Let $r, s \in \openint a b$ be arbitrarily selected such that $r < s$.
We are to show that:
:$\map {f'} r > \map {f'} s$
Let $x_1, x_2, x_3 \in \openint a b$ be chosen such that:
:$r < x_1 < x_2... | Real Function is Strictly Concave iff Derivative is Strictly Decreasing | https://proofwiki.org/wiki/Real_Function_is_Strictly_Concave_iff_Derivative_is_Strictly_Decreasing | https://proofwiki.org/wiki/Real_Function_is_Strictly_Concave_iff_Derivative_is_Strictly_Decreasing | [
"Differential Calculus",
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Strictly Concave Real Function",
"Definition:Derivative/Real Function/Derivative on Interval",
"Definition:Strictly Decreasing/Real Function"
] | [
"Definition:Strictly Concave Real Function",
"Definition:Strictly Concave Real Function",
"Inequality Rule for Real Sequences",
"Inequality Rule for Real Sequences",
"Definition:Strictly Decreasing/Real Function",
"Definition:Strictly Decreasing/Real Function",
"Definition:Strictly Decreasing/Real Funct... |
proofwiki-8905 | Inverse of Strictly Decreasing Convex Real Function is Convex | Let $f$ be a real function which is convex on the open interval $I$.
Let $J = f \left[{I}\right]$.
If $f$ be strictly decreasing on $I$, then $f^{-1}$ is convex on $J$. | Let:
:$X = f \left({x}\right) \in J$
:$Y = f \left({y}\right) \in J$.
From the definition of convex:
: $\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) \le \alpha f \left({x}\right) + \beta f \left({y}\right)$
Let $f$ be strictly decreasing on $I$.
Then from Inverse of Strictl... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Convex Real Function|convex]] on the [[Definition:Open Real Interval|open interval]] $I$.
Let $J = f \left[{I}\right]$.
If $f$ be [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on $I$, then $f^{-1}$ is [[Definition:Co... | Let:
:$X = f \left({x}\right) \in J$
:$Y = f \left({y}\right) \in J$.
From the definition of [[Definition:Convex Real Function|convex]]:
: $\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) \le \alpha f \left({x}\right) + \beta f \left({y}\right)$
Let $f$ be [[Definition:Stri... | Inverse of Strictly Decreasing Convex Real Function is Convex | https://proofwiki.org/wiki/Inverse_of_Strictly_Decreasing_Convex_Real_Function_is_Convex | https://proofwiki.org/wiki/Inverse_of_Strictly_Decreasing_Convex_Real_Function_is_Convex | [
"Convex Real Functions"
] | [
"Definition:Real Function",
"Definition:Convex Real Function",
"Definition:Real Interval/Open",
"Definition:Strictly Decreasing/Real Function",
"Definition:Convex Real Function"
] | [
"Definition:Convex Real Function",
"Definition:Strictly Decreasing/Real Function",
"Inverse of Strictly Monotone Function",
"Definition:Strictly Decreasing/Real Function",
"Definition:Convex Real Function"
] |
proofwiki-8906 | Inverse of Strictly Increasing Concave Real Function is Convex | Let $f$ be a real function which is concave on the open interval $I$.
Let $J = f \sqbrk I$.
If $f$ is strictly increasing on $I$, then $f^{-1}$ is convex on $J$. | Let $X = \map f x \in J, Y = \map f y \in J$.
From the definition of concave:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \ge \alpha \map f x + \beta \map f y$
Let $f$ be strictly increasing on $I$.
From Inverse of Strictly Monotone Function, it follows that $f^{-1}$ is strictly... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Concave Real Function|concave]] on the [[Definition:Open Real Interval|open interval]] $I$.
Let $J = f \sqbrk I$.
If $f$ is [[Definition:Strictly Increasing Real Function|strictly increasing]] on $I$, then $f^{-1}$ is [[Definition:Concave ... | Let $X = \map f x \in J, Y = \map f y \in J$.
From the definition of [[Definition:Concave Real Function|concave]]:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} \ge \alpha \map f x + \beta \map f y$
Let $f$ be [[Definition:Strictly Increasing Real Function|strictly increasing]]... | Inverse of Strictly Increasing Concave Real Function is Convex | https://proofwiki.org/wiki/Inverse_of_Strictly_Increasing_Concave_Real_Function_is_Convex | https://proofwiki.org/wiki/Inverse_of_Strictly_Increasing_Concave_Real_Function_is_Convex | [
"Convex Real Functions",
"Concave Real Functions",
"Strictly Increasing Real Functions"
] | [
"Definition:Real Function",
"Definition:Concave Real Function",
"Definition:Real Interval/Open",
"Definition:Strictly Increasing/Real Function",
"Definition:Concave Real Function"
] | [
"Definition:Concave Real Function",
"Definition:Strictly Increasing/Real Function",
"Inverse of Strictly Monotone Function",
"Definition:Strictly Increasing/Real Function",
"Definition:Convex Real Function"
] |
proofwiki-8907 | Inverse of Strictly Decreasing Concave Real Function is Concave | Let $f$ be a real function which is concave on the open interval $I$.
Let $J = f \left[{I}\right]$.
If $f$ be strictly decreasing on $I$, then $f^{-1}$ is concave on $J$. | Let:
:$X = f \left({x}\right) \in J$
:$Y = f \left({y}\right) \in J$.
From the definition of concave:
: $\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) \ge \alpha f \left({x}\right) + \beta f \left({y}\right)$
Let $f$ be strictly decreasing on $I$.
From Inverse of Strictly Mo... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Concave Real Function|concave]] on the [[Definition:Open Real Interval|open interval]] $I$.
Let $J = f \left[{I}\right]$.
If $f$ be [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on $I$, then $f^{-1}$ is [[Definition:... | Let:
:$X = f \left({x}\right) \in J$
:$Y = f \left({y}\right) \in J$.
From the definition of [[Definition:Concave Real Function|concave]]:
: $\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) \ge \alpha f \left({x}\right) + \beta f \left({y}\right)$
Let $f$ be [[Definition:St... | Inverse of Strictly Decreasing Concave Real Function is Concave | https://proofwiki.org/wiki/Inverse_of_Strictly_Decreasing_Concave_Real_Function_is_Concave | https://proofwiki.org/wiki/Inverse_of_Strictly_Decreasing_Concave_Real_Function_is_Concave | [
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Concave Real Function",
"Definition:Real Interval/Open",
"Definition:Strictly Decreasing/Real Function",
"Definition:Concave Real Function"
] | [
"Definition:Concave Real Function",
"Definition:Strictly Decreasing/Real Function",
"Inverse of Strictly Monotone Function",
"Definition:Strictly Decreasing/Real Function",
"Definition:Concave Real Function"
] |
proofwiki-8908 | Mean Value of Concave Real Function | Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.
Let $f$ be concave on $\openint a b$.
Then:
:$\forall \xi \in \openint a b: \map f x - \map f \xi \le \map {f'} \xi \paren {x - \xi}$ | By the Mean Value Theorem:
:$\exists \eta \in \openint x \xi: \map {f'} \eta = \dfrac {\map f x - \map f \xi} {x - \xi}$
From Real Function is Concave iff Derivative is Decreasing, the derivative of $f$ is decreasing.
Thus:
:$x > \xi \implies \map {f'} \eta \le \map {f'} \xi$
:$x < \xi \implies \map {f'} \eta \ge \map ... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous on Interval|continuous]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ and [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$... | By the [[Mean Value Theorem]]:
:$\exists \eta \in \openint x \xi: \map {f'} \eta = \dfrac {\map f x - \map f \xi} {x - \xi}$
From [[Real Function is Concave iff Derivative is Decreasing]], the [[Definition:Derivative|derivative]] of $f$ is [[Definition:Decreasing Real Function|decreasing]].
Thus:
:$x > \xi \implies \... | Mean Value of Concave Real Function | https://proofwiki.org/wiki/Mean_Value_of_Concave_Real_Function | https://proofwiki.org/wiki/Mean_Value_of_Concave_Real_Function | [
"Concave Real Functions",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Concave Real Function"
] | [
"Mean Value Theorem",
"Real Function is Concave iff Derivative is Decreasing",
"Definition:Derivative",
"Definition:Decreasing/Real Function"
] |
proofwiki-8909 | Differentiable Bounded Concave Real Function is Constant | Let $f$ be a real function which is:
:$(1): \quad$ differentiable on $\R$
:$(2): \quad$ bounded on $\R$
:$(3): \quad$ concave on $\R$.
Then $f$ is constant. | Let $f$ be differentiable and bounded on $\R$.
Let $f$ be concave on $\R$.
Let $\xi \in \R$.
{{AimForCont}} $\map {f'} \xi > 0$.
Then by Mean Value of Concave Real Function it follows that:
:$\map f x \le \map f \xi + \map {f'} \xi \paren {x - \xi} \to -\infty$ as $x \to -\infty$
and therefore is not bounded.
Similarly... | Let $f$ be a [[Definition:Real Function|real function]] which is:
:$(1): \quad$ [[Definition:Everywhere Differentiable Real Function|differentiable]] on $\R$
:$(2): \quad$ [[Definition:Bounded Real-Valued Function|bounded]] on $\R$
:$(3): \quad$ [[Definition:Concave Real Function|concave]] on $\R$.
Then $f$ is [[Defi... | Let $f$ be [[Definition:Everywhere Differentiable Real Function|differentiable]] and [[Definition:Bounded Real-Valued Function|bounded]] on $\R$.
Let $f$ be [[Definition:Concave Real Function|concave]] on $\R$.
Let $\xi \in \R$.
{{AimForCont}} $\map {f'} \xi > 0$.
Then by [[Mean Value of Concave Real Function]] it ... | Differentiable Bounded Concave Real Function is Constant | https://proofwiki.org/wiki/Differentiable_Bounded_Concave_Real_Function_is_Constant | https://proofwiki.org/wiki/Differentiable_Bounded_Concave_Real_Function_is_Constant | [
"Concave Real Functions",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Real Number Line",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Concave Real Function",
"Definition:Constant Mapping"
] | [
"Definition:Differentiable Mapping/Real Function/Real Number Line",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Concave Real Function",
"Mean Value of Concave Real Function",
"Definition:Bounded Mapping/Real-Valued",
"Mean Value of Concave Real Function",
"Definition:Bounded Mapping/Real-Value... |
proofwiki-8910 | Real Function with Strictly Negative Second Derivative is Strictly Concave | Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$ such that:
:$\map {f' '} x < 0$ for each $x \in \openint a b$.
Then $f$ is strictly concave on $\openint a b$. | From Real Function is Strictly Concave iff Derivative is Strictly Decreasing, $f$ is strictly concave {{iff}} $f'$ is strictly decreasing.
Since $f' ' < 0$, we have that $f'$ is strictly decreasing from Real Function with Strictly Negative Derivative is Strictly Decreasing.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Second Derivative|twice differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$ such that:
:$\map {f' '} x < 0$ for each $x \in \openint a b$.
Then $f$ is [[Definition:Strictly Concave Real Function|strictl... | From [[Real Function is Strictly Concave iff Derivative is Strictly Decreasing]], $f$ is [[Definition:Strictly Concave Real Function|strictly concave]] {{iff}} $f'$ is [[Definition:Strictly Decreasing Real Function|strictly decreasing]].
Since $f' ' < 0$, we have that $f'$ is [[Definition:Strictly Decreasing Real Func... | Real Function with Strictly Negative Second Derivative is Strictly Concave | https://proofwiki.org/wiki/Real_Function_with_Strictly_Negative_Second_Derivative_is_Strictly_Concave | https://proofwiki.org/wiki/Real_Function_with_Strictly_Negative_Second_Derivative_is_Strictly_Concave | [
"Differential Calculus",
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Real Interval/Open",
"Definition:Strictly Concave Real Function"
] | [
"Real Function is Strictly Concave iff Derivative is Strictly Decreasing",
"Definition:Strictly Concave Real Function",
"Definition:Strictly Decreasing/Real Function",
"Definition:Strictly Decreasing/Real Function",
"Real Function with Strictly Negative Derivative is Strictly Decreasing"
] |
proofwiki-8911 | Real Function with Strictly Positive Second Derivative is Strictly Convex | Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$ such that:
:$\map {f' '} x > 0$ for each $x \in \openint a b$.
Then $f$ is strictly convex on $\openint a b$. | From Real Function is Strictly Convex iff Derivative is Strictly Increasing, $f$ is strictly convex {{iff}} $f'$ is strictly increasing.
Since $f' ' > 0$, we have that $f'$ is strictly increasing from Real Function with Strictly Positive Derivative is Strictly Increasing.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Second Derivative|twice differentiable]] on the [[Definition:Open Real Interval|open interval]] $\openint a b$ such that:
:$\map {f' '} x > 0$ for each $x \in \openint a b$.
Then $f$ is [[Definition:Strictly Convex Real Function|strictly ... | From [[Real Function is Strictly Convex iff Derivative is Strictly Increasing]], $f$ is [[Definition:Strictly Convex Real Function|strictly convex]] {{iff}} $f'$ is [[Definition:Strictly Increasing Real Function|strictly increasing]].
Since $f' ' > 0$, we have that $f'$ is [[Definition:Strictly Increasing Real Functio... | Real Function with Strictly Positive Second Derivative is Strictly Convex | https://proofwiki.org/wiki/Real_Function_with_Strictly_Positive_Second_Derivative_is_Strictly_Convex | https://proofwiki.org/wiki/Real_Function_with_Strictly_Positive_Second_Derivative_is_Strictly_Convex | [
"Convex Real Functions",
"Differential Calculus"
] | [
"Definition:Real Function",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Real Interval/Open",
"Definition:Strictly Convex Real Function"
] | [
"Real Function is Strictly Convex iff Derivative is Strictly Increasing",
"Definition:Strictly Convex Real Function",
"Definition:Strictly Increasing/Real Function",
"Definition:Strictly Increasing/Real Function",
"Real Function with Strictly Positive Derivative is Strictly Increasing"
] |
proofwiki-8912 | Inverse of Strictly Decreasing Strictly Convex Real Function is Strictly Convex | Let $f$ be a real function which is strictly convex on the open interval $I$.
Let $J = f \sqbrk I$.
If $f$ be strictly decreasing on $I$, then $f^{-1}$ is strictly convex on $J$. | Let:
:$X = \map f x \in J$
:$Y = \map f y \in J$.
From the definition of strictly convex:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} < \alpha \map f x + \beta \map f y$
Let $f$ be strictly decreasing on $I$.
Then from Inverse of Strictly Monotone Function it follows that $f^{-1... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Strictly Convex Real Function|strictly convex]] on the [[Definition:Open Real Interval|open interval]] $I$.
Let $J = f \sqbrk I$.
If $f$ be [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on $I$, then $f^{-1}$ is [[Def... | Let:
:$X = \map f x \in J$
:$Y = \map f y \in J$.
From the definition of [[Definition:Strictly Convex Real Function|strictly convex]]:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} < \alpha \map f x + \beta \map f y$
Let $f$ be [[Definition:Strictly Decreasing Real Function|str... | Inverse of Strictly Decreasing Strictly Convex Real Function is Strictly Convex | https://proofwiki.org/wiki/Inverse_of_Strictly_Decreasing_Strictly_Convex_Real_Function_is_Strictly_Convex | https://proofwiki.org/wiki/Inverse_of_Strictly_Decreasing_Strictly_Convex_Real_Function_is_Strictly_Convex | [
"Convex Real Functions"
] | [
"Definition:Real Function",
"Definition:Strictly Convex Real Function",
"Definition:Real Interval/Open",
"Definition:Strictly Decreasing/Real Function",
"Definition:Strictly Convex Real Function"
] | [
"Definition:Strictly Convex Real Function",
"Definition:Strictly Decreasing/Real Function",
"Inverse of Strictly Monotone Function",
"Definition:Strictly Decreasing/Real Function",
"Definition:Strictly Convex Real Function"
] |
proofwiki-8913 | Inverse of Strictly Decreasing Strictly Concave Real Function is Strictly Concave | Let $f$ be a real function which is strictly concave on the open interval $I$.
Let $J = f \sqbrk I$.
If $f$ be strictly decreasing on $I$, then $f^{-1}$ is strictly concave on $J$. | Let:
:$X = \map f x \in J$
:$Y = \map f y \in J$.
From the definition of strictly concave:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} > \alpha \map f x + \beta \map f y$
Let $f$ be strictly decreasing on $I$.
From Inverse of Strictly Monotone Function, it follows that $f^{-1}$ ... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Strictly Concave Real Function|strictly concave]] on the [[Definition:Open Real Interval|open interval]] $I$.
Let $J = f \sqbrk I$.
If $f$ be [[Definition:Strictly Decreasing Real Function|strictly decreasing]] on $I$, then $f^{-1}$ is [[D... | Let:
:$X = \map f x \in J$
:$Y = \map f y \in J$.
