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-21500 | Real Elementary Functions are Continuous | The elementary functions on the real numbers $\R$ are all continuous at every point of their domain. | In turn:
:Real Polynomial Function is Continuous
:Real Trigonometric Functions are Continuous
:Real Logarithm Function is Continuous
:Real Exponential Function is Continuous
It remains to show that:
:All real functions that are compositions of the above
:All real functions obtained by adding, subtracting, multiplying a... | The [[Definition:Elementary Function|elementary functions]] on the [[Definition:Real Number|real numbers]] $\R$ are all [[Definition:Continuous Real Function at Point|continuous]] at every [[Definition:Point|point]] of their [[Definition:Domain of Mapping|domain]]. | In turn:
:[[Real Polynomial Function is Continuous]]
:[[Real Trigonometric Functions are Continuous]]
:[[Real Logarithm Function is Continuous]]
:[[Real Exponential Function is Continuous]]
It remains to show that:
:All [[Definition:Real Function|real functions]] that are [[Definition:Composition of Mappings|compositi... | Real Elementary Functions are Continuous | https://proofwiki.org/wiki/Real_Elementary_Functions_are_Continuous | https://proofwiki.org/wiki/Real_Elementary_Functions_are_Continuous | [
"Continuous Real Functions",
"Elementary Functions"
] | [
"Definition:Elementary Function",
"Definition:Real Number",
"Definition:Continuous Real Function/Point",
"Definition:Point",
"Definition:Domain (Set Theory)/Mapping"
] | [
"Real Polynomial Function is Continuous",
"Real Trigonometric Functions are Continuous",
"Real Logarithm Function is Continuous",
"Real Exponential Function is Continuous",
"Definition:Real Function",
"Definition:Composition of Mappings",
"Definition:Real Function",
"Definition:Continuous Real Functio... |
proofwiki-21501 | Real Trigonometric Functions are Continuous | The trigonometric functions on the real numbers $\R$:
:sine
:cosine
:tangent
:cotangent
:secant
:cosecant
are all continuous at every point of their domain. | In turn:
:Real Sine Function is Continuous
:Real Cosine Function is Continuous
:Real Tangent Function is Continuous
:Real Cotangent Function is Continuous
:Real Secant Function is Continuous
:Real Cosecant Function is Continuous
{{qed}} | The [[Definition:Trigonometric Function|trigonometric functions]] on the [[Definition:Real Number|real numbers]] $\R$:
:[[Definition:Real Sine Function|sine]]
:[[Definition:Real Cosine Function|cosine]]
:[[Definition:Real Tangent Function|tangent]]
:[[Definition:Real Cotangent Function|cotangent]]
:[[Definition:Real Se... | In turn:
:[[Real Sine Function is Continuous]]
:[[Real Cosine Function is Continuous]]
:[[Real Tangent Function is Continuous]]
:[[Real Cotangent Function is Continuous]]
:[[Real Secant Function is Continuous]]
:[[Real Cosecant Function is Continuous]]
{{qed}} | Real Trigonometric Functions are Continuous | https://proofwiki.org/wiki/Real_Trigonometric_Functions_are_Continuous | https://proofwiki.org/wiki/Real_Trigonometric_Functions_are_Continuous | [
"Continuous Real Functions",
"Trigonometric Functions"
] | [
"Definition:Trigonometric Function",
"Definition:Real Number",
"Definition:Sine/Real Function",
"Definition:Cosine/Real Function",
"Definition:Tangent Function/Real",
"Definition:Cotangent/Real Function",
"Definition:Secant Function/Real",
"Definition:Cosecant/Real Function",
"Definition:Continuous ... | [
"Real Sine Function is Continuous",
"Real Cosine Function is Continuous",
"Real Tangent Function is Continuous",
"Real Cotangent Function is Continuous",
"Real Secant Function is Continuous",
"Real Cosecant Function is Continuous"
] |
proofwiki-21502 | Continuous Function is not necessarily Differentiable | Let $f$ be a real function defined on an interval $I$.
Let $x_0 \in I$ such that $f$ is continuous at $x_0$.
Then it is not necessarily the case that $f$ is differentiable at $x_0$. | ;Proof by Counterexample
Consider the real function $f: \R \to \R$ defined as:
:$\forall x \in \R: \map f x = \size x$
where $\size x$ denotes the absolute value function.
From Absolute Value Function is Continuous, $f$ is continuous on $\R$.
From Derivative of Absolute Value Function, $f$ is not differentiable at $x =... | Let $f$ be a [[Definition:Real Function|real function]] defined on an [[Definition:Real Interval|interval]] $I$.
Let $x_0 \in I$ such that $f$ is [[Definition:Continuous Real Function at Point|continuous]] at $x_0$.
Then it is not necessarily the case that $f$ is [[Definition:Differentiable Real Function at Point|di... | ;[[Proof by Counterexample]]
Consider the [[Definition:Real Function|real function]] $f: \R \to \R$ defined as:
:$\forall x \in \R: \map f x = \size x$
where $\size x$ denotes the [[Definition:Absolute Value|absolute value function]].
From [[Absolute Value Function is Continuous]], $f$ is [[Definition:Continuous Rea... | Continuous Function is not necessarily Differentiable | https://proofwiki.org/wiki/Continuous_Function_is_not_necessarily_Differentiable | https://proofwiki.org/wiki/Continuous_Function_is_not_necessarily_Differentiable | [
"Differentiable Real Functions",
"Continuous Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Continuous Real Function/Point",
"Definition:Differentiable Mapping/Real Function/Point"
] | [
"Proof by Counterexample",
"Definition:Real Function",
"Definition:Absolute Value",
"Absolute Value Function is Continuous",
"Definition:Continuous Real Function",
"Derivative of Absolute Value Function",
"Definition:Differentiable Mapping/Real Function/Point",
"Definition:Real Function",
"Definitio... |
proofwiki-21503 | Combination Theorem for Continuous Functions/Complex/Difference Rule | :$f - g$ is continuous on $S$. | We have that:
:$\map {\paren {f - g} } x = \map {\paren {f + \paren {-g} } } x$
From Multiple Rule for Continuous Complex Functions:
:$-g$ is continuous on $S$.
From Sum Rule for Continuous Complex Functions:
:$f + \paren {-g}$ is continuous on $S$.
The result follows.
{{qed}}
Category:Combination Theorems for Continuo... | :$f - g$ is [[Definition:Continuous Complex Function|continuous]] on $S$. | We have that:
:$\map {\paren {f - g} } x = \map {\paren {f + \paren {-g} } } x$
From [[Multiple Rule for Continuous Complex Functions]]:
:$-g$ is [[Definition:Continuous Complex Function|continuous]] on $S$.
From [[Sum Rule for Continuous Complex Functions]]:
:$f + \paren {-g}$ is [[Definition:Continuous Complex Fun... | Combination Theorem for Continuous Functions/Complex/Difference Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Difference_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Continuous_Functions/Complex/Difference_Rule | [
"Combination Theorems for Continuous Functions"
] | [
"Definition:Continuous Complex Function"
] | [
"Combination Theorem for Continuous Functions/Complex/Multiple Rule",
"Definition:Continuous Complex Function",
"Combination Theorem for Continuous Functions/Complex/Sum Rule",
"Definition:Continuous Complex Function",
"Category:Combination Theorems for Continuous Functions"
] |
proofwiki-21504 | Real Function both Convex and Concave is Linear | Let $f$ be a real function which is both convex and concave.
Then $f$ is a linear function. | Let $f$ be both convex and concave on a subset $S \subseteq \R$ of the real numbers $\R$.
Then by definition:
{{begin-eqn}}
{{eqn | q = \forall x, y \in S: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1
| l = \map f {\alpha x + \beta y}
| o = \le
| r = \alpha \map f x + \beta \map f y
| c... | Let $f$ be a [[Definition:Real Function|real function]] which is both [[Definition:Convex Real Function|convex]] and [[Definition:Concave Real Function|concave]].
Then $f$ is a [[Definition:Linear Real Function|linear function]]. | Let $f$ be both [[Definition:Convex Real Function|convex]] and [[Definition:Concave Real Function|concave]] on a [[Definition:Subset|subset]] $S \subseteq \R$ of the [[Definition:Real Numbers|real numbers]] $\R$.
Then by definition:
{{begin-eqn}}
{{eqn | q = \forall x, y \in S: \forall \alpha, \beta \in \R_{>0}, \alp... | Real Function both Convex and Concave is Linear | https://proofwiki.org/wiki/Real_Function_both_Convex_and_Concave_is_Linear | https://proofwiki.org/wiki/Real_Function_both_Convex_and_Concave_is_Linear | [
"Convex Real Functions",
"Concave Real Functions",
"Linear Real Functions"
] | [
"Definition:Real Function",
"Definition:Convex Real Function",
"Definition:Concave Real Function",
"Definition:Linear Real Function"
] | [
"Definition:Convex Real Function",
"Definition:Concave Real Function",
"Definition:Subset",
"Definition:Real Number"
] |
proofwiki-21505 | Equivalence of Definitions of Convex Hull | {{TFAE|def = Convex Hull}}
Let $V$ be a vector space over $\R$.
Let $U \subseteq V$. | === Definition $(1)$ iff Definition $(3)$ ===
This is demonstrated in Convex Hull is Smallest Convex Set containing Set.
{{qed}} | {{TFAE|def = Convex Hull}}
Let $V$ be a [[Definition:Vector Space|vector space]] over $\R$.
Let $U \subseteq V$. | === Definition $(1)$ iff Definition $(3)$ ===
This is demonstrated in [[Convex Hull is Smallest Convex Set containing Set]].
{{qed}} | Equivalence of Definitions of Convex Hull | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convex_Hull | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convex_Hull | [
"Convex Hulls"
] | [
"Definition:Vector Space"
] | [
"Convex Hull is Smallest Convex Set containing Set"
] |
proofwiki-21506 | Euler Triangle Formula/Corollary | Let $\rho$ be the inradius of a triangle.