From the definition of [[Definition:Strictly Concave Real Function|strictly concave]]:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} > \alpha \map f x + \beta \map f y$
Let $f$ be [[Definition:Strictly Decreasing Real Function|s... | Inverse of Strictly Decreasing Strictly Concave Real Function is Strictly Concave | https://proofwiki.org/wiki/Inverse_of_Strictly_Decreasing_Strictly_Concave_Real_Function_is_Strictly_Concave | https://proofwiki.org/wiki/Inverse_of_Strictly_Decreasing_Strictly_Concave_Real_Function_is_Strictly_Concave | [
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Strictly Concave Real Function",
"Definition:Real Interval/Open",
"Definition:Strictly Decreasing/Real Function",
"Definition:Strictly Concave Real Function"
] | [
"Definition:Strictly Concave Real Function",
"Definition:Strictly Decreasing/Real Function",
"Inverse of Strictly Monotone Function",
"Definition:Strictly Decreasing/Real Function",
"Definition:Strictly Concave Real Function"
] |
proofwiki-8914 | Inverse of Strictly Increasing Strictly Convex Real Function is Strictly Concave | Let $f$ be a real function which is strictly convex on the open interval $I$.
Let $J = f \sqbrk I$.
If $f$ be strictly increasing on $I$, then $f^{-1}$ is strictly concave on $J$. | Let:
:$X = \map f x \in J$
:$Y = \map f y \in J$
From the definition of strictly convex:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} < \alpha \, \map f x + \beta \, \map f y$
Let $f$ be strictly increasing on $I$.
From Inverse of Strictly Monotone Function it follows that, $f^{-... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Strictly Convex Real Function|strictly convex]] on the [[Definition:Open Real Interval|open interval]] $I$.
Let $J = f \sqbrk I$.
If $f$ be [[Definition:Strictly Increasing Real Function|strictly increasing]] on $I$, then $f^{-1}$ is [[Def... | Let:
:$X = \map f x \in J$
:$Y = \map f y \in J$
From the definition of [[Definition:Strictly Convex Real Function|strictly convex]]:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} < \alpha \, \map f x + \beta \, \map f y$
Let $f$ be [[Definition:Strictly Increasing Real Functio... | Inverse of Strictly Increasing Strictly Convex Real Function is Strictly Concave | https://proofwiki.org/wiki/Inverse_of_Strictly_Increasing_Strictly_Convex_Real_Function_is_Strictly_Concave | https://proofwiki.org/wiki/Inverse_of_Strictly_Increasing_Strictly_Convex_Real_Function_is_Strictly_Concave | [
"Strictly Increasing Real Functions",
"Convex Real Functions",
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Strictly Convex Real Function",
"Definition:Real Interval/Open",
"Definition:Strictly Increasing/Real Function",
"Definition:Strictly Concave Real Function"
] | [
"Definition:Strictly Convex Real Function",
"Definition:Strictly Increasing/Real Function",
"Inverse of Strictly Monotone Function",
"Definition:Strictly Increasing/Real Function",
"Definition:Strictly Concave Real Function"
] |
proofwiki-8915 | Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex | Let $f$ be a real function which is strictly concave on the open interval $I$.
Let $J = f \sqbrk I$.
If $f$ be strictly increasing on $I$, then $f^{-1}$ is strictly convex on $J$. | Let $X = \map f x \in J, Y = \map f y \in J$.
From the definition of Strictly concave:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} > \alpha \map f x + \beta \map f y$
Let $f$ be strictly increasing on $I$.
From Inverse of Strictly Monotone Function, it follows that $f^{-1}$ is s... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Strictly Concave Real Function|strictly concave]] on the [[Definition:Open Real Interval|open interval]] $I$.
Let $J = f \sqbrk I$.
If $f$ be [[Definition:Strictly Increasing Real Function|strictly increasing]] on $I$, then $f^{-1}$ is [[D... | Let $X = \map f x \in J, Y = \map f y \in J$.
From the definition of [[Definition:Strictly Concave Real Function|Strictly concave]]:
:$\forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: \map f {\alpha x + \beta y} > \alpha \map f x + \beta \map f y$
Let $f$ be [[Definition:Strictly Increasing Real Function|stric... | Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex | https://proofwiki.org/wiki/Inverse_of_Strictly_Increasing_Strictly_Concave_Real_Function_is_Strictly_Convex | https://proofwiki.org/wiki/Inverse_of_Strictly_Increasing_Strictly_Concave_Real_Function_is_Strictly_Convex | [
"Strictly Increasing Real Functions",
"Convex Real Functions",
"Concave Real Functions"
] | [
"Definition:Real Function",
"Definition:Strictly Concave Real Function",
"Definition:Real Interval/Open",
"Definition:Strictly Increasing/Real Function",
"Definition:Strictly Concave Real Function"
] | [
"Definition:Strictly Concave Real Function",
"Definition:Strictly Increasing/Real Function",
"Inverse of Strictly Monotone Function",
"Definition:Strictly Increasing/Real Function",
"Definition:Strictly Convex Real Function"
] |
proofwiki-8916 | Gamma Function as Integral of Natural Logarithm | Let $x \in \R_{>0}$ be a strictly positive real number.
Then:
:$\ds \map \Gamma x = \int_{\to 0}^1 \paren {\ln \frac 1 t}^{x - 1} \rd t$
where $\Gamma$ denotes the Gamma function. | By definition of the Gamma function:
:$\ds \map \Gamma x = \int_0^{\to \infty} t^{x - 1} e^{-t} \rd t$
In order to allow the limits to be evaluated, this is to be expressed as:
:$\ds \map \Gamma x = \lim_{\delta \mathop \to 0^+, \ \Delta \mathop \to +\infty} \int_\delta^\Delta t^{x - 1} e^{-t} \rd t$
where $0 < \delta ... | Let $x \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Then:
:$\ds \map \Gamma x = \int_{\to 0}^1 \paren {\ln \frac 1 t}^{x - 1} \rd t$
where $\Gamma$ denotes the [[Definition:Gamma Function|Gamma function]]. | By definition of the [[Definition:Gamma Function|Gamma function]]:
:$\ds \map \Gamma x = \int_0^{\to \infty} t^{x - 1} e^{-t} \rd t$
In order to allow the [[Definition:Limits of Integration|limits]] to be evaluated, this is to be expressed as:
:$\ds \map \Gamma x = \lim_{\delta \mathop \to 0^+, \ \Delta \mathop \to +\... | Gamma Function as Integral of Natural Logarithm | https://proofwiki.org/wiki/Gamma_Function_as_Integral_of_Natural_Logarithm | https://proofwiki.org/wiki/Gamma_Function_as_Integral_of_Natural_Logarithm | [
"Gamma Function"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Gamma Function"
] | [
"Definition:Gamma Function",
"Definition:Definite Integral/Limits of Integration",
"Derivative of Logarithm Function",
"Derivative of Composite Function",
"Exponential of Natural Logarithm",
"Integration by Substitution",
"Reversal of Limits of Definite Integral",
"Exponential of Natural Logarithm",
... |
proofwiki-8917 | Reversal of Limits of Definite Integral | Let $a \le b$.
Then:
:$\ds \int_a^b \map f x \rd x = -\int_b^a \map f x \rd x$ | {{begin-eqn}}
{{eqn | l = \int_a^b \map f x \rd x + \int_b^a \map f x \rd x
| r = \int_a^a \map f x \rd x
| c = Sum of Integrals on Adjacent Intervals for Integrable Functions
}}
{{eqn | r = 0
| c = Definite Integral on Zero Interval
}}
{{eqn | ll= \leadsto
| l = \int_a^b \map f x \rd x
| ... | Let $a \le b$.
Then:
:$\ds \int_a^b \map f x \rd x = -\int_b^a \map f x \rd x$ | {{begin-eqn}}
{{eqn | l = \int_a^b \map f x \rd x + \int_b^a \map f x \rd x
| r = \int_a^a \map f x \rd x
| c = [[Sum of Integrals on Adjacent Intervals for Integrable Functions]]
}}
{{eqn | r = 0
| c = [[Definite Integral on Zero Interval]]
}}
{{eqn | ll= \leadsto
| l = \int_a^b \map f x \rd x
... | Reversal of Limits of Definite Integral | https://proofwiki.org/wiki/Reversal_of_Limits_of_Definite_Integral | https://proofwiki.org/wiki/Reversal_of_Limits_of_Definite_Integral | [
"Definite Integrals"
] | [] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Definite Integral on Zero Interval",
"Sum of Integrals on Adjacent Intervals for Integrable Functions"
] |
proofwiki-8918 | Beta Function is Defined for Positive Reals | Let $x, y \in \R$ be real numbers.
Let $\map \Beta {x, y}$ be the Beta function:
:$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$
Then $\map \Beta {x, y}$ exists provided that $x, y > 0$. | Consider the following inequalities, valid for $0 < t < 1$:
{{begin-eqn}}
{{eqn | l = t^{x - 1} \paren {1 - t}^{y - 1}
| o = <
| r = t^{x - 1}
| c =
}}
{{eqn | ll= \leadsto
| l = t^{x - 1} \paren {1 - t}^{y - 1}
| o = <
| r = \paren {1 - t}^{y - 1}
| c =
}}
{{end-eqn}}
Then:
... | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\map \Beta {x, y}$ be the [[Definition:Beta Function/Definition 1|Beta function]]:
:$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$
Then $\map \Beta {x, y}$ exists provided that $x, y > 0$. | Consider the following inequalities, valid for $0 < t < 1$:
{{begin-eqn}}
{{eqn | l = t^{x - 1} \paren {1 - t}^{y - 1}
| o = <
| r = t^{x - 1}
| c =
}}
{{eqn | ll= \leadsto
| l = t^{x - 1} \paren {1 - t}^{y - 1}
| o = <
| r = \paren {1 - t}^{y - 1}
| c =
}}
{{end-eqn}}
Then... | Beta Function is Defined for Positive Reals | https://proofwiki.org/wiki/Beta_Function_is_Defined_for_Positive_Reals | https://proofwiki.org/wiki/Beta_Function_is_Defined_for_Positive_Reals | [
"Beta Function"
] | [
"Definition:Real Number",
"Definition:Beta Function/Definition 1"
] | [
"Comparison Test for Improper Integral"
] |
proofwiki-8919 | Comparison Test for Improper Integral | Let $I = \openint a b$ be an open real interval.
Let $\phi$ be a real function which is continuous on $I$ and also non-negative on $I$.
Let $f$ be a real function which is continuous on $I$.
Let $f$ satisfy:
:$\forall x \in I: \size {\map f x} \le \map \phi x$
If the improper integral of $\phi$ over $I$ exists, then so... | {{WLOG}}, we consider the case $I = \openint 0 \to$ such that $\ds l = \int_0^{\mathop \to +\infty} \map \phi x \rd x$ exists.
Let:
:$\ds a_n = \int_{n - 1}^n \map f x \rd x$
:$\ds b_n = \int_{n - 1}^n \map \phi x \rd x$
for $n = 1, 2, \ldots$
Then the series:
:$\ds \sum_{n \mathop = 1}^\infty b_n$
is a convergent seri... | Let $I = \openint a b$ be an [[Definition:Open Real Interval|open real interval]].
Let $\phi$ be a [[Definition:Real Function|real function]] which is [[Definition:Continuous Real Function on Open Interval|continuous on $I$]] and also [[Definition:Positive Real Function|non-negative]] on $I$.
Let $f$ be a [[Definitio... | {{WLOG}}, we consider the case $I = \openint 0 \to$ such that $\ds l = \int_0^{\mathop \to +\infty} \map \phi x \rd x$ exists.
Let:
:$\ds a_n = \int_{n - 1}^n \map f x \rd x$
:$\ds b_n = \int_{n - 1}^n \map \phi x \rd x$
for $n = 1, 2, \ldots$
Then the [[Definition:Series|series]]:
:$\ds \sum_{n \mathop = 1}^\infty b... | Comparison Test for Improper Integral | https://proofwiki.org/wiki/Comparison_Test_for_Improper_Integral | https://proofwiki.org/wiki/Comparison_Test_for_Improper_Integral | [
"Comparison Test",
"Convergence Tests",
"Improper Integrals"
] | [
"Definition:Real Interval/Open",
"Definition:Real Function",
"Definition:Continuous Real Function/Open Interval",
"Definition:Positive Real Function",
"Definition:Real Function",
"Definition:Continuous Real Function/Open Interval",
"Definition:Improper Integral/Open Interval"
] | [
"Definition:Series",
"Definition:Convergent Series",
"Definition:Positive/Real Number",
"Definition:Convergent Series",
"Comparison Test",
"Definition:Natural Numbers"
] |
proofwiki-8920 | Beta Function is Continuous and Positive on Positive Reals | Let $x, y \in \R$ be real numbers.
Let $\map \Beta {x, y}$ be the Beta function:
:$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$
Let $y \in \R_{>0}$ be given.
Then $\map \Beta {x, y}$ is a positive and continuous function of $x$ on $\R_{>0}$. | For each $x > 0$, we have for all $t$ with $0 < t < 1$ that:
:$t^{x - 1} \paren {1 - t}^{y - 1} > 0$
from which it is immediate that $\map \Beta {x, y} > 0$.
{{ProofWanted|Continuity to be established.}} | Let $x, y \in \R$ be [[Definition:Real Number|real numbers]].
Let $\map \Beta {x, y}$ be the [[Definition:Beta Function/Definition 1|Beta function]]:
:$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$
Let $y \in \R_{>0}$ be given.
Then $\map \Beta {x, y}$ is a [[... | For each $x > 0$, we have for all $t$ with $0 < t < 1$ that:
:$t^{x - 1} \paren {1 - t}^{y - 1} > 0$
from which it is immediate that $\map \Beta {x, y} > 0$.