Then:
:$\dfrac 1 \rho = \dfrac 1 {R - d} + \dfrac 1 {R + d}$
where:
:$R$ is the circumradius
:$d$ is the distance between the incenter and the circumcenter. | {{begin-eqn}}
{{eqn | l = d^2
| r = R \paren {R - 2 \rho}
| c = Euler Triangle Formula
}}
{{eqn | ll= \leadsto
| l = d^2
| r = R^2 - 2 R \rho
| c =
}}
{{eqn | ll= \leadsto
| l = 2 R \rho
| r = R^2 - d^2
| c =
}}
{{eqn | ll= \leadsto
| l = \rho
| r = \dfrac {... | Let $\rho$ be the [[Definition:Inradius of Triangle|inradius]] of a [[Definition:Triangle (Geometry)|triangle]].
Then:
:$\dfrac 1 \rho = \dfrac 1 {R - d} + \dfrac 1 {R + d}$
where:
:$R$ is the [[Definition:Circumradius of Triangle|circumradius]]
:$d$ is the [[Definition:Distance between Points|distance]] between the [... | {{begin-eqn}}
{{eqn | l = d^2
| r = R \paren {R - 2 \rho}
| c = [[Euler Triangle Formula]]
}}
{{eqn | ll= \leadsto
| l = d^2
| r = R^2 - 2 R \rho
| c =
}}
{{eqn | ll= \leadsto
| l = 2 R \rho
| r = R^2 - d^2
| c =
}}
{{eqn | ll= \leadsto
| l = \rho
| r = \dfr... | Euler Triangle Formula/Corollary | https://proofwiki.org/wiki/Euler_Triangle_Formula/Corollary | https://proofwiki.org/wiki/Euler_Triangle_Formula/Corollary | [
"Euler Triangle Formula"
] | [
"Definition:Incircle of Triangle/Inradius",
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumradius",
"Definition:Distance between Points",
"Definition:Incircle of Triangle/Incenter",
"Definition:Circumcircle of Triangle/Circumcenter"
] | [
"Euler Triangle Formula",
"Category:Euler Triangle Formula"
] |
proofwiki-21507 | Condition for Cross-Ratio to be Harmonic | Let $A$, $B$, $C$ and $D$ be points on a straight line.
Let their '''cross-ratio''' $\set {A, B; C, D}$ be a harmonic ratio.
Then:
:$\set {A, B; C, D} = \set {B, A; C, D}$ | Let $\set {A, B; C, D}$ form a harmonic ratio.
We have:
{{begin-eqn}}
{{eqn | l = \set {A, B; C, D}
| r = -1
| c = {{Defof|Harmonic Ratio}}
}}
{{eqn | ll= \leadsto
| l = \dfrac {AC \cdot DB} {AD \cdot CB}
| r = -1
| c = {{Defof|Cross-Ratio of Points on Line}}
}}
{{eqn | ll= \leadsto
... | Let $A$, $B$, $C$ and $D$ be [[Definition:Point|points]] on a [[Definition:Straight Line|straight line]].
Let their '''[[Definition:Cross-Ratio of Points on Line|cross-ratio]]''' $\set {A, B; C, D}$ be a [[Definition:Harmonic Ratio|harmonic ratio]].
Then:
:$\set {A, B; C, D} = \set {B, A; C, D}$ | Let $\set {A, B; C, D}$ form a [[Definition:Harmonic Ratio|harmonic ratio]].
We have:
{{begin-eqn}}
{{eqn | l = \set {A, B; C, D}
| r = -1
| c = {{Defof|Harmonic Ratio}}
}}
{{eqn | ll= \leadsto
| l = \dfrac {AC \cdot DB} {AD \cdot CB}
| r = -1
| c = {{Defof|Cross-Ratio of Points on Line}... | Condition for Cross-Ratio to be Harmonic | https://proofwiki.org/wiki/Condition_for_Cross-Ratio_to_be_Harmonic | https://proofwiki.org/wiki/Condition_for_Cross-Ratio_to_be_Harmonic | [
"Definitions/Cross-Ratios",
"Definitions/Harmonic Ratios"
] | [
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Cross-Ratio/Points on Line",
"Definition:Harmonic Ratio"
] | [
"Definition:Harmonic Ratio",
"Definition:Directed Line Segment",
"Definition:Factor",
"Definition:Reciprocal",
"Real Multiplication is Commutative"
] |
proofwiki-21508 | Predecessor is Infimum | Let $\struct{S, \preceq}$ be an ordered set.
Let $x, y \in S$.
Then:
:$x \preceq y$ {{iff}} $x = \map \inf {x, y}$
where $\map \inf {x, y}$ is the infimum of $x$ and $y$ | This is the dual statement of Successor is Supremum by Dual Pairs (Order Theory).
The result follows from the Duality Principle.
{{qed}}
Category:Orderings
Category:Infima
gtst9yw29m63l4pccqo3fj81owkfysa | Let $\struct{S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y \in S$.
Then:
:$x \preceq y$ {{iff}} $x = \map \inf {x, y}$
where $\map \inf {x, y}$ is the [[Definition:Infimum of Set|infimum]] of $x$ and $y$ | This is the [[Definition:Dual Statement (Order Theory)|dual statement]] of [[Successor is Supremum]] by [[Dual Pairs (Order Theory)]].
The result follows from the [[Duality Principle (Order Theory)|Duality Principle]].
{{qed}}
[[Category:Orderings]]
[[Category:Infima]]
gtst9yw29m63l4pccqo3fj81owkfysa | Predecessor is Infimum | https://proofwiki.org/wiki/Predecessor_is_Infimum | https://proofwiki.org/wiki/Predecessor_is_Infimum | [
"Orderings",
"Infima"
] | [
"Definition:Ordered Set",
"Definition:Infimum of Set"
] | [
"Definition:Dual Statement (Order Theory)",
"Successor is Supremum",
"Dual Pairs (Order Theory)",
"Duality Principle (Order Theory)",
"Category:Orderings",
"Category:Infima"
] |
proofwiki-21509 | Successor is Supremum | Let $\struct {S, \preceq}$ be an ordered set.
Let $x, y \in S$.
Then:
:$x \preceq y$ {{iff}} $y = \map \sup {x, y}$
where $\map \sup {x, y}$ is the supremum of $x$ and $y$ | === Necessary Condition ===
Let $x \preceq y$.
By {{Ordering-axiom|1}}:
:$y \preceq y$
Hence $y$ is an upper bound of $\set {x, y}$ by definition.
Let $z$ be an upper bound of $\set {x, y}$.
By definition of upper bound:
:$y \preceq z$
Hence $y$ is the supremum of $x$ and $y$ by definition.
{{qed|lemma}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y \in S$.
Then:
:$x \preceq y$ {{iff}} $y = \map \sup {x, y}$
where $\map \sup {x, y}$ is the [[Definition:Supremum of Set|supremum]] of $x$ and $y$ | === Necessary Condition ===
Let $x \preceq y$.
By {{Ordering-axiom|1}}:
:$y \preceq y$
Hence $y$ is an [[Definition:Upper Bound|upper bound]] of $\set {x, y}$ by definition.
Let $z$ be an [[Definition:Upper Bound|upper bound]] of $\set {x, y}$.
By definition of [[Definition:Upper Bound|upper bound]]:
:$y \preceq... | Successor is Supremum | https://proofwiki.org/wiki/Successor_is_Supremum | https://proofwiki.org/wiki/Successor_is_Supremum | [
"Orderings",
"Suprema"
] | [
"Definition:Ordered Set",
"Definition:Supremum of Set"
] | [
"Definition:Upper Bound",
"Definition:Upper Bound",
"Definition:Upper Bound",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set"
] |
proofwiki-21510 | Cubical Parabola has Point of Inflection | The '''cubical parabola''' has a point of inflection.
When the equation for the '''cubical parabola''' is in the form $y = a x^3$, this point of inflection is the origin. | {{begin-eqn}}
{{eqn | l = y
| r = a x^3
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d y} {\d x}
| r = 3 a x^2
| c = Derivative of Power
}}
{{end-eqn}}
By definition, there is a point of inflection at a point where $x = \xi$ {{iff}} the first derivative has either a local maximum or a loc... | The '''[[Definition:Cubical Parabola|cubical parabola]]''' has a [[Definition:Point of Inflection|point of inflection]].
When the [[Definition:Equation|equation]] for the '''[[Definition:Cubical Parabola|cubical parabola]]''' is in the form $y = a x^3$, this [[Definition:Point of Inflection|point of inflection]] is t... | {{begin-eqn}}
{{eqn | l = y
| r = a x^3
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d y} {\d x}
| r = 3 a x^2
| c = [[Derivative of Power]]
}}
{{end-eqn}}
By definition, there is a [[Definition:Point of Inflection|point of inflection]] at a [[Definition:Point|point]] where $x = \xi$ {{i... | Cubical Parabola has Point of Inflection | https://proofwiki.org/wiki/Cubical_Parabola_has_Point_of_Inflection | https://proofwiki.org/wiki/Cubical_Parabola_has_Point_of_Inflection | [
"Cubical Parabola",
"Points of Inflection"
] | [
"Definition:Cubical Parabola",
"Definition:Point of Inflection",
"Definition:Equation",
"Definition:Cubical Parabola",
"Definition:Point of Inflection",
"Definition:Coordinate System/Origin"
] | [
"Power Rule for Derivatives",
"Definition:Point of Inflection",
"Definition:Point",
"Definition:Derivative",
"Definition:Maximum Value of Real Function/Local",
"Definition:Minimum Value of Real Function/Local",
"Definition:Equation",
"Definition:Parabola",
"Square of Non-Zero Real Number is Strictly... |
proofwiki-21511 | Euler Theorem on Curvature of Surface | Let $S$ be a surface in space.
The principal directions of the '''curvature''' of $S$ are perpendicular to each other. | {{ProofWanted}}
{{Namedfor|Leonhard Paul Euler|cat = Euler}} | Let $S$ be a [[Definition:Surface|surface]] in [[Definition:Ordinary Space|space]].
The [[Definition:Principal Directions of Curvature of Surface|principal directions]] of the '''[[Definition:Curvature|curvature]]''' of $S$ are [[Definition:Perpendicular|perpendicular]] to each other. | {{ProofWanted}}
{{Namedfor|Leonhard Paul Euler|cat = Euler}} | Euler Theorem on Curvature of Surface | https://proofwiki.org/wiki/Euler_Theorem_on_Curvature_of_Surface | https://proofwiki.org/wiki/Euler_Theorem_on_Curvature_of_Surface | [
"Euler's Theorems",
"Curvature",
"Solid Geometry"
] | [
"Definition:Surface",
"Definition:Ordinary Space",
"Definition:Principal Directions of Curvature of Surface",
"Definition:Curvature",
"Definition:Right Angle/Perpendicular"
] | [] |
proofwiki-21512 | Volume of Cylinder/Height and Base Area | Let $\CC$ be a cylinder such that:
:the bases of $\CC$ have area $A$
:the height of $\CC$ is $h$.
The volume $\VV$ of $\CC$ is given by the formula:
:$\VV = A h$ | :600px
Consider a cylinder $C$ whose base is of area $A$ and whose height is $h$.
Consider a cuboid $K$ whose height is $h$ and whose base is also $A$.
Let $C$ be positioned with its base in the same plane as the base of $K$.
By Cavalieri's Principle $C$ and $K$ have the same volume.
From Volume of Cuboid, $K$ has volu... | Let $\CC$ be a [[Definition:Cylinder|cylinder]] such that:
:the [[Definition:Base of Cylinder|bases]] of $\CC$ have [[Definition:Area|area]] $A$
:the [[Definition:Height of Cylinder|height]] of $\CC$ is $h$.
The [[Definition:Volume|volume]] $\VV$ of $\CC$ is given by the formula:
:$\VV = A h$ | :[[File:VolumeOfCylinder.png|600px]]
Consider a [[Definition:Cylinder|cylinder]] $C$ whose [[Definition:Base of Cylinder|base]] is of [[Definition:Area|area]] $A$ and whose [[Definition:Height of Cylinder|height]] is $h$.
Consider a [[Definition:Cuboid|cuboid]] $K$ whose [[Definition:Height of Cylinder|height]] is $h... | Volume of Cylinder/Height and Base Area | https://proofwiki.org/wiki/Volume_of_Cylinder/Height_and_Base_Area | https://proofwiki.org/wiki/Volume_of_Cylinder/Height_and_Base_Area | [
"Volume of Cylinder"
] | [
"Definition:Cylinder",
"Definition:Cylinder/Base",
"Definition:Area",
"Definition:Cylinder/Height",
"Definition:Volume"
] | [
"File:VolumeOfCylinder.png",
"Definition:Cylinder",
"Definition:Cylinder/Base",
"Definition:Area",
"Definition:Cylinder/Height",
"Definition:Cuboid",
"Definition:Cylinder/Height",
"Definition:Base of Solid Figure",
"Definition:Cylinder/Base",
"Definition:Plane Surface",
"Definition:Base of Solid... |
proofwiki-21513 | Finite Connected Simple Graph with Size One Less than Order has no Circuits | Let $G$ be a finite connected simple graph of order $n$.
Let the size of $T$ be $n - 1$.
Then $G$ has no circuits. | Let $G$ be a connected simple graph of order $n$ with $n - 1$ edges.
By definition:
:the order of a finite tree is how many nodes it has
:the size of a finite tree is how many edges it has.
{{AimForCont}} $G$ contains at least one circuit.
From Condition for Edge to be Bridge:
:there exists at least one edge $e$ in $G$... | Let $G$ be a [[Definition:Finite Graph|finite]] [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]] of [[Definition:Order of Graph|order]] $n$.
Let the [[Definition:Size of Graph|size]] of $T$ be $n - 1$.
Then $G$ has no [[Definition:Circuit (Graph Theory)|circuits]]. | Let $G$ be a [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]] of [[Definition:Order of Graph|order]] $n$ with $n - 1$ [[Definition:Edge of Graph|edges]].
By definition:
:the [[Definition:Order of Graph|order]] of a [[Definition:Finite Tree|finite tree]] is how many [[Definition:Node of... | Finite Connected Simple Graph with Size One Less than Order has no Circuits | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_with_Size_One_Less_than_Order_has_no_Circuits | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_with_Size_One_Less_than_Order_has_no_Circuits | [
"Finite Graphs",
"Connectedness (Graph Theory)",
"Simple Graphs",
"Circuits (Graph Theory)"
] | [
"Definition:Finite Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"Definition:Circuit (Graph Theory)"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Order",
"Definition:Tree (Graph Theory)/Finite",
"Definition:Tree (Graph Theory)/Node",
"Definition:Graph (Graph The... |
proofwiki-21514 | Characteristics of Finite Tree/Condition 1 | :$T$ is a finite tree of order $n$ {{iff}} $T$ has $n - 1$ edges and has no circuits. | First we recall that by definition of a tree, $T$ is:
:a simple connected graph
:with no circuits. | :$T$ is a [[Definition:Finite Tree|finite tree]] of [[Definition:Order of Graph|order]] $n$ {{iff}} $T$ has $n - 1$ [[Definition:Edge of Graph|edges]] and has no [[Definition:Circuit (Graph Theory)|circuits]]. | First we recall that by definition of a [[Definition:Tree (Graph Theory)|tree]], $T$ is:
:a [[Definition:Simple Graph|simple]] [[Definition:Connected Graph|connected graph]]
:with no [[Definition:Circuit (Graph Theory)|circuits]]. | Characteristics of Finite Tree/Condition 1 | https://proofwiki.org/wiki/Characteristics_of_Finite_Tree/Condition_1 | https://proofwiki.org/wiki/Characteristics_of_Finite_Tree/Condition_1 | [
"Characteristics of Finite Tree"
] | [
"Definition:Tree (Graph Theory)/Finite",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Circuit (Graph Theory)"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Simple Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Circuit (Graph Theory)",
"Definition:Circuit (Graph Theory)",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Tree (Graph Theory)"
] |
proofwiki-21515 | Characteristics of Finite Tree/Condition 2 | :$T$ is a finite tree of order $n$ {{iff}} $T$ has $n - 1$ edges and is connected. | This is an instance of:
:Finite Connected Simple Graph is Tree iff Size is One Less than Order.
{{qed}}
Category:Characteristics of Finite Tree
bgl3f96rpnzfu3eih511kjz9pomd9bv | :$T$ is a [[Definition:Finite Tree|finite tree]] of [[Definition:Order of Graph|order]] $n$ {{iff}} $T$ has $n - 1$ [[Definition:Edge of Graph|edges]] and is [[Definition:Connected Graph|connected]]. | This is an instance of:
:[[Finite Connected Simple Graph is Tree iff Size is One Less than Order]].
{{qed}}
[[Category:Characteristics of Finite Tree]]
bgl3f96rpnzfu3eih511kjz9pomd9bv | Characteristics of Finite Tree/Condition 2 | https://proofwiki.org/wiki/Characteristics_of_Finite_Tree/Condition_2 | https://proofwiki.org/wiki/Characteristics_of_Finite_Tree/Condition_2 | [
"Characteristics of Finite Tree"
] | [
"Definition:Tree (Graph Theory)/Finite",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Connected (Graph Theory)/Graph"
] | [
"Finite Connected Simple Graph is Tree iff Size is One Less than Order",
"Category:Characteristics of Finite Tree"
] |
proofwiki-21516 | Characteristics of Finite Tree/Condition 3 | :$T$ is a finite tree {{iff}} two arbitrary vertices of $T$ are connected by exactly one path. | This is an instance of:
:Path in Tree is Unique.
{{qed}}
Category:Characteristics of Finite Tree
t8zmbtb1155wc42mbeamf4bjkq0pgle | :$T$ is a [[Definition:Finite Tree|finite tree]] {{iff}} two arbitrary [[Definition:Vertex of Graph|vertices]] of $T$ are [[Definition:Connected Vertices|connected]] by exactly one [[Definition:Path (Graph Theory)|path]]. | This is an instance of:
:[[Path in Tree is Unique]].
{{qed}}
[[Category:Characteristics of Finite Tree]]
t8zmbtb1155wc42mbeamf4bjkq0pgle | Characteristics of Finite Tree/Condition 3 | https://proofwiki.org/wiki/Characteristics_of_Finite_Tree/Condition_3 | https://proofwiki.org/wiki/Characteristics_of_Finite_Tree/Condition_3 | [
"Characteristics of Finite Tree"
] | [
"Definition:Tree (Graph Theory)/Finite",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Connected (Graph Theory)/Vertices",
"Definition:Path (Graph Theory)"
] | [
"Path in Tree is Unique",
"Category:Characteristics of Finite Tree"
] |
proofwiki-21517 | Characteristics of Finite Tree/Condition 4 | :$T$ is a finite tree {{iff}} $T$ has no circuits, but adding one edge creates a cycle. | === Sufficient Condition ===
Let $T = \struct {V, E}$ be a finite tree.
Then by definition $T$ is connected and has no circuits.
Let $u, v \in V$ be any two vertices of $T$.
Let $P = \tuple {u, u_1, u_2, \ldots, u_{n - 1}, v}$ be a path from $u$ to $v$.
Let a new edges $\set {u, v}$ be added.