{{ProofWanted|Continuity to be established.}} | Beta Function is Continuous and Positive on Positive Reals | https://proofwiki.org/wiki/Beta_Function_is_Continuous_and_Positive_on_Positive_Reals | https://proofwiki.org/wiki/Beta_Function_is_Continuous_and_Positive_on_Positive_Reals | [
"Beta Function"
] | [
"Definition:Real Number",
"Definition:Beta Function/Definition 1",
"Definition:Positive Real Function",
"Definition:Continuous Real Function",
"Definition:Real Function"
] | [] |
proofwiki-8921 | Gamma Function of One Half | :$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ | From the definition of the Beta function:
:$\map \Beta {x, y} := \dfrac {\map \Gamma x \map \Gamma y} {\map \Gamma {x + y} }$
Setting $x = y = \dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \map \Beta {\dfrac 1 2, \dfrac 1 2}
| r = \frac {\map \Gamma {\dfrac 1 2} \map \Gamma {\dfrac 1 2} } {\map \Gamma {\dfrac 1 2 + \dfr... | :$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ | From the definition of the [[Definition:Beta Function/Definition 3|Beta function]]:
:$\map \Beta {x, y} := \dfrac {\map \Gamma x \map \Gamma y} {\map \Gamma {x + y} }$
Setting $x = y = \dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \map \Beta {\dfrac 1 2, \dfrac 1 2}
| r = \frac {\map \Gamma {\dfrac 1 2} \map \Gamma {... | Gamma Function of One Half/Proof 1 | https://proofwiki.org/wiki/Gamma_Function_of_One_Half | https://proofwiki.org/wiki/Gamma_Function_of_One_Half/Proof_1 | [
"Gamma Function of One Half",
"Examples of Gamma Function Values"
] | [] | [
"Definition:Beta Function/Definition 3",
"Beta Function of Half with Half"
] |
proofwiki-8922 | Gamma Function of One Half | :$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ | From Euler's Reflection Formula:
:$\forall z \notin \Z: \map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$
Setting $z = \dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \map \Gamma {\frac 1 2} \map \Gamma {\frac 1 2}
| r = \frac \pi {\map \sin {\frac \pi 2} }
| c =
}}
{{eqn | r = \frac \pi 1
... | :$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ | From [[Euler's Reflection Formula]]:
:$\forall z \notin \Z: \map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$
Setting $z = \dfrac 1 2$:
{{begin-eqn}}
{{eqn | l = \map \Gamma {\frac 1 2} \map \Gamma {\frac 1 2}
| r = \frac \pi {\map \sin {\frac \pi 2} }
| c =
}}
{{eqn | r = \frac \pi 1... | Gamma Function of One Half/Proof 2 | https://proofwiki.org/wiki/Gamma_Function_of_One_Half | https://proofwiki.org/wiki/Gamma_Function_of_One_Half/Proof_2 | [
"Gamma Function of One Half",
"Examples of Gamma Function Values"
] | [] | [
"Euler's Reflection Formula",
"Sine of Right Angle",
"Definition:Gamma Function",
"Definition:Negative/Real Number",
"Definition:Square Root"
] |
proofwiki-8923 | Gamma Function of One Half | :$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {\dfrac 1 2}
| r = \int_0^{\to \infty} t^{-\frac 1 2} e^{-t} \rd t
| c = {{Defof|Integral Form of Gamma Function|Gamma Function}}
}}
{{eqn | r = \int_0^{\to \infty} u^{-1} e^{-u^2} 2 u \rd u
| c = Integration by Substitution, $\map \phi u = u^2$
}}
{{eqn | r = 2 \... | :$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {\dfrac 1 2}
| r = \int_0^{\to \infty} t^{-\frac 1 2} e^{-t} \rd t
| c = {{Defof|Integral Form of Gamma Function|Gamma Function}}
}}
{{eqn | r = \int_0^{\to \infty} u^{-1} e^{-u^2} 2 u \rd u
| c = [[Integration by Substitution]], $\map \phi u = u^2$
}}
{{eqn | r =... | Gamma Function of One Half/Proof 3 | https://proofwiki.org/wiki/Gamma_Function_of_One_Half | https://proofwiki.org/wiki/Gamma_Function_of_One_Half/Proof_3 | [
"Gamma Function of One Half",
"Examples of Gamma Function Values"
] | [] | [
"Integration by Substitution",
"Linear Combination of Integrals/Definite",
"Definite Integral of Even Function",
"Gaussian Integral"
] |
proofwiki-8924 | Gamma Function of One Half | :$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {\frac 1 2}
| r = \frac {0!} {2^0 0!} \sqrt \pi
| c = Gamma Function of Positive Half-Integer
}}
{{eqn | r = \sqrt \pi
| c = Factorial of Zero
}}
{{end-eqn}}
{{qed}} | :$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {\frac 1 2}
| r = \frac {0!} {2^0 0!} \sqrt \pi
| c = [[Gamma Function of Positive Half-Integer]]
}}
{{eqn | r = \sqrt \pi
| c = [[Factorial of Zero]]
}}
{{end-eqn}}
{{qed}} | Gamma Function of One Half/Proof 4 | https://proofwiki.org/wiki/Gamma_Function_of_One_Half | https://proofwiki.org/wiki/Gamma_Function_of_One_Half/Proof_4 | [
"Gamma Function of One Half",
"Examples of Gamma Function Values"
] | [] | [
"Gamma Function of Positive Half-Integer",
"Factorial/Examples/0"
] |
proofwiki-8925 | Gamma Function of One Half | :$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ | {{begin-eqn}}
{{eqn | l = \map \Gamma 1 \, \map \Gamma {\frac 1 2}
| r = 2^0 \sqrt \pi \ \map \Gamma 1
| c = Legendre's Duplication Formula
}}
{{eqn | ll= \leadsto
| l = 0! \, \map \Gamma {\frac 1 2}
| r = 0! \, \sqrt \pi
| c = {{Defof|Gamma Function}}
}}
{{eqn | ll= \leadsto
... | :$\map \Gamma {\dfrac 1 2} = \sqrt \pi$ | {{begin-eqn}}
{{eqn | l = \map \Gamma 1 \, \map \Gamma {\frac 1 2}
| r = 2^0 \sqrt \pi \ \map \Gamma 1
| c = [[Legendre's Duplication Formula]]
}}
{{eqn | ll= \leadsto
| l = 0! \, \map \Gamma {\frac 1 2}
| r = 0! \, \sqrt \pi
| c = {{Defof|Gamma Function}}
}}
{{eqn | ll= \leadsto
... | Gamma Function of One Half/Proof 5 | https://proofwiki.org/wiki/Gamma_Function_of_One_Half | https://proofwiki.org/wiki/Gamma_Function_of_One_Half/Proof_5 | [
"Gamma Function of One Half",
"Examples of Gamma Function Values"
] | [] | [
"Legendre's Duplication Formula",
"Factorial/Examples/0"
] |
proofwiki-8926 | Integral of Exponent of Half Square over Reals | :$\ds \int_{\mathop \to -\infty}^{\mathop \to +\infty} e^{- x^2 / 2} \rd x = \sqrt {2 \pi}$ | Let $t = \dfrac {x^2} 2$.
Then:
{{begin-eqn}}
{{eqn | l = \int_0^{\mathop \to +\infty} e^{- x^2 / 2} \rd x
| r = \int_0^{\mathop \to +\infty} \paren {2 t} e^{-t} \rd t
| c = Integration by Substitution
}}
{{eqn | r = \frac 1 {\sqrt 2} \map \Gamma {\frac 1 2}
| c = {{Defof|Gamma Function|subdef = Integ... | :$\ds \int_{\mathop \to -\infty}^{\mathop \to +\infty} e^{- x^2 / 2} \rd x = \sqrt {2 \pi}$ | Let $t = \dfrac {x^2} 2$.
Then:
{{begin-eqn}}
{{eqn | l = \int_0^{\mathop \to +\infty} e^{- x^2 / 2} \rd x
| r = \int_0^{\mathop \to +\infty} \paren {2 t} e^{-t} \rd t
| c = [[Integration by Substitution]]
}}
{{eqn | r = \frac 1 {\sqrt 2} \map \Gamma {\frac 1 2}
| c = {{Defof|Gamma Function|subdef = ... | Integral of Exponent of Half Square over Reals | https://proofwiki.org/wiki/Integral_of_Exponent_of_Half_Square_over_Reals | https://proofwiki.org/wiki/Integral_of_Exponent_of_Half_Square_over_Reals | [
"Gaussian Integral"
] | [] | [
"Integration by Substitution",
"Gamma Function of One Half",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Even Function"
] |
proofwiki-8927 | Sum of Reciprocals of Primes is Divergent/Proof 2 | The series:
:$\ds \sum_{p \mathop \in \Bbb P} \frac 1 p$
where:
:$\Bbb P$ is the set of all prime numbers
is divergent. | Let $n \in \N$ be a natural number.
Let $p_n$ denote the $n$th prime number.
Consider the continued product:
:$\ds \prod_{k \mathop = 1}^n \frac 1 {1 - 1 / p_k}$
By Sum of Infinite Geometric Sequence:
{{begin-eqn}}
{{eqn | l = \frac 1 {1 - \frac 1 2}
| r = 1 + \frac 1 2 + \frac 1 {2^2} + \cdots
| c =
}}
{{... | The [[Definition:Series|series]]:
:$\ds \sum_{p \mathop \in \Bbb P} \frac 1 p$
where:
:$\Bbb P$ is the [[Definition:Set|set]] of all [[Definition:Prime Number|prime numbers]]
is [[Definition:Divergent Series|divergent]]. | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $p_n$ denote the $n$th [[Definition:Prime Number|prime number]].
Consider the [[Definition:Continued Product|continued product]]:
:$\ds \prod_{k \mathop = 1}^n \frac 1 {1 - 1 / p_k}$
By [[Sum of Infinite Geometric Sequence]]:
{{begin-eqn}}
{{eqn ... | Sum of Reciprocals of Primes is Divergent/Proof 2 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent/Proof_2 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent/Proof_2 | [
"Sum of Reciprocals of Primes is Divergent"
] | [
"Definition:Series",
"Definition:Set",
"Definition:Prime Number",
"Definition:Divergent Series"
] | [
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Continued Product",
"Sum of Infinite Geometric Sequence",
"Definition:Series",
"Definition:Series",
"Definition:Series",
"Definition:Series",
"Definition:Convergent Series",
"Fundamental Theorem of Arithmetic",
"Definition:Inte... |
proofwiki-8928 | Sum of Reciprocals of Primes is Divergent/Lemma | Let $C \in \R_{>0}$ be a (strictly) positive real number.
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {\map \ln {\ln n} - C} = + \infty$ | Fix $c \in \R$.
It is sufficient to show there exists $N \in \N$, such that:
:$(1): \quad n \ge N \implies \map \ln {\ln n} - C > c$
Proceed as follows:
{{begin-eqn}}
{{eqn | l = \map \ln {\ln n} - C
| o = >
| r = c
}}
{{eqn | ll= \leadstoandfrom
| l = \ln n
| o = >
| r = \map \exp {c + C}... | Let $C \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {\map \ln {\ln n} - C} = + \infty$ | Fix $c \in \R$.
It is sufficient to show there exists $N \in \N$, such that:
:$(1): \quad n \ge N \implies \map \ln {\ln n} - C > c$
Proceed as follows:
{{begin-eqn}}
{{eqn | l = \map \ln {\ln n} - C
| o = >
| r = c
}}
{{eqn | ll= \leadstoandfrom
| l = \ln n
| o = >
| r = \map \exp {c... | Sum of Reciprocals of Primes is Divergent/Lemma | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent/Lemma | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Primes_is_Divergent/Lemma | [
"Number Theory"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Logarithm is Strictly Increasing"
] |
proofwiki-8929 | Strictly Increasing Infinite Sequence of Integers is Cofinal in Natural Numbers | Let $S = \left\langle{x_n}\right\rangle$ be an infinite sequence of integers which is strictly increasing.
Then $S$ is a cofinal subset of $\left({\Z, \le}\right)$ where $\le$ is the usual ordering on the integers. | {{ProofWanted}}
Category:Integers
4lesc476t30awzjfte4szdcdh1hi33j | Let $S = \left\langle{x_n}\right\rangle$ be an [[Definition:Infinite Sequence|infinite sequence]] of [[Definition:Integer|integers]] which is [[Definition:Strictly Increasing Sequence|strictly increasing]].
Then $S$ is a [[Definition:Cofinal Subset|cofinal subset of $\left({\Z, \le}\right)$]] where $\le$ is the [[De... | {{ProofWanted}}
[[Category:Integers]]
4lesc476t30awzjfte4szdcdh1hi33j | Strictly Increasing Infinite Sequence of Integers is Cofinal in Natural Numbers | https://proofwiki.org/wiki/Strictly_Increasing_Infinite_Sequence_of_Integers_is_Cofinal_in_Natural_Numbers | https://proofwiki.org/wiki/Strictly_Increasing_Infinite_Sequence_of_Integers_is_Cofinal_in_Natural_Numbers | [
"Integers"
] | [
"Definition:Sequence/Infinite Sequence",
"Definition:Integer",
"Definition:Strictly Increasing/Sequence",
"Definition:Cofinal Subset",
"Definition:Usual Ordering",
"Definition:Integer"
] | [
"Category:Integers"
] |
proofwiki-8930 | Product of Even and Odd Functions | Let $\OO$ be an odd real function defined on some symmetric set $S$.
Let $\EE$ be an even real function defined on some symmetric set $S'$.
Let $\OO \EE$ be their pointwise product, defined on the intersection of the domains of $\OO$ and $\EE$.
Then $\OO \EE$ is odd.
That is:
:$\forall x \in S \cap S': \map {\paren {\... | {{begin-eqn}}
{{eqn | l = \map {\paren {\OO \EE} } {-x}
| r = \map \OO {-x} \map \EE {-x}
| c = {{Defof|Pointwise Multiplication of Real-Valued Functions}}
}}
{{eqn | r = -\map \OO x \map \EE x
| c = as $\OO$ is odd and $\EE$ is even
}}
{{eqn | r = -\map {\paren {\OO \EE} } x
}}
{{end-eqn}}
The result... | Let $\OO$ be an [[Definition:Odd Function|odd]] [[Definition:Real Function|real function]] defined on some [[Definition:Symmetric Set|symmetric set]] $S$.
Let $\EE$ be an [[Definition:Even Function|even]] [[Definition:Real Function|real function]] defined on some [[Definition:Symmetric Set|symmetric set]] $S'$.
Let $... | {{begin-eqn}}
{{eqn | l = \map {\paren {\OO \EE} } {-x}
| r = \map \OO {-x} \map \EE {-x}
| c = {{Defof|Pointwise Multiplication of Real-Valued Functions}}
}}
{{eqn | r = -\map \OO x \map \EE x
| c = as $\OO$ is [[Definition:Odd Function|odd]] and $\EE$ is [[Definition:Even Function|even]]
}}
{{eqn | ... | Product of Even and Odd Functions | https://proofwiki.org/wiki/Product_of_Even_and_Odd_Functions | https://proofwiki.org/wiki/Product_of_Even_and_Odd_Functions | [
"Real Analysis"
] | [
"Definition:Odd Function",
"Definition:Real Function",
"Definition:Symmetric Set",
"Definition:Even Function",
"Definition:Real Function",
"Definition:Symmetric Set",
"Definition:Pointwise Multiplication of Real-Valued Functions",
"Definition:Set Intersection",
"Definition:Domain (Set Theory)/Mappin... | [
"Definition:Odd Function",
"Definition:Even Function",
"Definition:Odd Function"
] |
proofwiki-8931 | Odd Bernoulli Numbers Vanish | Let $B_n$ denote the $n$th Bernoulli Number.
Then:
:$B_{2n + 1} = 0$
for $n \ge 1$. | By definition, the Bernoulli numbers are given by:
:$\ds \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$
We have:
{{begin-eqn}}
{{eqn | l = \frac x {e^x - 1}
| r = \frac x 2 \paren {\frac 2 {e^x - 1} }
| c = multiplying top and bottom by $2$
}}
{{eqn | r = \frac x 2 \paren {\frac {e^x... | Let $B_n$ denote the $n$th [[Definition:Bernoulli Numbers|Bernoulli Number]].
Then:
:$B_{2n + 1} = 0$
for $n \ge 1$. | By definition, the [[Definition:Bernoulli Numbers/Generating Function|Bernoulli numbers]] are given by:
:$\ds \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$
We have:
{{begin-eqn}}
{{eqn | l = \frac x {e^x - 1}
| r = \frac x 2 \paren {\frac 2 {e^x - 1} }
| c = multiplying [[Definiti... | Odd Bernoulli Numbers Vanish | https://proofwiki.org/wiki/Odd_Bernoulli_Numbers_Vanish | https://proofwiki.org/wiki/Odd_Bernoulli_Numbers_Vanish | [
"Bernoulli Numbers",
"Odd Bernoulli Numbers Vanish"
] | [
"Definition:Bernoulli Numbers"
] | [
"Definition:Bernoulli Numbers/Generating Function",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Even Function",
"Definition:Bernoulli Numbers/Generating Function",
"Definition:Even Function",
"Def... |
proofwiki-8932 | Gauss Multiplication Formula | Let $\Gamma$ denote the Gamma Function.
Let $n \in \N_{>0}$ where $\N_{>0}$ denotes the non-zero natural numbers.
Then:
:$\ds \forall z \notin \set {-\frac m n: m \in \N}: \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {z + \frac k n}
| r = \paren {z + \frac k n - 1} \map \Gamma {z + \frac k n - 1}
| c = Gamma Difference Equation
}}
{{eqn | r = \lim_{m \mathop \to \infty} \paren {z + \frac k n - 1} \frac {m! m^{z + k / n - 1} } {\paren {z + \frac k n - 1} \paren {z + \frac k n} \paren... | Let $\Gamma$ denote the [[Definition:Gamma Function|Gamma Function]].
Let $n \in \N_{>0}$ where $\N_{>0}$ denotes the [[Definition:Non-Zero Natural Numbers|non-zero natural numbers]].