Then $\tuple {u, u_1, u_2,... | :$T$ is a [[Definition:Finite Tree|finite tree]] {{iff}} $T$ has no [[Definition:Circuit (Graph Theory)|circuits]], but adding one [[Definition:Edge of Graph|edge]] creates a [[Definition:Cycle (Graph Theory)|cycle]]. | === Sufficient Condition ===
Let $T = \struct {V, E}$ be a [[Definition:Finite Tree|finite tree]].
Then by definition $T$ is [[Definition:Connected Graph|connected]] and has no [[Definition:Circuit (Graph Theory)|circuits]].
Let $u, v \in V$ be any two [[Definition:Vertex of Graph|vertices]] of $T$.
Let $P = \tuple... | Characteristics of Finite Tree/Condition 4 | https://proofwiki.org/wiki/Characteristics_of_Finite_Tree/Condition_4 | https://proofwiki.org/wiki/Characteristics_of_Finite_Tree/Condition_4 | [
"Characteristics of Finite Tree"
] | [
"Definition:Tree (Graph Theory)/Finite",
"Definition:Circuit (Graph Theory)",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Cycle (Graph Theory)"
] | [
"Definition:Tree (Graph Theory)/Finite",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Circuit (Graph Theory)",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Path (Graph Theory)",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Cycle (Graph Theory)",
"Definition:Circuit (Graph T... |
proofwiki-21518 | Simple Graph with no Circuits and Size One Less than Order is Connected | Let $G$ be a simple graph of order $n$ such that $G$ has no circuits.
Let the size of $G$ be $n - 1$.
Then $T$ is connected. | {{AimForCont}} $G$ is not connected.
Then $G$ has at least $2$ components.
By the Pigeonhole Principle, at least one of those components has at least as many edges as vertices.
Hence by Finite Connected Simple Graph is Tree iff Size is One Less than Order, that component is not a tree.
Hence as that component is itself... | Let $G$ be a [[Definition:Simple Graph|simple graph]] of [[Definition:Order of Graph|order]] $n$ such that $G$ has no [[Definition:Circuit (Graph Theory)|circuits]].
Let the [[Definition:Size of Graph|size]] of $G$ be $n - 1$.
Then $T$ is [[Definition:Connected Graph|connected]]. | {{AimForCont}} $G$ is not [[Definition:Connected Graph|connected]].
Then $G$ has at least $2$ [[Definition:Component of Graph|components]].
By the [[Pigeonhole Principle]], at least one of those [[Definition:Component of Graph|components]] has at least as many [[Definition:Edge of Graph|edges]] as [[Definition:Vertex... | Simple Graph with no Circuits and Size One Less than Order is Connected | https://proofwiki.org/wiki/Simple_Graph_with_no_Circuits_and_Size_One_Less_than_Order_is_Connected | https://proofwiki.org/wiki/Simple_Graph_with_no_Circuits_and_Size_One_Less_than_Order_is_Connected | [
"Simple Graphs",
"Connectedness (Graph Theory)"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Circuit (Graph Theory)",
"Definition:Graph (Graph Theory)/Size",
"Definition:Connected (Graph Theory)/Graph"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Component of Graph",
"Dirichlet's Box Principle/Corollary",
"Definition:Component of Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Vertex",
"Finite Connected Simple Graph is Tree iff Size is One Less than Order",... |
proofwiki-21519 | Propositional Calculus is Decidable | The '''propositional calculus''' is a decidable system. | The Method of Truth Tables is an effective procedure for determining the validity of propositional formulas with respect to boolean interpretations.
{{finish}} | The '''[[Definition:Propositional Calculus|propositional calculus]]''' is a [[Definition:Decidable Formal System|decidable]] system. | The [[Method of Truth Tables]] is an [[Definition:Effective Procedure|effective procedure]] for determining the [[Definition:Valid (Boolean Interpretation)|validity]] of [[Definition:Propositional Formula|propositional formulas]] with respect to [[Definition:Boolean Interpretation|boolean interpretations]].
{{finish}} | Propositional Calculus is Decidable | https://proofwiki.org/wiki/Propositional_Calculus_is_Decidable | https://proofwiki.org/wiki/Propositional_Calculus_is_Decidable | [
"Propositional Logic",
"Decision Problems",
"Decidability"
] | [
"Definition:Propositional Logic",
"Definition:Decidable/Formal System"
] | [
"Method of Truth Tables",
"Definition:Effective Procedure",
"Definition:Boolean Interpretation/Formal Semantics",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Boolean Interpretation"
] |
proofwiki-21520 | Join Semilattice has Smallest Element iff has Identity | Let $\struct{S, \vee, \preceq}$ be a join semilattice.
Let $s \in S$.
Then:
:$s$ is the smallest element of $S$
{{iff}}
:$s$ is the identity in $\struct{S, \vee}$. | By definition of smallest element:
:$s$ is the smallest element of $S$
{{iff}}:
:$\forall t \in S : s \preceq t$
We have:
{{begin-eqn}}
{{eqn | q = \forall t \in S
| l = s \preceq t
| o = \iff
| r = t = s \vee t
| c = Successor is Supremum
}}
{{eqn | o = \iff
| r = t = t \vee s
| c ... | Let $\struct{S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $s \in S$.
Then:
:$s$ is the [[Definition:Smallest Element|smallest element]] of $S$
{{iff}}
:$s$ is the [[Definition:Identity Element|identity]] in $\struct{S, \vee}$. | By definition of [[Definition:Smallest Element|smallest element]]:
:$s$ is the [[Definition:Smallest Element|smallest element]] of $S$
{{iff}}:
:$\forall t \in S : s \preceq t$
We have:
{{begin-eqn}}
{{eqn | q = \forall t \in S
| l = s \preceq t
| o = \iff
| r = t = s \vee t
| c = [[Successor... | Join Semilattice has Smallest Element iff has Identity | https://proofwiki.org/wiki/Join_Semilattice_has_Smallest_Element_iff_has_Identity | https://proofwiki.org/wiki/Join_Semilattice_has_Smallest_Element_iff_has_Identity | [
"Join Semilattices"
] | [
"Definition:Join Semilattice",
"Definition:Smallest Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Smallest Element",
"Definition:Smallest Element",
"Successor is Supremum",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Category:Join Semilattices"
] |
proofwiki-21521 | Join Semilattice is Dual to Meet Semilattice | Let $\struct {S, \preceq}$ be an ordered set.
The following are dual statements:
:$\struct{S, \vee, \preceq}$ is a join semilattice, where $\vee$ is the join for all $a,b \in S$
:$\struct{S, \wedge, \preceq}$ is a meet semilattice, where $\wedge$ is the meet for all $a,b \in S$ | By definition of join semilattice:
:$\struct{S, \vee, \preceq}$ is a join semilattice
{{iff}}:
:$\forall a,b \in S : a \vee b \in S$, where $a \vee b$ is the join of $a$ and $b$
The dual of this statement is:
:$\forall a,b \in S : a \wedge b \in S$, where $a \wedge b$ is the meet of $a$ and $b$
by Dual Pairs (Orde... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
The following are [[Definition:Dual Statement (Order Theory)|dual statements]]:
:$\struct{S, \vee, \preceq}$ is a [[Definition:Join Semilattice|join semilattice]], where $\vee$ is the [[Definition:Join (Order Theory)|join]] for all $a,b \in S$... | By definition of [[Definition:Join Semilattice|join semilattice]]:
:$\struct{S, \vee, \preceq}$ is a [[Definition:Join Semilattice|join semilattice]]
{{iff}}:
:$\forall a,b \in S : a \vee b \in S$, where $a \vee b$ is the [[Definition:Join (Order Theory)|join]] of $a$ and $b$
The [[Definition:Dual Statement (Or... | Join Semilattice is Dual to Meet Semilattice | https://proofwiki.org/wiki/Join_Semilattice_is_Dual_to_Meet_Semilattice | https://proofwiki.org/wiki/Join_Semilattice_is_Dual_to_Meet_Semilattice | [
"Join Semilattices",
"Meet Semilattices",
"Dual Pairs (Order Theory)"
] | [
"Definition:Ordered Set",
"Definition:Dual Statement (Order Theory)",
"Definition:Join Semilattice",
"Definition:Join (Order Theory)",
"Definition:Meet Semilattice",
"Definition:Meet (Order Theory)"
] | [
"Definition:Join Semilattice",
"Definition:Join Semilattice",
"Definition:Join (Order Theory)",
"Definition:Dual Statement (Order Theory)",
"Definition:Meet (Order Theory)",
"Dual Pairs (Order Theory)",
"Definition:Meet Semilattice",
"Definition:Join Semilattice"
] |
proofwiki-21522 | Meet Semilattice has Greatest Element iff has Identity | Let $\struct{S, \wedge, \preceq}$ be a meet semilattice.
Let $s \in S$.
Then:
:$s$ is the greatest element of $S$
{{iff}}
:$s$ is the identity in $\struct{S, \wedge}$. | This is the dual statement of Join Semilattice has Smallest Element iff has Identity by Dual Pairs (Order Theory).
The result follows from the Duality Principle.
{{qed}}
Category:Meet Semilattices
a4bqwqvlw5fgems8dbfq6rz4d2iwn3f | Let $\struct{S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $s \in S$.
Then:
:$s$ is the [[Definition:Greatest Element|greatest element]] of $S$
{{iff}}
:$s$ is the [[Definition:Identity Element|identity]] in $\struct{S, \wedge}$. | This is the [[Definition:Dual Statement (Order Theory)|dual statement]] of [[Join Semilattice has Smallest Element iff has Identity]] by [[Dual Pairs (Order Theory)]].
The result follows from the [[Duality Principle (Order Theory)|Duality Principle]].
{{qed}}
[[Category:Meet Semilattices]]
a4bqwqvlw5fgems8dbfq6rz4d2... | Meet Semilattice has Greatest Element iff has Identity | https://proofwiki.org/wiki/Meet_Semilattice_has_Greatest_Element_iff_has_Identity | https://proofwiki.org/wiki/Meet_Semilattice_has_Greatest_Element_iff_has_Identity | [
"Meet Semilattices"
] | [
"Definition:Meet Semilattice",
"Definition:Greatest Element",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Dual Statement (Order Theory)",
"Join Semilattice has Smallest Element iff has Identity",
"Dual Pairs (Order Theory)",
"Duality Principle (Order Theory)",
"Category:Meet Semilattices"
] |
proofwiki-21523 | Quotients of Homeomorphic Spaces are Homeomorphic | Let $X, Y$ be topological spaces.
Let $\phi : X \to Y$ be a homeomorphism from $X$ to $Y$.
Let:
:$\RR_X \subseteq X \times X$
:$\RR_Y \subseteq Y \times Y$
be equivalence relations on $X$ and $Y$, respectively.
Suppose that, for all $x, x' \in X$:
:$\map {\RR_X} {x, x'} \iff \map {\RR_Y} {\map \phi x, \map \phi {x'}}$
... | Let the mapping $\psi : X / \RR_X \to Y / \RR_Y$ be defined as:
:$\map \psi {\eqclass x {\RR_X}} = \eqclass {\map \phi x} {\RR_Y}$
In order for $\psi$ to be well-defined, the image needs to be independent of the choice of representative $x$.
But, for $x, x' \in X$ such that $\map {\RR_X} {x, x'}$, we have:
:$\map {\RR_... | Let $X, Y$ be [[Definition:Topological Space|topological spaces]].
Let $\phi : X \to Y$ be a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]] from $X$ to $Y$.
Let:
:$\RR_X \subseteq X \times X$
:$\RR_Y \subseteq Y \times Y$
be [[Definition:Equivalence Relation|equivalence relations]] on $X$ and $Y$, re... | Let the [[Definition:Mapping|mapping]] $\psi : X / \RR_X \to Y / \RR_Y$ be defined as:
:$\map \psi {\eqclass x {\RR_X}} = \eqclass {\map \phi x} {\RR_Y}$
In order for $\psi$ to be [[Definition:Well-Defined|well-defined]], the [[Definition:Image of Element under Mapping|image]] needs to be independent of the choice of ... | Quotients of Homeomorphic Spaces are Homeomorphic | https://proofwiki.org/wiki/Quotients_of_Homeomorphic_Spaces_are_Homeomorphic | https://proofwiki.org/wiki/Quotients_of_Homeomorphic_Spaces_are_Homeomorphic | [
"Homeomorphisms (Topological Spaces)",
"Quotient Spaces (Topology)"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Equivalence Relation",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Mapping",
"Definition:Well-Defined",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Equivalence Class/Representative",
"Definition:Inverse Mapping",
"Definition:Well-Defined",
"Definition:Inverse Mapping",
"Definition:Continuous Mapping",
"Definition:Open Set",
"Definitio... |
proofwiki-21524 | Joins of Homeomorphic Spaces are Homeomorphic | Let $X, Y, X', Y'$ be topological spaces.
Suppose that:
:$X \sim X'$
:$Y \sim Y'$
where $\sim$ denotes homeomorphic spaces.
Then:
:$(X \ast Y) \sim (X' \ast Y')$
where $\ast$ denotes the join. | By definition of homeomorphic, let:
:$\phi_X : X \to X'$
:$\phi_Y : Y \to Y'$
be homeomorphisms.
Additionally, let:
:$I_{\closedint 0 1} : \closedint 0 1 \to \closedint 0 1$
be the identity mapping.
By Identity Mapping is Homeomorphism, we have that $I_{\closedint 0 1}$ is a homeomorphism.
Let $\phi : X \times Y \times... | Let $X, Y, X', Y'$ be [[Definition:Topological Space|topological spaces]].
Suppose that:
:$X \sim X'$
:$Y \sim Y'$
where $\sim$ denotes [[Definition:Homeomorphic Topological Spaces|homeomorphic spaces]].
Then:
:$(X \ast Y) \sim (X' \ast Y')$
where $\ast$ denotes the [[Definition:Join (Topology)|join]]. | By definition of [[Definition:Homeomorphic Topological Spaces|homeomorphic]], let:
:$\phi_X : X \to X'$
:$\phi_Y : Y \to Y'$
be [[Definition:Homeomorphism (Topological Spaces)|homeomorphisms]].
Additionally, let:
:$I_{\closedint 0 1} : \closedint 0 1 \to \closedint 0 1$
be the [[Definition:Identity Mapping|identity ma... | Joins of Homeomorphic Spaces are Homeomorphic | https://proofwiki.org/wiki/Joins_of_Homeomorphic_Spaces_are_Homeomorphic | https://proofwiki.org/wiki/Joins_of_Homeomorphic_Spaces_are_Homeomorphic | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Join (Topology)"
] | [
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Identity Mapping",
"Identity Mapping is Homeomorphism",
"Definition:Homeomorphism/Topological Spaces",
"Products of Homeomorphic Spaces are Homeomorphic",
"Definition:Homeomorphism/Topological Space... |
proofwiki-21525 | Cones on Homeomorphic Spaces are Homeomorphic | Let $X, Y$ be topological spaces.
Suppose:
:$X \sim Y$
denoting that $X$ is homeomorphic to $Y$.
Then:
:$C X \sim C Y$
where $C X$ denotes the cone on $X$. | Let $T$ be the trivial topological space used in the definition of cone.
We have:
:$T \sim T$ by Homeomorphism Relation is Equivalence
:$X \sim Y$ {{hypothesis}}
Then, by Joins of Homeomorphic Spaces are Homeomorphic:
:$C X \sim C Y$
{{qed}}
Category:Homeomorphisms (Topological Spaces)
2nvzextr68ikw555lumqm0f4yb0dku7 | Let $X, Y$ be [[Definition:Topological Space|topological spaces]].
Suppose:
:$X \sim Y$
denoting that $X$ is [[Definition:Homeomorphic Topological Spaces|homeomorphic]] to $Y$.
Then:
:$C X \sim C Y$
where $C X$ denotes the [[Definition:Cone (Topology)|cone]] on $X$. | Let $T$ be the [[Definition:Trivial Topological Space|trivial topological space]] used in the definition of [[Definition:Cone (Topology)|cone]].
We have:
:$T \sim T$ by [[Homeomorphism Relation is Equivalence]]
:$X \sim Y$ {{hypothesis}}
Then, by [[Joins of Homeomorphic Spaces are Homeomorphic]]:
:$C X \sim C Y$
{{qe... | Cones on Homeomorphic Spaces are Homeomorphic | https://proofwiki.org/wiki/Cones_on_Homeomorphic_Spaces_are_Homeomorphic | https://proofwiki.org/wiki/Cones_on_Homeomorphic_Spaces_are_Homeomorphic | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Cone (Topology)"
] | [
"Definition:Trivial Topological Space",
"Definition:Cone (Topology)",
"Homeomorphism Relation is Equivalence",
"Joins of Homeomorphic Spaces are Homeomorphic",
"Category:Homeomorphisms (Topological Spaces)"
] |
proofwiki-21526 | Compact Convex Sets with Nonempty Interior are Homeomorphic | Let $n \in \N_{> 0}$.
Let $T, T' \subseteq \R^n$ be compact convex subsets of real Euclidean $n$-space.
Then, $T$ is homeomorphic to $T'$. | By Boundary of Compact Convex Set with Nonempty Interior is Homeomorphic to Sphere:
:$\partial T \sim \Bbb S^{n - 1}$
:$\partial T' \sim \Bbb S^{n - 1}$
Thus, by Homeomorphism Relation is Equivalence:
:$\partial T \sim \partial T'$
Hence, by Cones on Homeomorphic Spaces are Homeomorphic:
:$C \partial T \sim C \partial ... | Let $n \in \N_{> 0}$.
Let $T, T' \subseteq \R^n$ be [[Definition:Compact Set (Topology)|compact]] [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Subset|subsets]] of [[Definition:Real Euclidean Space|real Euclidean $n$-space]].
Then, $T$ is [[Definition:Homeomorphic Topological Spaces|homeomorphic]] to ... | By [[Boundary of Compact Convex Set with Nonempty Interior is Homeomorphic to Sphere]]:
:$\partial T \sim \Bbb S^{n - 1}$
:$\partial T' \sim \Bbb S^{n - 1}$
Thus, by [[Homeomorphism Relation is Equivalence]]:
:$\partial T \sim \partial T'$
Hence, by [[Cones on Homeomorphic Spaces are Homeomorphic]]:
:$C \partial T \s... | Compact Convex Sets with Nonempty Interior are Homeomorphic | https://proofwiki.org/wiki/Compact_Convex_Sets_with_Nonempty_Interior_are_Homeomorphic | https://proofwiki.org/wiki/Compact_Convex_Sets_with_Nonempty_Interior_are_Homeomorphic | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Euclidean Space/Real",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Boundary of Compact Convex Set with Nonempty Interior is Homeomorphic to Sphere",
"Homeomorphism Relation is Equivalence",
"Cones on Homeomorphic Spaces are Homeomorphic",
"Compact Convex Set with Nonempty Interior is Homeomorphic to Cone on Boundary",
"Homeomorphism Relation is Equivalence",
"Category:H... |
proofwiki-21527 | First Order ODE/y' = x | The first order ODE:
:$\dfrac {\d y} {\d x} = x$
has the general solution:
:$y = \dfrac {x^2} 2 + C$ | {{begin-eqn}}
{{eqn | l = \dfrac {\d y} {\d x}
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \rd y
| r = \int x \rd x
| c = Solution to Separable Differential Equation
}}
{{eqn | ll= \leadsto
| l = y
| r = \dfrac {x^2} 2 + C
| c = Primitive of Constant, Primitive of P... | The [[Definition:First Order ODE|first order ODE]]:
:$\dfrac {\d y} {\d x} = x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = \dfrac {x^2} 2 + C$ | {{begin-eqn}}
{{eqn | l = \dfrac {\d y} {\d x}
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \rd y
| r = \int x \rd x
| c = [[Solution to Separable Differential Equation]]
}}
{{eqn | ll= \leadsto
| l = y
| r = \dfrac {x^2} 2 + C
| c = [[Primitive of Constant]], [[Prim... | First Order ODE/y' = x | https://proofwiki.org/wiki/First_Order_ODE/y'_=_x | https://proofwiki.org/wiki/First_Order_ODE/y'_=_x | [
"Examples of Linear First Order ODEs",
"Examples of Solutions to Separable Differential Equation"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Solution to Separable Differential Equation",
"Primitive of Constant",
"Primitive of Power"
] |
proofwiki-21528 | Fixed-Point Property is Topological | Let $T \sim T'$ be homeomorphic topological spaces.