Then:
:$\ds \forall z \notin \set {-\frac m n: m \in \N}: \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^... | {{begin-eqn}}
{{eqn | l = \map \Gamma {z + \frac k n}
| r = \paren {z + \frac k n - 1} \map \Gamma {z + \frac k n - 1}
| c = [[Gamma Difference Equation]]
}}
{{eqn | r = \lim_{m \mathop \to \infty} \paren {z + \frac k n - 1} \frac {m! m^{z + k / n - 1} } {\paren {z + \frac k n - 1} \paren {z + \frac k n} \p... | Gauss Multiplication Formula | https://proofwiki.org/wiki/Gauss_Multiplication_Formula | https://proofwiki.org/wiki/Gauss_Multiplication_Formula | [
"Gamma Function"
] | [
"Definition:Gamma Function",
"Definition:Natural Numbers/Non-Zero"
] | [
"Gamma Difference Equation",
"Stirling's Formula",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sum of Arithmetic Sequence",
"Stirling's Formula",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Gamma Difference Equation"
] |
proofwiki-8933 | Dirichlet Integral | :$\ds \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$ | By Fubini's Theorem:
:$\ds \int_0^\infty \paren {\int_0^\infty e^{- x y} \sin x \rd y} \rd x = \int_0^\infty \paren {\int_0^\infty e^{- x y} \sin x \rd x} \rd y$
Then:
{{begin-eqn}}
{{eqn | l = \int_0^\infty e^{- x y} \sin x \rd y
| r = \intlimits {-e^{- x y} \frac {\sin x} x} 0 \infty
| c = Primitive of $... | :$\ds \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$ | By [[Fubini's Theorem]]:
:$\ds \int_0^\infty \paren {\int_0^\infty e^{- x y} \sin x \rd y} \rd x = \int_0^\infty \paren {\int_0^\infty e^{- x y} \sin x \rd x} \rd y$
Then:
{{begin-eqn}}
{{eqn | l = \int_0^\infty e^{- x y} \sin x \rd y
| r = \intlimits {-e^{- x y} \frac {\sin x} x} 0 \infty
| c = [[Primi... | Dirichlet Integral/Proof 1 | https://proofwiki.org/wiki/Dirichlet_Integral | https://proofwiki.org/wiki/Dirichlet_Integral/Proof_1 | [
"Dirichlet Integral",
"Sine Integral Function",
"Definite Integrals involving Sine Function",
"Integral Calculus"
] | [] | [
"Fubini's Theorem",
"Primitive of Exponential of a x",
"Primitive of Exponential of a x by Sine of b x",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-8934 | Dirichlet Integral | :$\ds \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$ | By Modulus of Sine of x Less Than or Equal To Absolute Value of x:
:$\ds \size {\frac {e^{-\alpha x} \sin x} x} \le e^{-\alpha x}$
From Laplace Transform of Real Power:
:$\ds \int_0^\infty e^{-\alpha x} x^0 \rd x = \frac {\map \Gamma {0 + 1} } {\alpha^{0 + 1} } = \frac 1 \alpha$
Hence by Comparison Test for Improper I... | :$\ds \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$ | By [[Modulus of Sine of x Less Than or Equal To Absolute Value of x]]:
:$\ds \size {\frac {e^{-\alpha x} \sin x} x} \le e^{-\alpha x}$
From [[Laplace Transform of Real Power]]:
:$\ds \int_0^\infty e^{-\alpha x} x^0 \rd x = \frac {\map \Gamma {0 + 1} } {\alpha^{0 + 1} } = \frac 1 \alpha$
Hence by [[Comparison Test ... | Dirichlet Integral/Proof 2 | https://proofwiki.org/wiki/Dirichlet_Integral | https://proofwiki.org/wiki/Dirichlet_Integral/Proof_2 | [
"Dirichlet Integral",
"Sine Integral Function",
"Definite Integrals involving Sine Function",
"Integral Calculus"
] | [] | [
"Modulus of Sine of x Less Than or Equal To Absolute Value of x",
"Laplace Transform of Real Power",
"Comparison Test for Improper Integral",
"Definition:Convergent Series",
"Definition:Real Function",
"Improper Integral of Partial Derivative",
"Leibniz's Integral Rule",
"Derivative of Exponential Fun... |
proofwiki-8935 | Dirichlet Integral | :$\ds \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$ | Let:
:$\map f x = \begin {cases} \dfrac {e^{i x} - 1} x & x \ne 0 \\ i & x = 0 \end {cases}$
We have, by Euler's Formula, for $x \in \R$:
:$\map \Im {\map f x} = \begin {cases} \dfrac {\sin x} x & x \ne 0 \\ 1 & x = 0 \end {cases}$
So:
:$\ds \map \Im {\int_0^\infty \dfrac {e^{i x} - 1} x \rd x} = \int_0^\infty \dfrac ... | :$\ds \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$ | Let:
:$\map f x = \begin {cases} \dfrac {e^{i x} - 1} x & x \ne 0 \\ i & x = 0 \end {cases}$
We have, by [[Euler's Formula]], for $x \in \R$:
:$\map \Im {\map f x} = \begin {cases} \dfrac {\sin x} x & x \ne 0 \\ 1 & x = 0 \end {cases}$
So:
:$\ds \map \Im {\int_0^\infty \dfrac {e^{i x} - 1} x \rd x} = \int_0^\inft... | Dirichlet Integral/Proof 3 | https://proofwiki.org/wiki/Dirichlet_Integral | https://proofwiki.org/wiki/Dirichlet_Integral/Proof_3 | [
"Dirichlet Integral",
"Sine Integral Function",
"Definite Integrals involving Sine Function",
"Integral Calculus"
] | [] | [
"Euler's Formula",
"Definition:Circle/Arc",
"Definition:Radius",
"Definition:Coordinate System/Origin",
"Definition:Anticlockwise",
"Contour Integral of Concatenation of Contours",
"Linear Combination of Contour Integrals",
"Definition:Holomorphic Function",
"Cauchy-Goursat Theorem",
"Jordan's Lem... |
proofwiki-8936 | Dirichlet Integral | :$\ds \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$ | From Integral to Infinity of Function over Argument:
:$\ds \int_0^\infty {\dfrac {\map f x} x} = \int_0^{\to \infty} \map F u \rd u$
for a real function $f$ and its Laplace transform $\laptrans f = F$, provided they exist.
Let $\map f x := \sin x$.
Then from Laplace Transform of Sine:
:$\laptrans {\map f x} = \dfrac 1 ... | :$\ds \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$ | From [[Integral to Infinity of Function over Argument]]:
:$\ds \int_0^\infty {\dfrac {\map f x} x} = \int_0^{\to \infty} \map F u \rd u$
for a [[Definition:Real Function|real function]] $f$ and its [[Definition:Laplace Transform|Laplace transform]] $\laptrans f = F$, provided they exist.
Let $\map f x := \sin x$.
T... | Dirichlet Integral/Proof 4 | https://proofwiki.org/wiki/Dirichlet_Integral | https://proofwiki.org/wiki/Dirichlet_Integral/Proof_4 | [
"Dirichlet Integral",
"Sine Integral Function",
"Definite Integrals involving Sine Function",
"Integral Calculus"
] | [] | [
"Integral to Infinity of Function over Argument",
"Definition:Real Function",
"Definition:Laplace Transform",
"Laplace Transform of Sine",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-8937 | Dirichlet Integral | :$\ds \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$ | Let $M \in \R_{>0}$.
Define a real function $I_M : \R \to \R$ by:
:$\ds \map {I_M} \alpha := \int_0^M \dfrac {\sin x} x e^{-\alpha x} \rd x$
Then, for $\alpha > 0$:
{{begin-eqn}}
{{eqn | l = \size {\map {I_M} \alpha}
| o = \le
| r = \int_0^M \size {\dfrac {\sin x} x e^{-\alpha x} } \rd x
| c = Absolut... | :$\ds \int_0^\infty \frac {\sin x} x \rd x = \frac \pi 2$ | Let $M \in \R_{>0}$.
Define a [[Definition:Real Function|real function]] $I_M : \R \to \R$ by:
:$\ds \map {I_M} \alpha := \int_0^M \dfrac {\sin x} x e^{-\alpha x} \rd x$
Then, for $\alpha > 0$:
{{begin-eqn}}
{{eqn | l = \size {\map {I_M} \alpha}
| o = \le
| r = \int_0^M \size {\dfrac {\sin x} x e^{-\alpha... | Dirichlet Integral/Proof 5 | https://proofwiki.org/wiki/Dirichlet_Integral | https://proofwiki.org/wiki/Dirichlet_Integral/Proof_5 | [
"Dirichlet Integral",
"Sine Integral Function",
"Definite Integrals involving Sine Function",
"Integral Calculus"
] | [] | [
"Definition:Real Function",
"Triangle Inequality for Integrals/Real",
"Sine Inequality",
"Primitive of Exponential of a x",
"Definite Integral of Partial Derivative",
"Primitive of Exponential of a x",
"Primitive of Exponential of a x by Sine of b x",
"Fundamental Theorem of Calculus",
"Linear Combi... |
proofwiki-8938 | Riemann Zeta Function and Prime Counting Function | For $\map \Re s > 1$:
:$\ds \log \map \zeta s = s \int_0^{\mathop \to \infty} \frac {\map \pi x} {x \paren {x^s - 1} } \rd x$
where:
:$\zeta$ denotes the Riemann Zeta Function
:$\pi$ denotes the Prime-Counting Function. | From the definition of the Riemann Zeta Function:
{{begin-eqn}}
{{eqn | l = \map \zeta s
| r = \prod_p \frac 1 {1 - p^{-s} }
| c =
}}
{{eqn | ll= \leadsto
| l = \log \map \zeta s
| r = \log \prod_p \frac 1 {1 - p^{-s} }
| c =
}}
{{eqn | r = \sum_p \map \log {\frac 1 {1 - p^{-s} } }
... | For $\map \Re s > 1$:
:$\ds \log \map \zeta s = s \int_0^{\mathop \to \infty} \frac {\map \pi x} {x \paren {x^s - 1} } \rd x$
where:
:$\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann Zeta Function]]
:$\pi$ denotes the [[Definition:Prime-Counting Function|Prime-Counting Function]]. | From the definition of the [[Definition:Riemann Zeta Function|Riemann Zeta Function]]:
{{begin-eqn}}
{{eqn | l = \map \zeta s
| r = \prod_p \frac 1 {1 - p^{-s} }
| c =
}}
{{eqn | ll= \leadsto
| l = \log \map \zeta s
| r = \log \prod_p \frac 1 {1 - p^{-s} }
| c =
}}
{{eqn | r = \sum_p \... | Riemann Zeta Function and Prime Counting Function | https://proofwiki.org/wiki/Riemann_Zeta_Function_and_Prime_Counting_Function | https://proofwiki.org/wiki/Riemann_Zeta_Function_and_Prime_Counting_Function | [
"Riemann Zeta Function",
"Number Theory"
] | [
"Definition:Riemann Zeta Function",
"Definition:Prime-Counting Function"
] | [
"Definition:Riemann Zeta Function",
"Sum of Logarithms",
"Logarithm of Power",
"Derivative of Logarithm Function",
"Derivative of Composite Function",
"Fundamental Theorem of Calculus/Second Part",
"Category:Riemann Zeta Function",
"Category:Number Theory"
] |
proofwiki-8939 | Faulhaber's Formula | Let $n, p \in \Z_{>0}$ be (strictly) positive integers.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n k^p
| r = 1^p + 2^p + \cdots + n^p
| c =
}}
{{eqn | r = \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}
| c =
}}
{{eqn | r = \frac {n^{p + 1} } {p + ... | Let $x \ge 0$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} e^{k x}
| r = \sum_{k \mathop = 0}^{n - 1} \sum_{p \mathop = 0}^\infty \frac {\paren {k x}^p} {p!}
| c = Power Series Expansion for Exponential Function
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \paren {\sum_{k \mathop = 0}^{n - 1} k^p}... | Let $n, p \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n k^p
| r = 1^p + 2^p + \cdots + n^p
| c =
}}
{{eqn | r = \frac 1 {p + 1} \sum_{i \mathop = 0}^p \paren {-1}^i \binom {p + 1} i B_i n^{p + 1 - i}
| ... | Let $x \ge 0$.
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{n - 1} e^{k x}
| r = \sum_{k \mathop = 0}^{n - 1} \sum_{p \mathop = 0}^\infty \frac {\paren {k x}^p} {p!}
| c = [[Power Series Expansion for Exponential Function]]
}}
{{eqn | r = \sum_{p \mathop = 0}^\infty \paren {\sum_{k \mathop = 0}^{n - 1}... | Faulhaber's Formula | https://proofwiki.org/wiki/Faulhaber's_Formula | https://proofwiki.org/wiki/Faulhaber's_Formula | [
"Faulhaber's Formula",
"Bernoulli Numbers",
"Sums of Sequences",
"Number Theory"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Bernoulli Numbers"
] | [
"Power Series Expansion for Exponential Function",
"Tonelli's Theorem",
"Sum of Geometric Sequence",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Power Series Expansion for Exponential Function",
"Factorial/Examples/0",
"Definition:Power (Algebra)/Integer",
"Translation of In... |
proofwiki-8940 | Complex Exponential Tends to Zero | Let $\exp z$ be the complex exponential function.
Then:
:$\ds \lim_{\map \Re z \mathop \to +\infty} e^{-z} = 0$
where $\map \Re z$ denotes the real part of $z$. | Let $z = x + iy$.
Let $\epsilon > 0$.
By the definition of limits at infinity, we need to show that there is some $M > 0$ such that:
:$x > M \implies \size {e^{-z} - 0} < \epsilon$
But:
{{begin-eqn}}
{{eqn | l = \size {e^{-z} - 0}
| r = \size {e^{-z} }
}}
{{eqn | r = \size {e^{-x} }
| c = Modulus of Exponen... | Let $\exp z$ be the [[Definition:Complex Exponential Function|complex exponential function]].
Then:
:$\ds \lim_{\map \Re z \mathop \to +\infty} e^{-z} = 0$
where $\map \Re z$ denotes the [[Definition:Real Part|real part]] of $z$. | Let $z = x + iy$.
Let $\epsilon > 0$.
By the definition of [[Definition:Limit at Infinity|limits at infinity]], we need to show that there is some $M > 0$ such that:
:$x > M \implies \size {e^{-z} - 0} < \epsilon$
But:
{{begin-eqn}}
{{eqn | l = \size {e^{-z} - 0}
| r = \size {e^{-z} }
}}
{{eqn | r = \size {e... | Complex Exponential Tends to Zero | https://proofwiki.org/wiki/Complex_Exponential_Tends_to_Zero | https://proofwiki.org/wiki/Complex_Exponential_Tends_to_Zero | [
"Exponential Function"
] | [
"Definition:Exponential Function/Complex",
"Definition:Complex Number/Real Part"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Positive",
"Modulus of Exponential is Exponential of Real Part",
"Definition:Limit of Real Function/Limit at Infinity/Positive",
"Definition:Exponential Function/Real",
"Exponential Tends to Zero and Infinity",
"Category:Exponential Function"
] |
proofwiki-8941 | Modulus of Exponential is Exponential of Real Part | Let $z \in \C$ be a complex number.
Let $\exp z$ denote the complex exponential function.
Let $\cmod {\, \cdot \,}$ denote the complex modulus
Then:
:$\cmod {\exp z} = \map \exp {\map \Re z}$
where $\map \Re z$ denotes the real part of $z$. | Let $z = x + iy$.
{{begin-eqn}}
{{eqn | l = \cmod {\exp z}
| r = \cmod {\map \exp {x + iy} }
}}
{{eqn | r = \cmod {\paren {\exp x} \paren {\exp i y} }
| c = Exponential of Sum
}}
{{eqn | r = \cmod {\exp x} \cmod {\exp i y}
| c = Modulus of Product
}}
{{eqn | r = \cmod {\exp x}
| c = Modulus of E... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\exp z$ denote the [[Definition:Complex Exponential Function|complex exponential function]].
Let $\cmod {\, \cdot \,}$ denote the [[Definition:Complex Modulus|complex modulus]]
Then:
:$\cmod {\exp z} = \map \exp {\map \Re z}$
where $\map \Re ... | Let $z = x + iy$.
{{begin-eqn}}
{{eqn | l = \cmod {\exp z}
| r = \cmod {\map \exp {x + iy} }
}}
{{eqn | r = \cmod {\paren {\exp x} \paren {\exp i y} }
| c = [[Exponential of Sum/Complex Numbers|Exponential of Sum]]
}}
{{eqn | r = \cmod {\exp x} \cmod {\exp i y}
| c = [[Complex Modulus of Product of C... | Modulus of Exponential is Exponential of Real Part | https://proofwiki.org/wiki/Modulus_of_Exponential_is_Exponential_of_Real_Part | https://proofwiki.org/wiki/Modulus_of_Exponential_is_Exponential_of_Real_Part | [
"Complex Modulus",
"Exponential Function"
] | [
"Definition:Complex Number",
"Definition:Exponential Function/Complex",
"Definition:Complex Modulus",
"Definition:Complex Number/Real Part"
] | [
"Exponential of Sum/Complex Numbers",
"Complex Modulus of Product of Complex Numbers",
"Modulus of Exponential of Imaginary Number is One",
"Exponential of Real Number is Strictly Positive"
] |
proofwiki-8942 | Laplace Transform of Exponential/Real Argument | Let $\laptrans f$ denote the Laplace transform of a function $f$.
Let $e^x$ be the real exponential.
Then:
:$\map {\laptrans {e^{a t} } } s = \dfrac 1 {s - a}$
where $a \in \R$ is constant, and $\map \Re s > \map \Re a$. | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{a t} } } s
| r = \int_0^{\to +\infty} e^{-s t} e^{a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} e^{\paren {a - s} t} \rd t
| c = Exponential of Sum
}}
{{eqn | r = \lim_{L \mathop \to \infty} \int_0^L e^{\paren {a - s} t} ... | Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Function|function]] $f$.
Let $e^x$ be the [[Definition:Real Exponential Function|real exponential]].