Suppose every continuous $f : T \to T$ has a fixed point.
Then, every continuous $g : T' \to T'$ also has a fixed point.
{{improve|It's customary on {{ProofWiki}} to utilise the terminology of the title in the page, otherwise a casual reader would say "What has being ... | Let $\phi : T \to T'$ be a homeomorphism.
Let $g : T' \to T'$ be an arbitrary continuous mapping.
Consider $f_g : T \to T$ defined as:
:$\map {f_g} x = \map {\phi^{-1}} {\map g {\map \phi x}}$
By definition of homeomorphism and Composite of Continuous Mappings is Continuous:
:$f_g$ is continuous
Therefore, {{hypothesis... | Let $T \sim T'$ be [[Definition:Homeomorphic Topological Spaces|homeomorphic]] [[Definition:Topological Space|topological spaces]].
Suppose every [[Definition:Everywhere Continuous Mapping (Topology)|continuous]] $f : T \to T$ has a [[Definition:Fixed Point|fixed point]].
Then, every [[Definition:Everywhere Continuo... | Let $\phi : T \to T'$ be a [[Definition:Homeomorphism|homeomorphism]].
Let $g : T' \to T'$ be an arbitrary [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]].
Consider $f_g : T \to T$ defined as:
:$\map {f_g} x = \map {\phi^{-1}} {\map g {\map \phi x}}$
By definition of [[Definition:Homeomorp... | Fixed-Point Property is Topological | https://proofwiki.org/wiki/Fixed-Point_Property_is_Topological | https://proofwiki.org/wiki/Fixed-Point_Property_is_Topological | [
"Fixed Point Theorems",
"Topological Properties"
] | [
"Definition:Homeomorphism/Topological Spaces",
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Fixed Point",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Fixed Point"
] | [
"Definition:Homeomorphism",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Homeomorphism",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Fixed Point",
"Definition:Fixed Point",
"Category:Fixed Point Theorems",
... |
proofwiki-21529 | Paths Crossing Rectangle Necessarily Meet | Let $R = \closedint a b \times \closedint c d$ be a closed rectangle.
Let $p, q : \closedint {-1} 1 \to R$ be paths in $R$.
Suppose:
* $\map {p_1} {-1} = a$
* $\map {p_1} 1 = b$
* $\map {q_2} {-1} = c$
* $\map {q_2} 1 = d$
where:
:$\map p t = \tuple {\map {p_1} t, \map {p_2} t}$
:$\map q t = \tuple {\map {q_1} t, \map ... | {{AimForCont}}, suppose there are no such $t_p, t_q$.
Define $N : \closedint {-1} 1^2 \to \R_{\ge 0}$ as:
:$\map N {s, t} = \map \max {\size {\map {p_1} s - \map {q_1} t}, \size {\map {p_2} s - \map {q_2} t}}$
By our assumption:
:$\map p s \ne \map q t$
for all $s, t \in \closedint {-1} 1$.
Therefore, for each $s, t$, ... | Let $R = \closedint a b \times \closedint c d$ be a [[Definition:Closed Rectangle|closed rectangle]].
Let $p, q : \closedint {-1} 1 \to R$ be [[Definition:Path (Topology)|paths]] in $R$.
Suppose:
* $\map {p_1} {-1} = a$
* $\map {p_1} 1 = b$
* $\map {q_2} {-1} = c$
* $\map {q_2} 1 = d$
where:
:$\map p t = \tuple {\map... | {{AimForCont}}, suppose there are no such $t_p, t_q$.
Define $N : \closedint {-1} 1^2 \to \R_{\ge 0}$ as:
:$\map N {s, t} = \map \max {\size {\map {p_1} s - \map {q_1} t}, \size {\map {p_2} s - \map {q_2} t}}$
By our assumption:
:$\map p s \ne \map q t$
for all $s, t \in \closedint {-1} 1$.
Therefore, for each $s, t... | Paths Crossing Rectangle Necessarily Meet | https://proofwiki.org/wiki/Paths_Crossing_Rectangle_Necessarily_Meet | https://proofwiki.org/wiki/Paths_Crossing_Rectangle_Necessarily_Meet | [
"Paths (Topology)"
] | [
"Definition:Closed Rectangle",
"Definition:Path (Topology)"
] | [
"Combination Theorem for Continuous Mappings/Metric Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Well-Defined/Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Absolute Value Function is Completely Multiplicative",
"Brouwer's Fixed Point Theorem/General Case",
"Co... |
proofwiki-21530 | Sum of nth Fibonacci Number over nth Power of 2 | :$\ds \sum_{n \mathop = 0}^\infty \frac {F_n} {2^n} = 2$
where $F_n$ is the $n$th Fibonacci number. | Let us define a sample space which satisfies the Kolmogorov Axioms such that it is the set of all combinations of flipping a fair coin until you receive two heads in a row.
Let $X_n$ be the event of some outcome from flipping $n$ fair coins in a row, then $\map \Pr {X_n} = \dfrac 1 {2^n}$.
In the sample space defined ... | :$\ds \sum_{n \mathop = 0}^\infty \frac {F_n} {2^n} = 2$
where $F_n$ is the $n$th [[Definition:Fibonacci Number|Fibonacci number]]. | Let us define a [[Definition:Sample Space|sample space]] which satisfies the [[Definition:Probability Measure/Definition 3|Kolmogorov Axioms]] such that it is the set of all combinations of [[Definition:Coin-Tossing|flipping]] a fair [[Definition:Coin|coin]] until you receive two [[Definition:Head of Coin|heads]] in a... | Sum of nth Fibonacci Number over nth Power of 2/Proof 1 | https://proofwiki.org/wiki/Sum_of_nth_Fibonacci_Number_over_nth_Power_of_2 | https://proofwiki.org/wiki/Sum_of_nth_Fibonacci_Number_over_nth_Power_of_2/Proof_1 | [
"Sum of nth Fibonacci Number over nth Power of 2",
"Fibonacci Numbers"
] | [
"Definition:Fibonacci Number"
] | [
"Definition:Sample Space",
"Definition:Probability Measure/Definition 3",
"Definition:Coin/Coin-Tossing",
"Definition:Coin",
"Definition:Coin/Head",
"Definition:Coin/Coin-Tossing",
"Definition:Coin",
"Definition:Sample Space",
"Definition:Sample Space",
"Definition:Matrix/Row",
"Definition:Sampl... |
proofwiki-21531 | Sum of nth Fibonacci Number over nth Power of 2 | :$\ds \sum_{n \mathop = 0}^\infty \frac {F_n} {2^n} = 2$
where $F_n$ is the $n$th Fibonacci number. | From the Euler-Binet Formula, we have:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5}$
Therefore:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \frac{F_n} {2^n}
| r = \sum_{n \mathop = 0}^\infty \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5 \times 2^n}
| c = Euler-Binet Formula
}}
{{eqn | ... | :$\ds \sum_{n \mathop = 0}^\infty \frac {F_n} {2^n} = 2$
where $F_n$ is the $n$th [[Definition:Fibonacci Number|Fibonacci number]]. | From the [[Euler-Binet Formula]], we have:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5}$
Therefore:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \frac{F_n} {2^n}
| r = \sum_{n \mathop = 0}^\infty \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5 \times 2^n}
| c = [[Euler-Binet Formula]]
}... | Sum of nth Fibonacci Number over nth Power of 2/Proof 2 | https://proofwiki.org/wiki/Sum_of_nth_Fibonacci_Number_over_nth_Power_of_2 | https://proofwiki.org/wiki/Sum_of_nth_Fibonacci_Number_over_nth_Power_of_2/Proof_2 | [
"Sum of nth Fibonacci Number over nth Power of 2",
"Fibonacci Numbers"
] | [
"Definition:Fibonacci Number"
] | [
"Euler-Binet Formula",
"Euler-Binet Formula",
"Exponent Combination Laws/Product of Powers",
"Sum of Infinite Geometric Sequence",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-21532 | Sum of nth Fibonacci Number over nth Power of 2 | :$\ds \sum_{n \mathop = 0}^\infty \frac {F_n} {2^n} = 2$
where $F_n$ is the $n$th Fibonacci number. | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{\infty} F_k z^k
| r = \dfrac z {1 - z - z^2}
| c = Generating Function for Fibonacci Numbers holds for all $\cmod {z} < \dfrac 1 {\phi} \approx 0.62$
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 0}^{\infty} \frac {F_k} {2^k}
| r = \dfrac {\dfrac... | :$\ds \sum_{n \mathop = 0}^\infty \frac {F_n} {2^n} = 2$
where $F_n$ is the $n$th [[Definition:Fibonacci Number|Fibonacci number]]. | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^{\infty} F_k z^k