Then:
:$\map {\laptrans {e^{a t} } } s = \dfrac 1 {s - a}$
where $a \in \R$ is [[Definition:Constant|constant]], and $\... | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{a t} } } s
| r = \int_0^{\to +\infty} e^{-s t} e^{a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to +\infty} e^{\paren {a - s} t} \rd t
| c = [[Exponential of Sum]]
}}
{{eqn | r = \lim_{L \mathop \to \infty} \int_0^L e^{\paren {a - s}... | Laplace Transform of Exponential/Real Argument/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential/Real_Argument | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential/Real_Argument/Proof_1 | [
"Laplace Transform of Exponential"
] | [
"Definition:Laplace Transform",
"Definition:Function",
"Definition:Exponential Function/Real",
"Definition:Constant"
] | [
"Exponential of Sum",
"Primitive of Exponential Function",
"Integration by Substitution",
"Exponential of Zero",
"Exponential Tends to Zero and Infinity"
] |
proofwiki-8943 | Laplace Transform of Exponential/Real Argument | Let $\laptrans f$ denote the Laplace transform of a function $f$.
Let $e^x$ be the real exponential.
Then:
:$\map {\laptrans {e^{a t} } } s = \dfrac 1 {s - a}$
where $a \in \R$ is constant, and $\map \Re s > \map \Re a$. | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{a t} } } s
| r = \map {\laptrans {1 \times e^{a t} } } s
| c =
}}
{{eqn | r = \map {\laptrans 1} {s - a}
| c = First Translation Property of Laplace Transforms
}}
{{eqn | r = \frac 1 {s - a}
| c = Laplace Transform of Constant Mapping
}}
{{end-eqn}}... | Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Function|function]] $f$.
Let $e^x$ be the [[Definition:Real Exponential Function|real exponential]].
Then:
:$\map {\laptrans {e^{a t} } } s = \dfrac 1 {s - a}$
where $a \in \R$ is [[Definition:Constant|constant]], and $\... | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{a t} } } s
| r = \map {\laptrans {1 \times e^{a t} } } s
| c =
}}
{{eqn | r = \map {\laptrans 1} {s - a}
| c = [[First Translation Property of Laplace Transforms]]
}}
{{eqn | r = \frac 1 {s - a}
| c = [[Laplace Transform of Constant Mapping]]
}}
{{e... | Laplace Transform of Exponential/Real Argument/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential/Real_Argument | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential/Real_Argument/Proof_2 | [
"Laplace Transform of Exponential"
] | [
"Definition:Laplace Transform",
"Definition:Function",
"Definition:Exponential Function/Real",
"Definition:Constant"
] | [
"First Translation Property of Laplace Transforms",
"Laplace Transform of Constant Mapping"
] |
proofwiki-8944 | Laplace Transform of Cosine | Let $\cos$ be the real cosine function.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | Let $\map f t := \map \Ci t = \ds \int_t^\infty \dfrac {\cos u} u \rd u$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = -\dfrac {\cos t} t
| c =
}}
{{eqn | ll= \leadsto
| l = t \map {f'} t
| r = -\cos t
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {t \map {f'} t}
| r = -... | Let $\cos$ be the [[Definition:Real Cosine Function|real cosine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|cons... | Let $\map f t := \map \Ci t = \ds \int_t^\infty \dfrac {\cos u} u \rd u$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = -\dfrac {\cos t} t
| c =
}}
{{eqn | ll= \leadsto
| l = t \map {f'} t
| r = -\cos t
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {t \map {f'} t}
| r = ... | Laplace Transform of Cosine Integral Function/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine_Integral_Function/Proof_1 | [
"Laplace Transform of Cosine",
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Cosine Function"
] | [
"Definition:Cosine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Laplace Transform of Cosine",
"Derivative of Laplace Transform",
"Laplace Transform of Derivative",
"Primitive of x over x squared plus a squared",
"Initial Value Theorem of Laplace Transform"
] |
proofwiki-8945 | Laplace Transform of Cosine | Let $\cos$ be the real cosine function.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | Let $\map f t = \sin \sqrt t$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = \dfrac {\cos \sqrt t} {2 \sqrt t}
| c =
}}
{{eqn | l = \map f 0
| r = 0
| c =
}}
{{end-eqn}}
So:
{{begin-eqn}}
{{eqn | l = \laptrans {\map {f'} t}
| r = \dfrac 1 2 \laptrans {\dfrac {\cos \sqrt t} {\sqrt t} ... | Let $\cos$ be the [[Definition:Real Cosine Function|real cosine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|cons... | Let $\map f t = \sin \sqrt t$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = \dfrac {\cos \sqrt t} {2 \sqrt t}
| c =
}}
{{eqn | l = \map f 0
| r = 0
| c =
}}
{{end-eqn}}
So:
{{begin-eqn}}
{{eqn | l = \laptrans {\map {f'} t}
| r = \dfrac 1 2 \laptrans {\dfrac {\cos \sqrt t} {\sqrt... | Laplace Transform of Cosine of Root over Root/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine_of_Root_over_Root/Proof_1 | [
"Laplace Transform of Cosine",
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Cosine Function"
] | [
"Definition:Cosine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Laplace Transform of Derivative",
"Laplace Transform of Sine of Root"
] |
proofwiki-8946 | Laplace Transform of Cosine | Let $\cos$ be the real cosine function.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\cos {a t} } } s
| r = \int_0^{\to +\infty} e^{-s t} \cos {a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} \int_0^L e^{-s t} \cos {a t} \rd t
| c = {{Defof|Improper Integral}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} ... | Let $\cos$ be the [[Definition:Real Cosine Function|real cosine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|cons... | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\cos {a t} } } s
| r = \int_0^{\to +\infty} e^{-s t} \cos {a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} \int_0^L e^{-s t} \cos {a t} \rd t
| c = {{Defof|Improper Integral}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} ... | Laplace Transform of Cosine/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine/Proof_1 | [
"Laplace Transform of Cosine",
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Cosine Function"
] | [
"Definition:Cosine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Primitive of Exponential of a x by Cosine of b x",
"Exponential Tends to Zero and Infinity",
"Sine of Zero is Zero",
"Cosine of Zero is One"
] |
proofwiki-8947 | Laplace Transform of Cosine | Let $\cos$ be the real cosine function.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | {{begin-eqn}}
{{eqn | l = \laptrans {e^{i a t} }
| r = \frac 1 {s - i a}
| c = Laplace Transform of Exponential
}}
{{eqn | r = \frac {s + i a} {s^2 + a^2}
| c = multiply top and bottom by $s + i a$
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = \laptrans {e^{i a t} }
| r = \laptrans {\cos a t +... | Let $\cos$ be the [[Definition:Real Cosine Function|real cosine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|cons... | {{begin-eqn}}
{{eqn | l = \laptrans {e^{i a t} }
| r = \frac 1 {s - i a}
| c = [[Laplace Transform of Exponential]]
}}
{{eqn | r = \frac {s + i a} {s^2 + a^2}
| c = multiply top and bottom by $s + i a$
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = \laptrans {e^{i a t} }
| r = \laptrans {\cos... | Laplace Transform of Cosine/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine/Proof_2 | [
"Laplace Transform of Cosine",
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Cosine Function"
] | [
"Definition:Cosine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Laplace Transform of Exponential",
"Euler's Formula",
"Linear Combination of Laplace Transforms"
] |
proofwiki-8948 | Laplace Transform of Cosine | Let $\cos$ be the real cosine function.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | {{begin-eqn}}
{{eqn | l = \laptrans {\cos a t}
| r = \laptrans {\frac {e^{i a t} + e^{-i a t} } 2}
| c = Euler's Cosine Identity
}}
{{eqn | r = \frac 1 2 \paren {\laptrans {e^{i a t} } + \laptrans {e^{-i a t} } }
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \frac 1 2 \paren {\frac 1 {... | Let $\cos$ be the [[Definition:Real Cosine Function|real cosine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|cons... | {{begin-eqn}}
{{eqn | l = \laptrans {\cos a t}
| r = \laptrans {\frac {e^{i a t} + e^{-i a t} } 2}
| c = [[Euler's Cosine Identity]]
}}
{{eqn | r = \frac 1 2 \paren {\laptrans {e^{i a t} } + \laptrans {e^{-i a t} } }
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \frac 1 2 \paren {\... | Laplace Transform of Cosine/Proof 3 | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine/Proof_3 | [
"Laplace Transform of Cosine",
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Cosine Function"
] | [
"Definition:Cosine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Euler's Cosine Identity",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential"
] |
proofwiki-8949 | Laplace Transform of Cosine | Let $\cos$ be the real cosine function.
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > a$. | By definition of the Laplace Transform:
:$\ds \laptrans {\cos at} = \int_0^{\to +\infty} e^{-s t} \cos at \rd t$
From Integration by Parts:
:$\ds \int f g' \rd t = f g - \int f'g \rd t$
Here:
{{begin-eqn}}
{{eqn | l = f
| r = \cos at
}}
{{eqn | ll= \leadsto
| l = f'
| r = -a \sin a t
| c = Deriv... | Let $\cos$ be the [[Definition:Real Cosine Function|real cosine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\cos a t} = \dfrac s {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|cons... | By definition of the [[Definition:Laplace Transform|Laplace Transform]]:
:$\ds \laptrans {\cos at} = \int_0^{\to +\infty} e^{-s t} \cos at \rd t$
From [[Integration by Parts]]:
:$\ds \int f g' \rd t = f g - \int f'g \rd t$
Here:
{{begin-eqn}}
{{eqn | l = f
| r = \cos at
}}
{{eqn | ll= \leadsto
| l = f... | Laplace Transform of Cosine/Proof 4 | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine/Proof_4 | [
"Laplace Transform of Cosine",
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Cosine Function"
] | [
"Definition:Cosine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Definition:Laplace Transform",
"Integration by Parts",
"Derivative of Cosine Function",
"Primitive of Exponential Function",
"Integration by Parts",
"Derivative of Sine Function",
"Primitive of Exponential Function",
"Boundedness of Real Sine and Cosine",
"Complex Exponential Tends to Zero"
] |
proofwiki-8950 | Laplace Transform of Sine | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$. | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = \dfrac {\sin t} t
| c =
}}
{{eqn | ll= \leadsto
| l = t \map {f'} t
| r = \sin t
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {t \map {f'} t}... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|constan... | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = \dfrac {\sin t} t
| c =
}}
{{eqn | ll= \leadsto
| l = t \map {f'} t
| r = \sin t
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {t \map {f'}... | Laplace Transform of Sine Integral Function/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_Integral_Function/Proof_1 | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Laplace Transform of Sine",
"Derivative of Laplace Transform",
"Laplace Transform of Derivative",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Initial Value Theorem of Laplace Transform",
"Sum of Arctangent and Arccotangent",
"Arctangent of Reciprocal equals Arccotangent"
] |
proofwiki-8951 | Laplace Transform of Sine | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$. | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map \Si t
| r = \int_0^t \dfrac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = \int_0^t \dfrac 1 u \paren {u - \dfrac {u^3} {3!} + \dfrac {u^5} {5!} - \dfrac {u^... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|constan... | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map \Si t
| r = \int_0^t \dfrac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = \int_0^t \dfrac 1 u \paren {u - \dfrac {u^3} {3!} + \dfrac {u^5} {5!} - \dfrac ... | Laplace Transform of Sine Integral Function/Proof 3 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_Integral_Function/Proof_3 | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Primitive of Power",
"Laplace Transform of Positive Integer Power",
"Power Series Expansion for Real Arctangent Function"
] |
proofwiki-8952 | Laplace Transform of Sine | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$. | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map \Si t
| r = \int_0^t \dfrac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = \int_0^1 \dfrac {\sin t v} v \rd v
| c = Integration by Substitution $u = t ... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|constan... | Let $\map f t := \map \Si t = \ds \int_0^t \dfrac {\sin u} u \rd u$.
Then:
:$\map f 0 = 0$
and:
{{begin-eqn}}
{{eqn | l = \map \Si t
| r = \int_0^t \dfrac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = \int_0^1 \dfrac {\sin t v} v \rd v
| c = [[Integration by Substitution]] ... | Laplace Transform of Sine Integral Function/Proof 4 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_Integral_Function/Proof_4 | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Integration by Substitution",
"Laplace Transform of Sine",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-8953 | Laplace Transform of Sine | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \sin \sqrt t
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {\sqrt t}^{2 n + 1} } {\paren {2 n + 1}!}
| c = {{Defof|Real Sine Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}!} t^{n + \frac 1 2}
| c =
}}
{{eqn | ll=... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|constan... | {{begin-eqn}}
{{eqn | l = \sin \sqrt t
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {\sqrt t}^{2 n + 1} } {\paren {2 n + 1}!}
| c = {{Defof|Real Sine Function}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}!} t^{n + \frac 1 2}
| c =
}}
{{eqn | ll=... | Laplace Transform of Sine of Root/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_of_Root/Proof_1 | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Laplace Transform of Power",
"Linear Combination of Laplace Transforms",
"Gamma Difference Equation",
"Gamma Function of Positive Half-Integer"
] |
proofwiki-8954 | Laplace Transform of Sine | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$. | Let $\map y t := \sin \sqrt t$.
Differentiating twice {{WRT|Differentiation}} $t$, we get:
:$(1): \quad 4 t y' ' + 2 y' ' + y = 0$
Let $\map Y s = \laptrans {\map t y}$ be the Laplace transform of $y$.
Then taking the Laplace transform of $(1)$:
{{begin-eqn}}
{{eqn | l = -4 \map {\dfrac \d {\d s} } {\laptrans {\map {y'... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|constan... | Let $\map y t := \sin \sqrt t$.
[[Definition:Differentiation|Differentiating]] twice {{WRT|Differentiation}} $t$, we get:
:$(1): \quad 4 t y' ' + 2 y' ' + y = 0$
Let $\map Y s = \laptrans {\map t y}$ be the [[Definition:Laplace Transform|Laplace transform]] of $y$.
Then taking the [[Definition:Laplace Transform|Lap... | Laplace Transform of Sine of Root/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_of_Root/Proof_2 | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Definition:Differentiation",
"Definition:Laplace Transform",
"Definition:Laplace Transform",
"Derivative of Laplace Transform",
"Laplace Transform of Derivative",
"Laplace Transform of Second Derivative"
] |
proofwiki-8955 | Laplace Transform of Sine | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\sin {a t} } } s
| r = \int_0^{\to +\infty} e^{-s t} \sin {a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} \int_0^L e^{-s t} \sin {a t} \rd t
| c = {{Defof|Improper Integral}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} ... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|constan... | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\sin {a t} } } s
| r = \int_0^{\to +\infty} e^{-s t} \sin {a t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} \int_0^L e^{-s t} \sin {a t} \rd t
| c = {{Defof|Improper Integral}}
}}
{{eqn | r = \lim_{L \mathop \to \infty} ... | Laplace Transform of Sine/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Sine/Proof_1 | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Primitive of Exponential of a x by Sine of b x",
"Exponential of Zero",
"Exponential Tends to Zero and Infinity",
"Sine of Zero is Zero",
"Cosine of Zero is One"
] |
proofwiki-8956 | Laplace Transform of Sine | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \laptrans {e^{i a t} }
| r = \frac 1 {s - i a}
| c = Laplace Transform of Exponential
}}
{{eqn | r = \frac {s + i a} {s^2 + a^2}
| c = multiplying top and bottom by $s + i a$
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn | l = \laptrans {e^{i a t} }
| r = \laptrans {\cos a ... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|constan... | {{begin-eqn}}
{{eqn | l = \laptrans {e^{i a t} }
| r = \frac 1 {s - i a}
| c = [[Laplace Transform of Exponential]]
}}
{{eqn | r = \frac {s + i a} {s^2 + a^2}
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $s + i a$
}}
{{end-eqn}}
Also:
{{begin-eqn}}
{{eqn |... | Laplace Transform of Sine/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Sine/Proof_2 | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Laplace Transform of Exponential",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Euler's Formula",
"Linear Combination of Laplace Transforms"
] |
proofwiki-8957 | Laplace Transform of Sine | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$. | {{begin-eqn}}
{{eqn | l = \laptrans {\sin a t}
| r = \laptrans {\frac {e^{i a t} - e^{-i a t} } {2 i} }
| c = Euler's Sine Identity
}}
{{eqn | r = \frac 1 {2 i} \paren {\laptrans {e^{i a t} } - \laptrans {e^{-i a t} } }
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \frac 1 {2 i} \paren... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|constan... | {{begin-eqn}}
{{eqn | l = \laptrans {\sin a t}
| r = \laptrans {\frac {e^{i a t} - e^{-i a t} } {2 i} }
| c = [[Euler's Sine Identity]]
}}
{{eqn | r = \frac 1 {2 i} \paren {\laptrans {e^{i a t} } - \laptrans {e^{-i a t} } }
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \frac 1 {2 i... | Laplace Transform of Sine/Proof 3 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Sine/Proof_3 | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Euler's Sine Identity",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential"
] |
proofwiki-8958 | Laplace Transform of Sine | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$. | By definition of the Laplace transform:
:$\ds \laptrans {\sin a t} = \int_0^{\to +\infty} e^{-s t} \sin a t \rd t$
From Integration by Parts:
:$\ds \int f g' \rd t = f g - \int f' g \rd t$
Here:
{{begin-eqn}}
{{eqn | l = f
| r = \sin a t
}}
{{eqn | ll= \leadsto
| l = f'
| r = a \cos a t
| c = De... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|constan... | By definition of the [[Definition:Laplace Transform|Laplace transform]]:
:$\ds \laptrans {\sin a t} = \int_0^{\to +\infty} e^{-s t} \sin a t \rd t$
From [[Integration by Parts]]:
:$\ds \int f g' \rd t = f g - \int f' g \rd t$
Here:
{{begin-eqn}}
{{eqn | l = f
| r = \sin a t
}}
{{eqn | ll= \leadsto
| l... | Laplace Transform of Sine/Proof 4 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Sine/Proof_4 | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Definition:Laplace Transform",
"Integration by Parts",
"Derivative of Sine Function/Corollary",
"Primitive of Exponential of a x",
"Integration by Parts",
"Derivative of Cosine Function",
"Primitive of Exponential Function",
"Boundedness of Real Sine and Cosine",
"Complex Exponential Tends to Zero"... |
proofwiki-8959 | Laplace Transform of Sine | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is constant, and $\map \Re s > 0$. | From Laplace Transform of Second Derivative:
:$(1): \quad \laptrans {\map {f' '} t} = s^2 \laptrans {\map f t} - s \, \map f 0 - \map {f'} 0$
under suitable conditions.