| r = \dfrac z {1 - z - z^2}
| c = [[Generating Function for Fibonacci Numbers]] holds for all $\cmod {z} < \dfrac 1 {\phi} \approx 0.62$
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 0}^{\infty} \frac {F_k} {2^k}
| r = \dfrac {\d... | Sum of nth Fibonacci Number over nth Power of 2/Proof 3 | https://proofwiki.org/wiki/Sum_of_nth_Fibonacci_Number_over_nth_Power_of_2 | https://proofwiki.org/wiki/Sum_of_nth_Fibonacci_Number_over_nth_Power_of_2/Proof_3 | [
"Sum of nth Fibonacci Number over nth Power of 2",
"Fibonacci Numbers"
] | [
"Definition:Fibonacci Number"
] | [
"Generating Function for Fibonacci Numbers"
] |
proofwiki-21533 | Sum of nth Fibonacci Number over nth Power of 2 | :$\ds \sum_{n \mathop = 0}^\infty \frac {F_n} {2^n} = 2$
where $F_n$ is the $n$th Fibonacci number. | First, let:
:$S = \ds \sum_{n \mathop = 0}^\infty \frac {F_n} {2^n}$
Some sundry results:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n F_k
| r = F_{n + 2} - 1
| c = Sum of Sequence of Fibonacci Numbers
}}
{{eqn | n = 1
| ll= \leadsto
| l = F_{n + 2}
| r = \paren {\sum_{k \mathop = 0}^n... | :$\ds \sum_{n \mathop = 0}^\infty \frac {F_n} {2^n} = 2$
where $F_n$ is the $n$th [[Definition:Fibonacci Number|Fibonacci number]]. | First, let:
:$S = \ds \sum_{n \mathop = 0}^\infty \frac {F_n} {2^n}$
Some sundry results:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n F_k
| r = F_{n + 2} - 1
| c = [[Sum of Sequence of Fibonacci Numbers]]
}}
{{eqn | n = 1
| ll= \leadsto
| l = F_{n + 2}
| r = \paren {\sum_{k \mathop ... | Sum of nth Fibonacci Number over nth Power of 2/Proof 4 | https://proofwiki.org/wiki/Sum_of_nth_Fibonacci_Number_over_nth_Power_of_2 | https://proofwiki.org/wiki/Sum_of_nth_Fibonacci_Number_over_nth_Power_of_2/Proof_4 | [
"Sum of nth Fibonacci Number over nth Power of 2",
"Fibonacci Numbers"
] | [
"Definition:Fibonacci Number"
] | [
"Sum of Sequence of Fibonacci Numbers",
"Sum of Infinite Geometric Sequence",
"Translation of Index Variable of Summation",
"Definition:Absolutely Convergent Series",
"Ratio Test",
"Definition:Summation",
"Definition:Absolutely Convergent Series",
"Fubini's Theorem for Infinite Sums"
] |
proofwiki-21534 | Second Order ODE/y'' = f(x) | The second order ODE:
:$\dfrac {\d^2 y} {\d x^2} = \map f x$
has the general solution:
:$\ds y = \iint \map f x \rd x \rd x + C_1 x + C_2$
where $C_1$ and $C_2$ are arbitrary constants. | {{begin-eqn}}
{{eqn | l = \dfrac {\d^2 y} {\d x^2}
| r = \map f x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \dfrac {\d^2 y} {\d x^2} \rd y
| r = \int \map f x \rd x
| c = Solution to Separable Differential Equation
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d y} {\d x}
| r = \int \... | The [[Definition:Second Order ODE|second order ODE]]:
:$\dfrac {\d^2 y} {\d x^2} = \map f x$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$\ds y = \iint \map f x \rd x \rd x + C_1 x + C_2$
where $C_1$ and $C_2$ are [[Definition:Arbitrary Constant|arbitrary constants]]. | {{begin-eqn}}
{{eqn | l = \dfrac {\d^2 y} {\d x^2}
| r = \map f x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \dfrac {\d^2 y} {\d x^2} \rd y
| r = \int \map f x \rd x
| c = [[Solution to Separable Differential Equation]]
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d y} {\d x}
| r = \i... | Second Order ODE/y'' = f(x) | https://proofwiki.org/wiki/Second_Order_ODE/y''_=_f(x) | https://proofwiki.org/wiki/Second_Order_ODE/y''_=_f(x) | [
"Examples of Second Order ODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Arbitrary Constant"
] | [
"Solution to Separable Differential Equation",
"Definition:Arbitrary Constant",
"Solution to Separable Differential Equation",
"Linear Combination of Integrals/Indefinite",
"Primitive of Power",
"Definition:Arbitrary Constant"
] |
proofwiki-21535 | Recurrence Relation for Polygamma Function | :$\map {\psi_n} {z + 1} = \map {\psi_n} z + \paren {-1}^n n! z^{-n - 1}$
where:
:$\psi_n$ denote the $n$th polygamma function
:$z \in \C \setminus \Z_{\le 0}$. | {{begin-eqn}}
{{eqn | l = \map \Gamma {z + 1}
| r = z \map \Gamma z
| c = Gamma Difference Equation
}}
{{eqn | ll= \leadsto
| l = \map \ln {\map \Gamma {z + 1} }
| r = \map \ln {z \map \Gamma z}
| c = applying $\ln$ on both sides
}}
{{eqn | r = \ln z + \map \ln {\map \Gamma z}
| c = Sum of L... | :$\map {\psi_n} {z + 1} = \map {\psi_n} z + \paren {-1}^n n! z^{-n - 1}$
where:
:$\psi_n$ denote the $n$th [[Definition:Polygamma Function|polygamma function]]
:$z \in \C \setminus \Z_{\le 0}$. | {{begin-eqn}}
{{eqn | l = \map \Gamma {z + 1}
| r = z \map \Gamma z
| c = [[Gamma Difference Equation]]
}}
{{eqn | ll= \leadsto
| l = \map \ln {\map \Gamma {z + 1} }
| r = \map \ln {z \map \Gamma z}
| c = applying $\ln$ on both sides
}}
{{eqn | r = \ln z + \map \ln {\map \Gamma z}
| c = [[Su... | Recurrence Relation for Polygamma Function/Proof 2 | https://proofwiki.org/wiki/Recurrence_Relation_for_Polygamma_Function | https://proofwiki.org/wiki/Recurrence_Relation_for_Polygamma_Function/Proof_2 | [
"Recurrence Relation for Polygamma Function",
"Polygamma Function",
"Recurrence Relations"
] | [
"Definition:Polygamma Function"
] | [
"Gamma Difference Equation",
"Sum of Logarithms",
"Definition:Differentiation",
"Derivative of Natural Logarithm Function",
"Derivative of Composite Function",
"Definition:Derivative/Higher Derivatives"
] |
proofwiki-21536 | Second Order ODE/y'' = f(y) | The second order ODE:
:$\dfrac {\d^2 y} {\d x^2} = \map f y$
has the general solution:
:$\dfrac {\sqrt 2} 2 \ds \int \dfrac {\d y} {\sqrt {\ds \int \map f y \rd y + C_1} } = x + C_2$
where $C_1$ and $C_2$ are arbitrary constants. | {{begin-eqn}}
{{eqn | l = \dfrac {\d^2 y} {\d x^2}
| r = \map f y
| c =
}}
{{eqn | ll= \leadsto
| l = 2 \dfrac {\d y} {\d x} \dfrac {\d^2 y} {\d x^2}
| r = 2 \map f y \dfrac {\d y} {\d x}
| c =
}}
{{eqn | ll= \leadsto
| l = \int 2 \dfrac {\d y} {\d x} \dfrac {\d^2 y} {\d x^2} \rd x... | The [[Definition:Second Order ODE|second order ODE]]:
:$\dfrac {\d^2 y} {\d x^2} = \map f y$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$\dfrac {\sqrt 2} 2 \ds \int \dfrac {\d y} {\sqrt {\ds \int \map f y \rd y + C_1} } = x + C_2$
where $C_1$ and $C_2$ are [[Definition:Arbitrar... | {{begin-eqn}}
{{eqn | l = \dfrac {\d^2 y} {\d x^2}
| r = \map f y
| c =
}}
{{eqn | ll= \leadsto
| l = 2 \dfrac {\d y} {\d x} \dfrac {\d^2 y} {\d x^2}
| r = 2 \map f y \dfrac {\d y} {\d x}
| c =
}}
{{eqn | ll= \leadsto
| l = \int 2 \dfrac {\d y} {\d x} \dfrac {\d^2 y} {\d x^2} \rd x... | Second Order ODE/y'' = f(y) | https://proofwiki.org/wiki/Second_Order_ODE/y''_=_f(y) | https://proofwiki.org/wiki/Second_Order_ODE/y''_=_f(y) | [
"Examples of Second Order ODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Arbitrary Constant"
] | [
"Solution to Separable Differential Equation"
] |
proofwiki-21537 | Lie Algebra is Anticommutative | Let $L$ be a Lie algebra.
Then $L$ is anticommutative:
:$\forall a, b \in L: \sqbrk {a, b} = -\sqbrk {b, a}$
where $\sqbrk {\, \cdot, \cdot \,}$ denotes the bilinear mapping over $L$. | {{begin-eqn}}
{{eqn | l = \sqbrk {a + b, a + b}
| r = \sqbrk {a, a + b} + \sqbrk {b, a + b}
| c = {{Defof|Bilinear Mapping}}
}}
{{eqn | ll= \leadsto
| l = \sqbrk {a + b, a + b}
| r = \sqbrk {a, a} + \sqbrk {a, b} + \sqbrk{b, a} + \sqbrk {b, b}
| c = {{Defof|Bilinear Mapping}}
}}
{{eqn | ll... | Let $L$ be a [[Definition:Lie Algebra|Lie algebra]].