Then:
{{begin-eqn}}
{{eqn | l = \map f t
| r = \sin a t
| c =
}}
{{eqn | ll= \leadsto
| l = \map {f'} t
| r = a \cos a t
... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\sin at} = \dfrac a {s^2 + a^2}$
where $a \in \R_{>0}$ is [[Definition:Constant|constan... | From [[Laplace Transform of Second Derivative]]:
:$(1): \quad \laptrans {\map {f' '} t} = s^2 \laptrans {\map f t} - s \, \map f 0 - \map {f'} 0$
under suitable conditions.
Then:
{{begin-eqn}}
{{eqn | l = \map f t
| r = \sin a t
| c =
}}
{{eqn | ll= \leadsto
| l = \map {f'} t
| r = a \cos a... | Laplace Transform of Sine/Proof 5 | https://proofwiki.org/wiki/Laplace_Transform_of_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Sine/Proof_5 | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Trigonometric Functions",
"Examples of Laplace Transforms",
"Sine Function"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Constant"
] | [
"Laplace Transform of Second Derivative"
] |
proofwiki-8960 | Sum Rule for Derivatives/General Result | Let $\map {f_1} x, \map {f_2} x, \ldots, \map {f_n} x$ be real functions all differentiable.
Then for all $n \in \N_{>0}$:
:$\ds \map {D_x} {\sum_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \map {D_x} {\map {f_i} x}$ | The proof proceeds by induction.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \map {D_x} {\sum_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \map {D_x} {\map {f_i} x}$
$\map P 1$ is true, as this just says:
:$\map {D_x} {\map {f_1} x} = \map {D_x} {\map {f_1} x}$
which is trivially true... | Let $\map {f_1} x, \map {f_2} x, \ldots, \map {f_n} x$ be [[Definition:Real Function|real functions]] all [[Definition:Differentiable on Interval|differentiable]].
Then for all $n \in \N_{>0}$:
:$\ds \map {D_x} {\sum_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \map {D_x} {\map {f_i} x}$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \map {D_x} {\sum_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \map {D_x} {\map {f_i} x}$
$\map P 1$ is true, as this just says:
:$\map {D_... | Sum Rule for Derivatives/General Result | https://proofwiki.org/wiki/Sum_Rule_for_Derivatives/General_Result | https://proofwiki.org/wiki/Sum_Rule_for_Derivatives/General_Result | [
"Differential Calculus",
"Sum Rule for Derivatives"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-8961 | Product Rule for Derivatives/General Result | Let $\map {f_1} x, \map {f_2} x, \ldots, \map {f_n} x$ be real functions differentiable on the open interval $I$.
then:
:$\forall x \in I: \ds \map {D_x} {\prod_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \paren {\map {D_x} {\map {f_i} x} \prod_{j \mathop \ne i} \map {f_j} x}$ | Proof by Principle of Mathematical Induction:
For all $n \in \N_{\ge 1}$, let $\map P n$ be the proposition:
:$\ds \map {D_x} {\prod_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \paren {\map {D_x} {\map {f_i} x} \prod_{j \mathop \ne i} \map {f_j} x}$
$\map P 1$ is true, as this just says:
:$\map {D_x} {\map... | Let $\map {f_1} x, \map {f_2} x, \ldots, \map {f_n} x$ be [[Definition:Real Function|real functions]] [[Definition:Differentiable on Interval|differentiable]] on the [[Definition:Open Real Interval|open interval]] $I$.
then:
:$\forall x \in I: \ds \map {D_x} {\prod_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = ... | Proof by [[Principle of Mathematical Induction]]:
For all $n \in \N_{\ge 1}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \map {D_x} {\prod_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \paren {\map {D_x} {\map {f_i} x} \prod_{j \mathop \ne i} \map {f_j} x}$
$\map P 1$ is true, as ... | Product Rule for Derivatives/General Result | https://proofwiki.org/wiki/Product_Rule_for_Derivatives/General_Result | https://proofwiki.org/wiki/Product_Rule_for_Derivatives/General_Result | [
"Product Rule for Derivatives"
] | [
"Definition:Real Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Open"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-8962 | Derivative of Arccotangent Function | :$\dfrac {\map \d {\arccot x} } {\d x} = \dfrac {-1} {1 + x^2}$ | {{begin-eqn}}
{{eqn | l = y
| r = \arccot x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \cot y
| c = {{Defof|Real Arccotangent}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = -\csc^2 y
| c = Derivative of Cotangent Function
}}
{{eqn | r = -\paren {1 + \cot^2 y}... | :$\dfrac {\map \d {\arccot x} } {\d x} = \dfrac {-1} {1 + x^2}$ | {{begin-eqn}}
{{eqn | l = y
| r = \arccot x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \cot y
| c = {{Defof|Real Arccotangent}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = -\csc^2 y
| c = [[Derivative of Cotangent Function]]
}}
{{eqn | r = -\paren {1 + \cot^... | Derivative of Arccotangent Function/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Arccotangent_Function | https://proofwiki.org/wiki/Derivative_of_Arccotangent_Function/Proof_1 | [
"Derivatives of Inverse Trigonometric Functions",
"Arccotangent Function",
"Derivative of Arccotangent Function"
] | [] | [
"Derivative of Cotangent Function",
"Sum of Squares of Sine and Cosine/Corollary 2",
"Derivative of Inverse Function"
] |
proofwiki-8963 | Derivative of Arccotangent Function | :$\dfrac {\map \d {\arccot x} } {\d x} = \dfrac {-1} {1 + x^2}$ | {{begin-eqn}}
{{eqn | l = \frac {\map \d {\arccot x} } {\d x}
| r = \map {\frac \d {\d x} } {\frac \pi 2 - \arctan x}
| c = Tangent of Complement equals Cotangent
}}
{{eqn | r = -\frac 1 {1 + x^2}
| c = Derivative of Arctangent Function
}}
{{end-eqn}}
{{qed}} | :$\dfrac {\map \d {\arccot x} } {\d x} = \dfrac {-1} {1 + x^2}$ | {{begin-eqn}}
{{eqn | l = \frac {\map \d {\arccot x} } {\d x}
| r = \map {\frac \d {\d x} } {\frac \pi 2 - \arctan x}
| c = [[Tangent of Complement equals Cotangent]]
}}
{{eqn | r = -\frac 1 {1 + x^2}
| c = [[Derivative of Arctangent Function]]
}}
{{end-eqn}}
{{qed}} | Derivative of Arccotangent Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Arccotangent_Function | https://proofwiki.org/wiki/Derivative_of_Arccotangent_Function/Proof_2 | [
"Derivatives of Inverse Trigonometric Functions",
"Arccotangent Function",
"Derivative of Arccotangent Function"
] | [] | [
"Tangent of Complement equals Cotangent",
"Derivative of Arctangent Function"
] |
proofwiki-8964 | Derivative of Arcsecant Function/Corollary 2 | :$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {x^2 \sqrt {1 - \frac 1 {x^2} } }$ | From Derivative of Arcsecant Function:
:$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\size x \sqrt {x^2 - 1} }$
Since for all $x \in \R$, we have $\size x = \sqrt{x^2}$, we can write:
:$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\sqrt {x^2} \sqrt {x^2 - 1} }$
Multiplying the denominator by $1 = \dfrac {\sqrt... | :$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {x^2 \sqrt {1 - \frac 1 {x^2} } }$ | From [[Derivative of Arcsecant Function]]:
:$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\size x \sqrt {x^2 - 1} }$
Since for all $x \in \R$, we have $\size x = \sqrt{x^2}$, we can write:
:$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\sqrt {x^2} \sqrt {x^2 - 1} }$
Multiplying the denominator by $1 = \dfr... | Derivative of Arcsecant Function/Corollary 2 | https://proofwiki.org/wiki/Derivative_of_Arcsecant_Function/Corollary_2 | https://proofwiki.org/wiki/Derivative_of_Arcsecant_Function/Corollary_2 | [
"Derivative of Arcsecant Function"
] | [] | [
"Derivative of Arcsecant Function"
] |
proofwiki-8965 | Asymptotic Formula for Bernoulli Numbers | The Bernoulli numbers with even index can be approximated by the asymptotic formula:
:$B_{2 n} \sim \paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\dfrac n {\pi e} }^{2 n}$
where:
:$B_n$ denotes the $n$th Bernoulli number
:$\sim$ denotes asymptotically equal. | {{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \frac {B_{2 n} } {\paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\frac n {\pi e} }^{2 n} }
| r = \lim_{n \mathop \to \infty} \frac {\paren {-1}^{n + 1} \paren {2 n}! \map \zeta {2 n} } {2^{2 n - 1} \pi^{2 n} \paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\frac n {\... | The [[Definition:Bernoulli Numbers|Bernoulli numbers]] with [[Definition:Even Integer|even]] index can be approximated by the asymptotic formula:
:$B_{2 n} \sim \paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\dfrac n {\pi e} }^{2 n}$
where:
:$B_n$ denotes the $n$th [[Definition:Bernoulli Numbers|Bernoulli number]]
:$\si... | {{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \frac {B_{2 n} } {\paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\frac n {\pi e} }^{2 n} }
| r = \lim_{n \mathop \to \infty} \frac {\paren {-1}^{n + 1} \paren {2 n}! \map \zeta {2 n} } {2^{2 n - 1} \pi^{2 n} \paren {-1}^{n + 1} 4 \sqrt {\pi n} \paren {\frac n {\... | Asymptotic Formula for Bernoulli Numbers | https://proofwiki.org/wiki/Asymptotic_Formula_for_Bernoulli_Numbers | https://proofwiki.org/wiki/Asymptotic_Formula_for_Bernoulli_Numbers | [
"Asymptotic Formula for Bernoulli Numbers",
"Bernoulli Numbers",
"Asymptotics"
] | [
"Definition:Bernoulli Numbers",
"Definition:Even Integer",
"Definition:Bernoulli Numbers",
"Definition:Asymptotic Equality"
] | [
"Riemann Zeta Function at Even Integers",
"Stirling's Formula"
] |
proofwiki-8966 | Derivative of Hyperbolic Cotangent Function | :$\map {\dfrac \d {\d x} } {\coth u} = -\csch^2 u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\coth u}
| r = \map {\frac \d {\d u} } {\coth u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = -\csch^2 u \frac {\d u} {\d x}
| c = Derivative of Hyperbolic Cotangent
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\coth u} = -\csch^2 u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\coth u}
| r = \map {\frac \d {\d u} } {\coth u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = -\csch^2 u \frac {\d u} {\d x}
| c = [[Derivative of Hyperbolic Cotangent]]
}}
{{end-eqn}}
{{qed}} | Derivative of Hyperbolic Cotangent Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cotangent_Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cotangent_Function | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Cotangent Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Hyperbolic Cotangent"
] |
proofwiki-8967 | Derivative of Hyperbolic Cosecant Function | :$\map {\dfrac \d {\d x} } {\csch u} = -\csch u \coth u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\csch u}
| r = \map {\frac \d {\d u} } {\csch u} \frac {\d u} {\d x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = -\csch u \coth u \frac {\d u} {\d x}
| c = Derivative of Hyperbolic Cotangent
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\csch u} = -\csch u \coth u \dfrac {\d u} {\d x}$ | {{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\csch u}
| r = \map {\frac \d {\d u} } {\csch u} \frac {\d u} {\d x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = -\csch u \coth u \frac {\d u} {\d x}
| c = [[Derivative of Hyperbolic Cotangent]]
}}
{{end-eqn}}
{{qed}} | Derivative of Hyperbolic Cosecant Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosecant_Function | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cosecant_Function | [
"Derivatives of Hyperbolic Functions",
"Hyperbolic Cosecant Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Hyperbolic Cotangent"
] |
proofwiki-8968 | Derivative of Inverse Hyperbolic Sine | :$\map {\dfrac \d {\d x} } {\arsinh x} = \dfrac 1 {\sqrt {x^2 + 1} }$ | {{begin-eqn}}
{{eqn | l = y
| r = \arsinh x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \sinh y
| c = {{Defof|Real Inverse Hyperbolic Sine}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = \cosh y
| c = Derivative of Hyperbolic Sine
}}
{{eqn | ll= \leadsto
... | :$\map {\dfrac \d {\d x} } {\arsinh x} = \dfrac 1 {\sqrt {x^2 + 1} }$ | {{begin-eqn}}
{{eqn | l = y
| r = \arsinh x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \sinh y
| c = {{Defof|Real Inverse Hyperbolic Sine}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = \cosh y
| c = [[Derivative of Hyperbolic Sine]]
}}
{{eqn | ll= \leadsto
... | Derivative of Inverse Hyperbolic Sine | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Sine | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Sine | [
"Derivatives of Inverse Hyperbolic Functions",
"Inverse Hyperbolic Sine",
"Derivative of Inverse Hyperbolic Sine"
] | [] | [
"Derivative of Hyperbolic Sine",
"Derivative of Inverse Function",
"Difference of Squares of Hyperbolic Cosine and Sine"
] |
proofwiki-8969 | Derivative of Real Area Hyperbolic Cosine | :$\map {\dfrac \d {\d x} } {\arcosh x} = \dfrac 1 {\sqrt {x^2 - 1} }$ | {{begin-eqn}}
{{eqn | l = y
| r = \arcosh x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \cosh y
| c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = \sinh y
| c = Derivative of Hyperbolic Cosine
}}
{{eqn | ll= \leadsto
... | :$\map {\dfrac \d {\d x} } {\arcosh x} = \dfrac 1 {\sqrt {x^2 - 1} }$ | {{begin-eqn}}
{{eqn | l = y
| r = \arcosh x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \cosh y
| c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = \sinh y
| c = [[Derivative of Hyperbolic Cosine]]
}}
{{eqn | ll= \leadsto
... | Derivative of Real Area Hyperbolic Cosine | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosine | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosine | [
"Derivatives of Inverse Hyperbolic Functions",
"Inverse Hyperbolic Cosine",
"Derivative of Real Area Hyperbolic Cosine"
] | [] | [
"Derivative of Hyperbolic Cosine",
"Derivative of Inverse Function",
"Difference of Squares of Hyperbolic Cosine and Sine",
"Derivative of Hyperbolic Cosine",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Sign of Number",
"Real Area Hyperbolic Cosine is Strictly Increasing",
"Derivative of Mon... |
proofwiki-8970 | Derivative of Inverse Hyperbolic Tangent | :$\map {\dfrac \d {\d x} } {\tanh^{-1} x} = \dfrac 1 {1 - x^2}$ | {{begin-eqn}}
{{eqn | l = y
| r = \tanh^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \tanh y
| c = {{Defof|Real Inverse Hyperbolic Tangent}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = \sech^2 y
| c = Derivative of Hyperbolic Tangent
}}
{{eqn | ll= \lea... | :$\map {\dfrac \d {\d x} } {\tanh^{-1} x} = \dfrac 1 {1 - x^2}$ | {{begin-eqn}}
{{eqn | l = y
| r = \tanh^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \tanh y
| c = {{Defof|Real Inverse Hyperbolic Tangent}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = \sech^2 y
| c = [[Derivative of Hyperbolic Tangent]]
}}
{{eqn | ll= ... | Derivative of Inverse Hyperbolic Tangent | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Tangent | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Tangent | [
"Derivatives of Inverse Hyperbolic Functions",
"Inverse Hyperbolic Tangent",
"Derivative of Inverse Hyperbolic Tangent"
] | [] | [
"Derivative of Hyperbolic Tangent",
"Derivative of Inverse Function",
"Sum of Squares of Hyperbolic Secant and Tangent"
] |
proofwiki-8971 | Derivative of Inverse Hyperbolic Cotangent | :$\map {\dfrac \d {\d x} } {\coth^{-1} x} = \dfrac {-1} {x^2 - 1}$ | {{begin-eqn}}
{{eqn | l = y
| r = \coth^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \coth y
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = -\csch^2 y
| c = Derivative of Hyperbolic Cotangent
}}
{{eqn | ll=... | :$\map {\dfrac \d {\d x} } {\coth^{-1} x} = \dfrac {-1} {x^2 - 1}$ | {{begin-eqn}}
{{eqn | l = y
| r = \coth^{-1} x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \coth y
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = -\csch^2 y
| c = [[Derivative of Hyperbolic Cotangent]]
}}
{{eqn |... | Derivative of Inverse Hyperbolic Cotangent | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Cotangent | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Cotangent | [
"Derivative of Inverse Hyperbolic Cotangent",
"Derivatives of Inverse Hyperbolic Functions",
"Inverse Hyperbolic Cotangent"
] | [] | [
"Derivative of Hyperbolic Cotangent",
"Derivative of Inverse Function",
"Difference of Squares of Hyperbolic Cotangent and Cosecant"
] |
proofwiki-8972 | Derivative of Inverse Hyperbolic Secant | :$\map {\dfrac \d {\d x} } {\arsech x} = \dfrac {-1} {x \sqrt{1 - x^2} }$ | $\arsech x$ is defined only on the half-open real interval $\hointl 0 1$.
Thus on $\hointl 0 1$:
{{begin-eqn}}
{{eqn | l = y
| r = \arsech x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \sech y
| c = where $y \in \R_{>0}$
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = ... | :$\map {\dfrac \d {\d x} } {\arsech x} = \dfrac {-1} {x \sqrt{1 - x^2} }$ | $\arsech x$ is defined only on the [[Definition:Half-Open Real Interval|half-open real interval]] $\hointl 0 1$.
Thus on $\hointl 0 1$:
{{begin-eqn}}
{{eqn | l = y
| r = \arsech x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \sech y
| c = where $y \in \R_{>0}$
}}
{{eqn | ll= \leadsto
... | Derivative of Inverse Hyperbolic Secant | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Secant | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Secant | [
"Derivatives of Inverse Hyperbolic Functions",
"Inverse Hyperbolic Secant"
] | [] | [
"Definition:Real Interval/Half-Open",
"Derivative of Hyperbolic Secant",
"Derivative of Inverse Function",
"Sum of Squares of Hyperbolic Secant and Tangent",
"Sum of Squares of Hyperbolic Secant and Tangent"
] |
proofwiki-8973 | Derivative of Inverse Hyperbolic Cosecant | :$\map {\dfrac \d {\d x} } {\arcsch x} = \dfrac {-1} {\size x \sqrt {1 + x^2} }$ | 500pxright
Let $x > 1$.
Then we have:
{{begin-eqn}}
{{eqn | l = y
| r = \arcsch x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \csch y
| c = where $y \ne 0$
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = -\csch y \coth y
| c = Derivative of Hyperbolic Cosecant
}}... | :$\map {\dfrac \d {\d x} } {\arcsch x} = \dfrac {-1} {\size x \sqrt {1 + x^2} }$ | [[File:Arcsch.png|500px|right]]
Let $x > 1$.
Then we have:
{{begin-eqn}}
{{eqn | l = y
| r = \arcsch x
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \csch y
| c = where $y \ne 0$
}}
{{eqn | ll= \leadsto
| l = \frac {\d x} {\d y}
| r = -\csch y \coth y
| c = [[Derivative... | Derivative of Inverse Hyperbolic Cosecant | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Cosecant | https://proofwiki.org/wiki/Derivative_of_Inverse_Hyperbolic_Cosecant | [
"Derivatives of Inverse Hyperbolic Functions",
"Inverse Hyperbolic Cosecant"
] | [] | [
"File:Arcsch.png",
"Derivative of Hyperbolic Cosecant",
"Derivative of Inverse Function",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Inverse Hyperbolic Cosecant is Odd Function",
"Derivative of Odd Function is Even",
"Definition:Even Function"
] |
proofwiki-8974 | Sum of Cosecant and Cotangent | :$\csc x + \cot x = \cot {\dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = \csc x + \cot x
| r = \frac 1 {\sin x} + \frac {\cos x} {\sin x}
| c = {{Defof|Cosecant}} and {{Defof|Cotangent}}
}}
{{eqn | r = \frac {1 + \cos x} {\sin x}
| c =
}}
{{eqn | r = \frac {2 \cos^2 {\frac x 2} } {2 \sin {\frac x 2} \cos {\frac x 2} }
| c = Double Angle Form... | :$\csc x + \cot x = \cot {\dfrac x 2}$ | {{begin-eqn}}
{{eqn | l = \csc x + \cot x
| r = \frac 1 {\sin x} + \frac {\cos x} {\sin x}
| c = {{Defof|Cosecant}} and {{Defof|Cotangent}}
}}
{{eqn | r = \frac {1 + \cos x} {\sin x}
| c =
}}
{{eqn | r = \frac {2 \cos^2 {\frac x 2} } {2 \sin {\frac x 2} \cos {\frac x 2} }
| c = [[Double Angle Fo... | Sum of Cosecant and Cotangent | https://proofwiki.org/wiki/Sum_of_Cosecant_and_Cotangent | https://proofwiki.org/wiki/Sum_of_Cosecant_and_Cotangent | [
"Trigonometric Identities"
] | [] | [
"Double Angle Formulas/Sine",
"Category:Trigonometric Identities"
] |
proofwiki-8975 | Leibniz's Rule/One Variable/Second Derivative | Let $f$ and $g$ be real functions defined on the open interval $I$.
Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are twice differentiable.
Then:
:$\paren {\map f x \map g x}' ' = \map f x \map {g' '} x + 2 \map {f'} x \map {g'} x + \map {f' '} x \map g x$ | From Leibniz's Rule in One Variable:
:$\ds \paren {\map f x \map g x}^{\paren n} = \sum_{k \mathop = 0}^n \binom n k \map {f^{\paren k} } x \map {g^{\paren {n - k} } } x$
where $\paren n$ denotes the order of the derivative.
Setting $n = 2$:
{{begin-eqn}}
{{eqn | l = \paren {\map f x \map g x}' '
| r = \paren {\m... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] defined on the [[Definition:Open Real Interval|open interval]] $I$.
Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are [[Definition:Second Derivative|twice differentiable]].
Then:
:$\paren {\map f x \map g x}' ' = \map f x \map {g' '} x + 2 \m... | From [[Leibniz's Rule in One Variable]]:
:$\ds \paren {\map f x \map g x}^{\paren n} = \sum_{k \mathop = 0}^n \binom n k \map {f^{\paren k} } x \map {g^{\paren {n - k} } } x$
where $\paren n$ denotes the [[Definition:Order of Derivative|order of the derivative]].
Setting $n = 2$:
{{begin-eqn}}
{{eqn | l = \paren {\m... | Leibniz's Rule/One Variable/Second Derivative | https://proofwiki.org/wiki/Leibniz's_Rule/One_Variable/Second_Derivative | https://proofwiki.org/wiki/Leibniz's_Rule/One_Variable/Second_Derivative | [
"Leibniz's Rule"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open",
"Definition:Derivative/Higher Derivatives/Second Derivative"
] | [
"Leibniz's Rule/One Variable",
"Definition:Derivative/Higher Derivatives/Order of Derivative",
"Leibniz's Rule/One Variable"
] |
proofwiki-8976 | Leibniz's Rule/One Variable/Third Derivative | Let $f$ and $g$ be real functions defined on the open interval $I$.
Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are thrice differentiable.
Then:
:$\paren {\map f x \map g x}' ' ' = \map f x \map {g' ' '} x + 3 \map {f'} x \map {g' '} x + 3 \map {f' '} x \map {g'} x + \map {f' ' '} x \map g x$ | From Leibniz's Rule in One Variable:
:$\ds \paren {\map f x \map g x}^{\paren n} = \sum_{k \mathop = 0}^n \binom n k \map {f^{\paren k} } x \map {g^{\paren {n - k} } } x$
where $\paren n$ denotes the order of the derivative.
Setting $n = 3$:
{{begin-eqn}}
{{eqn | l = \paren {\map f x \map g x}' ' '
| r = \paren {... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] defined on the [[Definition:Open Real Interval|open interval]] $I$.
Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are [[Definition:Third Derivative|thrice differentiable]].
Then:
:$\paren {\map f x \map g x}' ' ' = \map f x \map {g' ' '} x + ... | From [[Leibniz's Rule in One Variable]]:
:$\ds \paren {\map f x \map g x}^{\paren n} = \sum_{k \mathop = 0}^n \binom n k \map {f^{\paren k} } x \map {g^{\paren {n - k} } } x$
where $\paren n$ denotes the [[Definition:Order of Derivative|order of the derivative]].
Setting $n = 3$:
{{begin-eqn}}
{{eqn | l = \paren {\m... | Leibniz's Rule/One Variable/Third Derivative | https://proofwiki.org/wiki/Leibniz's_Rule/One_Variable/Third_Derivative | https://proofwiki.org/wiki/Leibniz's_Rule/One_Variable/Third_Derivative | [
"Leibniz's Rule"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open",
"Definition:Derivative/Higher Derivatives/Third Derivative"
] | [
"Leibniz's Rule/One Variable",
"Definition:Derivative/Higher Derivatives/Order of Derivative",
"Leibniz's Rule/One Variable"
] |
proofwiki-8977 | Power Series Expansion for Tangent Function | The tangent function has a Taylor series expansion:
{{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 =
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {2 x^5} {15} + \frac {17 x^7} {315} + \cdots... | From Power Series Expansion for Cotangent Function:
:$(1): \quad \cot x = \ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}$
Then:
{{begin-eqn}}
{{eqn | l = \tan x
| r = \cot x - 2 \cot 2 x
| c = Double Angle Formula for Cotangent
}}
{{eqn | r = \sum_{n \m... | The [[Definition:Tangent Function|tangent function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
{{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 =
}}
{{eqn | r = x + \fra... | From [[Power Series Expansion for Cotangent Function]]:
:$(1): \quad \cot x = \ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}$
Then:
{{begin-eqn}}
{{eqn | l = \tan x
| r = \cot x - 2 \cot 2 x
| c = [[Double Angle Formula for Cotangent]]
}}
{{eqn | r =... | Power Series Expansion for Tangent Function/Proof 1 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Tangent_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Tangent_Function/Proof_1 | [
"Power Series Expansion for Tangent Function",
"Examples of Power Series",
"Bernoulli Numbers",
"Tangent Function"
] | [
"Definition:Tangent Function",
"Definition:Taylor Series",
"Definition:Bernoulli Numbers",
"Definition:Convergent Series"
] | [
"Power Series Expansion for Cotangent Function",
"Double Angle Formulas/Cotangent"
] |
proofwiki-8978 | Power Series Expansion for Tangent Function | The tangent function has a Taylor series expansion:
{{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 =
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {2 x^5} {15} + \frac {17 x^7} {315} + \cdots... | We have:
{{begin-eqn}}
{{eqn | l = \frac x {e^x - 1}
| r = \frac x 2 \paren {\frac 2 {e^x - 1} }
| c =
}}
{{eqn | r = \frac x 2 \paren {\frac {e^x - e^x + 2} {e^x - 1} }
| c =
}}
{{eqn | r = \frac x 2 \paren {\frac {\paren {e^x + 1} - \paren {e^x - 1} } {e^x - 1} }
| c =
}}
{{eqn | r = \frac... | The [[Definition:Tangent Function|tangent function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
{{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 =
}}
{{eqn | r = x + \fra... | We have:
{{begin-eqn}}
{{eqn | l = \frac x {e^x - 1}
| r = \frac x 2 \paren {\frac 2 {e^x - 1} }
| c =
}}
{{eqn | r = \frac x 2 \paren {\frac {e^x - e^x + 2} {e^x - 1} }
| c =
}}
{{eqn | r = \frac x 2 \paren {\frac {\paren {e^x + 1} - \paren {e^x - 1} } {e^x - 1} }
| c =
}}
{{eqn | r = \frac... | Power Series Expansion for Tangent Function/Proof 2 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Tangent_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Tangent_Function/Proof_2 | [
"Power Series Expansion for Tangent Function",
"Examples of Power Series",
"Bernoulli Numbers",
"Tangent Function"
] | [
"Definition:Tangent Function",
"Definition:Taylor Series",
"Definition:Bernoulli Numbers",
"Definition:Convergent Series"
] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Euler's Cotangent Identity",
"Double Angle Formulas/Cotangent"
] |
proofwiki-8979 | Power Series Expansion for Tangent Function | The tangent function has a Taylor series expansion:
{{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 =
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {2 x^5} {15} + \frac {17 x^7} {315} + \cdots... | By Combination Theorem for Limits of Real Functions we can deduce the following.
{{begin-eqn}}
{{eqn | o =
| r = \lim_{n \mathop \to \infty} \size {\frac {\frac {\paren {-1}^n 2^{2 n + 2} \paren {2^{2 n + 2} - 1} B_{2 n + 2} } {\paren {2 n + 2}!} x^{2 n + 1} } {\frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} ... | The [[Definition:Tangent Function|tangent function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
{{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 =
}}
{{eqn | r = x + \fra... | By [[Combination Theorem for Limits of Real Functions]] we can deduce the following.
{{begin-eqn}}
{{eqn | o =
| r = \lim_{n \mathop \to \infty} \size {\frac {\frac {\paren {-1}^n 2^{2 n + 2} \paren {2^{2 n + 2} - 1} B_{2 n + 2} } {\paren {2 n + 2}!} x^{2 n + 1} } {\frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{... | Power Series Expansion for Tangent Function/Proof of Convergence | https://proofwiki.org/wiki/Power_Series_Expansion_for_Tangent_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Tangent_Function/Proof_of_Convergence | [
"Power Series Expansion for Tangent Function",
"Examples of Power Series",
"Bernoulli Numbers",
"Tangent Function"
] | [
"Definition:Tangent Function",
"Definition:Taylor Series",
"Definition:Bernoulli Numbers",
"Definition:Convergent Series"
] | [
"Combination Theorem for Limits of Functions/Real",
"Asymptotic Formula for Bernoulli Numbers",
"Ratio Test",
"Definition:Convergent Series"
] |
proofwiki-8980 | Primitive of Constant Multiple of Function | Let $f$ be a real function which is integrable.
Let $c$ be a constant.
Then:
:$\ds \int c \map f x \rd x = c \int \map f x \rd x$ | From Linear Combination of Primitives:
:$\ds \int \paren {\lambda \map f x + \mu \map g x} \rd x = \lambda \int \map f x \rd x + \mu \int \map g x \rd x$
The result follows by setting $\lambda = c$ and $\mu = 0$.
{{qed}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]].
Let $c$ be a [[Definition:Constant|constant]].
Then:
:$\ds \int c \map f x \rd x = c \int \map f x \rd x$ | From [[Linear Combination of Primitives]]:
:$\ds \int \paren {\lambda \map f x + \mu \map g x} \rd x = \lambda \int \map f x \rd x + \mu \int \map g x \rd x$
The result follows by setting $\lambda = c$ and $\mu = 0$.
{{qed}} | Primitive of Constant Multiple of Function/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Constant_Multiple_of_Function | https://proofwiki.org/wiki/Primitive_of_Constant_Multiple_of_Function/Proof_1 | [
"Primitive of Constant Multiple of Function",
"Primitives"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Constant"
] | [
"Linear Combination of Integrals/Indefinite"
] |
proofwiki-8981 | Primitive of Constant Multiple of Function | Let $f$ be a real function which is integrable.
Let $c$ be a constant.
Then:
:$\ds \int c \map f x \rd x = c \int \map f x \rd x$ | From Derivative of Constant Multiple:
:$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$
The result follows from the definition of primitive. | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]].
Let $c$ be a [[Definition:Constant|constant]].
Then:
:$\ds \int c \map f x \rd x = c \int \map f x \rd x$ | From [[Derivative of Constant Multiple]]:
:$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$
The result follows from the definition of [[Definition:Primitive (Calculus)|primitive]]. | Primitive of Constant Multiple of Function/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Constant_Multiple_of_Function | https://proofwiki.org/wiki/Primitive_of_Constant_Multiple_of_Function/Proof_2 | [
"Primitive of Constant Multiple of Function",
"Primitives"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Constant"
] | [
"Derivative of Constant Multiple",
"Definition:Primitive (Calculus)"
] |
proofwiki-8982 | Primitive of Pointwise Sum of Functions | Let $f_1, f_2, \ldots, f_n$ be real functions which are integrable.
Then:
:$\ds \int \map {\paren {f_1 \pm f_2 \pm \, \cdots \pm f_n} } x \rd x = \int \map {f_1} x \rd x \pm \int \map {f_2} x \rd x \pm \, \cdots \pm \int \map {f_n} x \rd x$ | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \int \map {\paren {f_1 \pm f_2 \pm \, \cdots \pm f_n} } x \rd x = \int \map {f_1} x \rd x \pm \int \map {f_2} x \rd x \pm \, \cdots \pm \int \map {f_n} x \rd x$
$\map P 1$ is true, as this just says:
:$\ds \int \map {f_1} x \rd x = \... | Let $f_1, f_2, \ldots, f_n$ be [[Definition:Real Function|real functions]] which are [[Definition:Integrable Function|integrable]].
Then:
:$\ds \int \map {\paren {f_1 \pm f_2 \pm \, \cdots \pm f_n} } x \rd x = \int \map {f_1} x \rd x \pm \int \map {f_2} x \rd x \pm \, \cdots \pm \int \map {f_n} x \rd x$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \int \map {\paren {f_1 \pm f_2 \pm \, \cdots \pm f_n} } x \rd x = \int \map {f_1} x \rd x \pm \int \map {f_2} x \rd x \pm \, \cdots \pm \int \map {f_n} x \rd x$
$\... | Primitive of Pointwise Sum of Functions/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Pointwise_Sum_of_Functions | https://proofwiki.org/wiki/Primitive_of_Pointwise_Sum_of_Functions/Proof_1 | [
"Primitive of Pointwise Sum of Functions",
"Primitives"
] | [
"Definition:Real Function",
"Definition:Integrable Function"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Linear Combination of Integrals/Indefinite",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Primitive of Pointwise Sum of Functions",
"Primitive of Pointwise Sum of Functions",
"... |
proofwiki-8983 | Primitive of Pointwise Sum of Functions | Let $f_1, f_2, \ldots, f_n$ be real functions which are integrable.
Then:
:$\ds \int \map {\paren {f_1 \pm f_2 \pm \, \cdots \pm f_n} } x \rd x = \int \map {f_1} x \rd x \pm \int \map {f_2} x \rd x \pm \, \cdots \pm \int \map {f_n} x \rd x$ | From Sum Rule for Derivatives:
:$\ds \map {\dfrac \d {\d x} } {\sum_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \map {\dfrac \d {\d x} } {\map {f_i} x}$
The result follows by definition of primitive.
{{qed}} | Let $f_1, f_2, \ldots, f_n$ be [[Definition:Real Function|real functions]] which are [[Definition:Integrable Function|integrable]].
Then:
:$\ds \int \map {\paren {f_1 \pm f_2 \pm \, \cdots \pm f_n} } x \rd x = \int \map {f_1} x \rd x \pm \int \map {f_2} x \rd x \pm \, \cdots \pm \int \map {f_n} x \rd x$ | From [[Sum Rule for Derivatives/General Result|Sum Rule for Derivatives]]:
:$\ds \map {\dfrac \d {\d x} } {\sum_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \map {\dfrac \d {\d x} } {\map {f_i} x}$
The result follows by definition of [[Definition:Primitive (Calculus)|primitive]].
{{qed}} | Primitive of Pointwise Sum of Functions/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Pointwise_Sum_of_Functions | https://proofwiki.org/wiki/Primitive_of_Pointwise_Sum_of_Functions/Proof_2 | [
"Primitive of Pointwise Sum of Functions",
"Primitives"
] | [
"Definition:Real Function",
"Definition:Integrable Function"
] | [
"Sum Rule for Derivatives/General Result",
"Definition:Primitive (Calculus)"
] |
proofwiki-8984 | Hurwitz's Theorem (Number Theory) | Let $\xi$ be an irrational number.
Then there are infinitely many relatively prime integers $p, q \in \Z$ such that:
:$\size {\xi - \dfrac p q} < \dfrac 1 {\sqrt 5 \, q^2}$ | === Lemma 1 ===
{{:Hurwitz's Theorem (Number Theory)/Lemma 1}}{{qed|lemma}} | Let $\xi$ be an [[Definition:Irrational Number|irrational number]].
Then there are [[Definition:Infinite Set|infinitely]] many [[Definition:Coprime Integers|relatively prime integers]] $p, q \in \Z$ such that:
:$\size {\xi - \dfrac p q} < \dfrac 1 {\sqrt 5 \, q^2}$ | === [[Hurwitz's Theorem (Number Theory)/Lemma 1|Lemma 1]] ===
{{:Hurwitz's Theorem (Number Theory)/Lemma 1}}{{qed|lemma}} | Hurwitz's Theorem (Number Theory) | https://proofwiki.org/wiki/Hurwitz's_Theorem_(Number_Theory) | https://proofwiki.org/wiki/Hurwitz's_Theorem_(Number_Theory) | [
"Hurwitz's Theorem (Number Theory)",
"Hurwitz's Theorem",
"Number Theory"
] | [
"Definition:Irrational Number",
"Definition:Infinite Set",
"Definition:Coprime/Integers"
] | [
"Hurwitz's Theorem (Number Theory)/Lemma 1",
"Hurwitz's Theorem (Number Theory)/Lemma 1"
] |
proofwiki-8985 | Primitive of Function of Constant Multiple | Let $f$ be a real function which is integrable.
Let $c$ be a constant.
Then:
:$\ds \int \map f {c x} \rd x = \frac 1 c \int \map f u \d u$
where $u = c x$. | Let $u = c x$.
By {{Corollary|Derivative of Identity Function}}:
:$\dfrac {\d u} {\d x} = c$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map f {c x} \rd x
| r = \int \frac {\map f u} c \rd u
| c = Primitive of Composite Function
}}
{{eqn | r = \frac 1 c \int \map f u \rd u
| c = Primitive of Constant Multip... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]].
Let $c$ be a [[Definition:Constant|constant]].
Then:
:$\ds \int \map f {c x} \rd x = \frac 1 c \int \map f u \d u$
where $u = c x$. | Let $u = c x$.
By {{Corollary|Derivative of Identity Function}}:
:$\dfrac {\d u} {\d x} = c$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map f {c x} \rd x
| r = \int \frac {\map f u} c \rd u
| c = [[Primitive of Composite Function]]
}}
{{eqn | r = \frac 1 c \int \map f u \rd u
| c = [[Primitive of Consta... | Primitive of Function of Constant Multiple | https://proofwiki.org/wiki/Primitive_of_Function_of_Constant_Multiple | https://proofwiki.org/wiki/Primitive_of_Function_of_Constant_Multiple | [
"Primitives"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Constant"
] | [
"Primitive of Composite Function",
"Primitive of Constant Multiple of Function"
] |
proofwiki-8986 | Primitive of Composite Function | Let $f$ and $g$ be a real functions which are integrable.
Let the composite function $g \circ f$ also be integrable.
Then:
{{begin-eqn}}
{{eqn | l = \int \map {\paren {g \circ f} } x \rd x
| r = \int \map g u \frac {\d x} {\d u} \rd u
| c =
}}
{{eqn | r = \int \frac {\map g u} {\map {f'} x} \rd u
| c... | {{begin-eqn}}
{{eqn | l = \map F x
| r = \int \map {\paren {g \circ f} } x \rd x
}}
{{eqn | r = \int \map g {\map f x} \rd x
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \int \map g u \rd x
| c = where $u = \map f x$
}}
{{eqn | ll= \leadsto
| l = \frac {\d F} {\d x}
| r = \map g ... | Let $f$ and $g$ be a [[Definition:Real Function|real functions]] which are [[Definition:Integrable Function|integrable]].
Let the [[Definition:Composition of Mappings|composite function]] $g \circ f$ also be [[Definition:Integrable Function|integrable]].
Then:
{{begin-eqn}}
{{eqn | l = \int \map {\paren {g \circ f} ... | {{begin-eqn}}
{{eqn | l = \map F x
| r = \int \map {\paren {g \circ f} } x \rd x
}}
{{eqn | r = \int \map g {\map f x} \rd x
| c = {{Defof|Composition of Mappings}}
}}
{{eqn | r = \int \map g u \rd x
| c = where $u = \map f x$
}}
{{eqn | ll= \leadsto
| l = \frac {\d F} {\d x}
| r = \map g ... | Primitive of Composite Function | https://proofwiki.org/wiki/Primitive_of_Composite_Function | https://proofwiki.org/wiki/Primitive_of_Composite_Function | [
"Primitive of Composite Function",
"Primitives"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Composition of Mappings",
"Definition:Integrable Function"
] | [
"Derivative of Composite Function",
"Derivative of Inverse Function"
] |
proofwiki-8987 | Weierstrass Product Inequality | For $n \ge 1$:
:$\ds \prod_{i \mathop = 1}^n \paren {1 - a_i} \ge 1 - \sum_{i \mathop = 1}^n a_i$
where all of $a_i$ are in the closed interval $\closedint 0 1$. | For $n = 1$ we have:
:$1 - a_1 \ge 1 - a_1$
which is clearly true.
Suppose the proposition is true for $n = k$, that is:
:$\ds \prod_{i \mathop = 1}^k \paren {1 - a_i} \ge 1 - \sum_{i \mathop = 1}^k a_i$
Then:
{{begin-eqn}}
{{eqn | l = \prod_{i \mathop = 1}^{k + 1} \paren {1 - a_i}
| r = \paren {1 - a_{k + 1} }... | For $n \ge 1$:
:$\ds \prod_{i \mathop = 1}^n \paren {1 - a_i} \ge 1 - \sum_{i \mathop = 1}^n a_i$
where all of $a_i$ are in the [[Definition:Closed Real Interval|closed interval]] $\closedint 0 1$. | For $n = 1$ we have:
:$1 - a_1 \ge 1 - a_1$
which is clearly true.
Suppose the proposition is true for $n = k$, that is:
:$\ds \prod_{i \mathop = 1}^k \paren {1 - a_i} \ge 1 - \sum_{i \mathop = 1}^k a_i$
Then:
{{begin-eqn}}
{{eqn | l = \prod_{i \mathop = 1}^{k + 1} \paren {1 - a_i}
| r = \paren {1 - a_{k ... | Weierstrass Product Inequality | https://proofwiki.org/wiki/Weierstrass_Product_Inequality | https://proofwiki.org/wiki/Weierstrass_Product_Inequality | [
"Inequalities",
"Infinite Products"
] | [
"Definition:Real Interval/Closed"
] | [
"Principle of Mathematical Induction"
] |
proofwiki-8988 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Let $u = \sqrt {a x + b}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {\sqrt {a x + b} }^m \rd x
| r = \frac 2 a \int u \cdot u^m \rd x
| c = Primitive of Function of $\sqrt {a x + b}$
}}
{{eqn | r = \frac 2 a \int u^{m + 1} \rd x
| c = simplifying
}}
{{eqn | r = \frac 2 a \frac {u^{m + 2} } {m + 2} ... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Let $u = \sqrt {a x + b}$.
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {\sqrt {a x + b} }^m \rd x
| r = \frac 2 a \int u \cdot u^m \rd x
| c = [[Primitive of Function of Root of a x + b|Primitive of Function of $\sqrt {a x + b}$]]
}}
{{eqn | r = \frac 2 a \int u^{m + 1} \rd x
| c = simplifying
}}
{{e... | Primitive of Power of Root of a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_Root_of_a_x_+_b/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Function of Root of a x + b",
"Primitive of Power"
] |
proofwiki-8989 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \frac 1 a \int u^n \rd u
| c = Primitive of Function of $a x + b$
}}
{{eqn | r = \frac 1 a \frac {u^{n + 1} } {n + 1} + C
| c = Primitive of Power
}}
{{eqn | r = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} ... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \frac 1 a \int u^n \rd u
| c = [[Primitive of Function of a x + b|Primitive of Function of $a x + b$]]
}}
{{eqn | r = \frac 1 a \frac {u^{n + 1} } {n + 1} + C
| c = [[Primitive of Power]]
}}
{{eqn | r = \frac {\pa... | Primitive of Power of a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Function of a x + b",
"Primitive of Power"
] |
proofwiki-8990 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Let $u = a x + b$.
Then:
:$\dfrac {\d u} {\d x} = a$
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \int \dfrac {u^n} a \rd u
| c = Integration by Substitution
}}
{{eqn | r = \dfrac 1 a \dfrac {u^{n + 1} } {n + 1}
| c = Primitive of Power
}}
{{eqn | r = \frac {\paren {a x + b}^{n ... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Let $u = a x + b$.
Then:
:$\dfrac {\d u} {\d x} = a$
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \int \dfrac {u^n} a \rd u
| c = [[Integration by Substitution]]
}}
{{eqn | r = \dfrac 1 a \dfrac {u^{n + 1} } {n + 1}
| c = [[Primitive of Power]]
}}
{{eqn | r = \frac {\paren {a ... | Primitive of Power of a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-8991 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} }
| r = \dfrac {\paren {n + 1} \paren {a x + b}^n} {\paren {n + 1} a} \map {\dfrac \d {\d x} } {a x + b}
| c = Power Rule for Derivatives, Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {a \paren {n + ... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} }
| r = \dfrac {\paren {n + 1} \paren {a x + b}^n} {\paren {n + 1} a} \map {\dfrac \d {\d x} } {a x + b}
| c = [[Power Rule for Derivatives]], [[Chain Rule for Derivatives]]
}}
{{eqn | r = \dfrac {a \par... | Primitive of Power of a x + b/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_3 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Definition:Primitive (Calculus)"
] |
proofwiki-8992 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccsc x \rd x
| r = <nowiki>\begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\ \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \df... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccsc x \rd x
| r = <nowiki>\begin {cases}
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x + \dfrac 1 {m + 1} \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \\ \\
\ds \dfrac {x^{m + 1} } {m + 1} \arccsc x - \dfrac 1 {m + 1} \int \d... | Primitive of Power of x by Arccosecant of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x_over_a/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of x by Arccosecant of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-8993 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin {cases} \dfrac {-a} {x \s... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccsc \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin... | Primitive of Power of x by Arccosecant of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosecant_of_x_over_a/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Definition:Primitive",
"Derivative of Arccosecant Function/Corollary",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-8994 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccos x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = Primitive of $x^m \arccos x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arccos \frac x a \rd x
| r =... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccos x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccos x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt {1 - x^2} }
| c = [[Primitive of Power of x by Arccosine of x|Primitive of $x^m \arccos x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn |... | Primitive of Power of x by Arccosine of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x_over_a/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of x by Arccosine of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-8995 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqrt {a^2 - x^2} }
| c ... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccos \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-1} {\sqr... | Primitive of Power of x by Arccosine of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccosine_of_x_over_a/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Definition:Primitive",
"Derivative of Arccosine Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-8996 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccot x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}
| c = Primitive of $x^m \arccot x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \arccot \frac x a \rd x
| r = \int a^m... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arccot x \rd x
| r = \frac {x^{m + 1} } {m + 1} \arccot x + \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {x^2 + 1}
| c = [[Primitive of Power of x by Arccotangent of x|Primitive of $x^m \arccot x$]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \... | Primitive of Power of x by Arccotangent of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x_over_a/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of x by Arccotangent of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-8997 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 + a^2}
| c = Derivat... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arccot \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \frac {-a} {x^2 ... | Primitive of Power of x by Arccotangent of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arccotangent_of_x_over_a/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Definition:Primitive",
"Derivative of Arccotangent Function/Corollary",
"Primitive of Power",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
proofwiki-8998 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsec x \rd x
| r = <nowiki>\begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfrac... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | Recall:
{{begin-eqn}}
{{eqn | n = 1
| l = \int x^m \arcsec x \rd x
| r = <nowiki>\begin {cases}
\dfrac {x^{m + 1} } {m + 1} \arcsec x - \dfrac 1 {m + 1} \ds \int \dfrac {x^m \rd x} {\sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \\
\dfrac {x^{m + 1} } {m + 1} \arcsec x + \dfrac 1 {m + 1} \ds \int \dfra... | Primitive of Power of x by Arcsecant of x over a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x_over_a/Proof_1 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Primitive of Power of x by Arcsecant of x",
"Primitive of Constant Multiple of Function",
"Primitive of Function of Constant Multiple"
] |
proofwiki-8999 | Primitive of Power | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin {cases} \dfrac a {x \sqrt ... | Let $n \in \R: n \ne -1$.
Then:
:$\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = <nowiki> \begin ... | Primitive of Power of x by Arcsecant of x over a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power | https://proofwiki.org/wiki/Primitive_of_Power_of_x_by_Arcsecant_of_x_over_a/Proof_2 | [
"Primitive of Power",
"Primitives"
] | [
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Definition:Primitive",
"Derivative of Arcsecant Function/Corollary 1",
"Primitive of Power",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.