Then $L$ is [[Definition:Anticommutative|anticommutative]]:
:$\forall a, b \in L: \sqbrk {a, b} = -\sqbrk {b, a}$
where $\sqbrk {\, \cdot, \cdot \,}$ denotes the [[Definition:Bilinear Mapping|bilinear mapping]] over $L$. | {{begin-eqn}}
{{eqn | l = \sqbrk {a + b, a + b}
| r = \sqbrk {a, a + b} + \sqbrk {b, a + b}
| c = {{Defof|Bilinear Mapping}}
}}
{{eqn | ll= \leadsto
| l = \sqbrk {a + b, a + b}
| r = \sqbrk {a, a} + \sqbrk {a, b} + \sqbrk{b, a} + \sqbrk {b, b}
| c = {{Defof|Bilinear Mapping}}
}}
{{eqn | ll... | Lie Algebra is Anticommutative | https://proofwiki.org/wiki/Lie_Algebra_is_Anticommutative | https://proofwiki.org/wiki/Lie_Algebra_is_Anticommutative | [
"Lie Theory",
"Anticommutativity"
] | [
"Definition:Lie Algebra",
"Definition:Anticommutative",
"Definition:Bilinear Mapping"
] | [] |
proofwiki-21538 | Two Planes have Line in Common/Hilbert's Axioms | Two planes $\alpha, \beta$ have no point in common or a straight line $a$ in common. | {{tidy}}
{{MissingLinks}}
By Law of Excluded Middle, $\alpha$ and $\beta$ have no point in common, or some point $A$ in common.
In the former case, the theorem holds trivially.
In the latter, by Axiom $\text I, 6$, there is a second point $B$ lying on both $\alpha$ and $\beta$.
By Axiom $I, 1$, there is a straight line... | Two planes $\alpha, \beta$ have no point in common or a straight line $a$ in common. | {{tidy}}
{{MissingLinks}}
By [[Law of Excluded Middle]], $\alpha$ and $\beta$ have no point in common, or some point $A$ in common.
In the former case, the theorem holds trivially.
In the latter, by [[Axiom:Hilbert's Axioms/Connection|Axiom $\text I, 6$]], there is a second point $B$ lying on both $\alpha$ and $\bet... | Two Planes have Line in Common/Hilbert's Axioms | https://proofwiki.org/wiki/Two_Planes_have_Line_in_Common/Hilbert's_Axioms | https://proofwiki.org/wiki/Two_Planes_have_Line_in_Common/Hilbert's_Axioms | [
"Hilbert's Axioms"
] | [] | [
"Law of Excluded Middle",
"Axiom:Hilbert's Axioms/Connection",
"Axiom:Hilbert's Axioms/Connection",
"Axiom:Hilbert's Axioms/Connection"
] |
proofwiki-21539 | Relation between Direction Cosines | Let $\LL$ be a line embedded in a Cartesian $3$-space.
Let $l$, $m$ and $n$ be the direction cosines of $\LL$ {{WRT}} the $x$-axis, $y$-axis and $z$-axis respectively.
Then:
:$l^2 + m^2 + n^2 = 1$ | Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space that is parallel to $\LL$.
Let the direction angles which $\mathbf r$ makes with the $x$-axis, $y$-axis and $z$-axis be $\alpha$, $\beta$ and $\gamma$ respectively.
From Components of Vector in terms of Direction Cosines:
:$l = \cos \alpha = \dfrac ... | Let $\LL$ be a [[Definition:Line|line]] embedded in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]].
Let $l$, $m$ and $n$ be the [[Definition:Direction Cosines|direction cosines]] of $\LL$ {{WRT}} the [[Definition:X-Axis|$x$-axis]], [[Definition:Y-Axis|$y$-axis]] and [[Definition:Z-Axis|$z$-axis]] respectively.... | Let $\mathbf r$ be a [[Definition:Vector Quantity|vector quantity]] embedded in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]] that is parallel to $\LL$.
Let the [[Definition:Direction Angles|direction angles]] which $\mathbf r$ makes with the [[Definition:X-Axis|$x$-axis]], [[Definition:Y-Axis|$y$-axis]] and ... | Relation between Direction Cosines | https://proofwiki.org/wiki/Relation_between_Direction_Cosines | https://proofwiki.org/wiki/Relation_between_Direction_Cosines | [
"Direction Cosines",
"Components of Vector in terms of Direction Cosines"
] | [
"Definition:Line",
"Definition:Cartesian 3-Space",
"Definition:Direction Cosines",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/Z-Axis"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Direction Angles",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/Z-Axis",
"Components of Vector in terms of Direction Cosines",
"Definition:Axis/X-Axis"
] |
proofwiki-21540 | Integral Form of Polygamma Function | Let $z$ be a complex number with a positive real part.
Then:
:$\ds \map {\psi_n} z = \paren {-1}^{n + 1} \int_0^\infty \frac {t^n e^{-z t} } {1 - e^{-t} } \rd t$
where $\map {\psi_n} z$ denotes the $n$th polygamma function. | From Gauss's Integral Form of Digamma Function, we have:
:$\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t$
Therefore:
{{begin-eqn}}
{{eqn | l = \map \psi z
| r = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t
| c =
}}
... | Let $z$ be a [[Definition:Complex Number|complex number]] with a [[Definition:Positive Real Number|positive]] [[Definition:Real Part|real part]].
Then:
:$\ds \map {\psi_n} z = \paren {-1}^{n + 1} \int_0^\infty \frac {t^n e^{-z t} } {1 - e^{-t} } \rd t$
where $\map {\psi_n} z$ denotes the $n$th [[Definition:Polygam... | From [[Gauss's Integral Form of Digamma Function]], we have:
:$\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t$
Therefore:
{{begin-eqn}}
{{eqn | l = \map \psi z
| r = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t
| c ... | Integral Form of Polygamma Function | https://proofwiki.org/wiki/Integral_Form_of_Polygamma_Function | https://proofwiki.org/wiki/Integral_Form_of_Polygamma_Function | [
"Integral Form of Polygamma Function",
"Polygamma Function",
"Complex Analysis",
"Definite Integrals"
] | [
"Definition:Complex Number",
"Definition:Positive/Real Number",
"Definition:Complex Number/Real Part",
"Definition:Polygamma Function"
] | [
"Gauss's Integral Form of Digamma Function",
"Definition:Derivative"
] |
proofwiki-21541 | Equivalence of Definitions of Disconnected Space | {{TFAE|def = Disconnected Space}}
Let $T = \struct {S, \tau}$ be a topological space. | By definition $1$, $T$ is '''disconnected''' {{iff}} $T$ is not connected.
{{Recall|Connected Topological Space|connected topological space}}
{{:Definition:Connected Topological Space/Definition 7}}
Hence precisely a '''disconnected space''' by definition $2$.
{{qed}}
Category:Disconnected Spaces
oko18c2v3q9n2m9u8zo4x5... | {{TFAE|def = Disconnected Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]]. | By [[Definition:Disconnected Space/Definition 1|definition $1$]], $T$ is '''[[Definition:Disconnected Space|disconnected]]''' {{iff}} $T$ is not [[Definition:Connected Topological Space|connected]].
{{Recall|Connected Topological Space|connected topological space}}
{{:Definition:Connected Topological Space/Definition... | Equivalence of Definitions of Disconnected Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Disconnected_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Disconnected_Space | [
"Disconnected Spaces"
] | [
"Definition:Topological Space"
] | [
"Definition:Disconnected Space/Definition 1",
"Definition:Disconnected (Topology)/Topological Space",
"Definition:Connected Topological Space",
"Definition:Disconnected (Topology)/Topological Space",
"Definition:Disconnected Space/Definition 2",
"Category:Disconnected Spaces"
] |
proofwiki-21542 | Sum of Trigamma of n plus 3 over n plus 2 | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \dfrac {\map {\psi_1} {n + 3} } {\paren {n + 2} }
| r = \dfrac {\map {\psi_1} 3} 2 + \dfrac {\map {\psi_1} 4} 3 + \dfrac {\map {\psi_1} 5} 4 + \dfrac {\map {\psi_1} 6} 5 + \cdots
| c =
}}
{{eqn | r = \map \zeta 3 - \map \zeta 2 + 1
| c =
}}
{{en... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \dfrac {\map {\psi_1} {n + 3} } {\paren {n + 2} }
| r = \sum_{n \mathop = 1}^\infty \dfrac {\map {\psi_1} {n + 1} } n - \map {\psi_1} 2
| c = $n \to n - 2$
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \dfrac {\map {\psi_1} {n + 1} } n - \paren {\map {\p... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \dfrac {\map {\psi_1} {n + 3} } {\paren {n + 2} }
| r = \dfrac {\map {\psi_1} 3} 2 + \dfrac {\map {\psi_1} 4} 3 + \dfrac {\map {\psi_1} 5} 4 + \dfrac {\map {\psi_1} 6} 5 + \cdots
| c =
}}
{{eqn | r = \map \zeta 3 - \map \zeta 2 + 1
| c =
}}
{{en... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \dfrac {\map {\psi_1} {n + 3} } {\paren {n + 2} }
| r = \sum_{n \mathop = 1}^\infty \dfrac {\map {\psi_1} {n + 1} } n - \map {\psi_1} 2
| c = $n \to n - 2$
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \dfrac {\map {\psi_1} {n + 1} } n - \paren {\map {\p... | Sum of Trigamma of n plus 3 over n plus 2 | https://proofwiki.org/wiki/Sum_of_Trigamma_of_n_plus_3_over_n_plus_2 | https://proofwiki.org/wiki/Sum_of_Trigamma_of_n_plus_3_over_n_plus_2 | [
"Trigamma Function",
"Riemann Zeta Function",
"Apéry's Constant"
] | [
"Definition:Trigamma Function",
"Definition:Riemann Zeta Function",
"Definition:Apéry's Constant",
"Definition:Riemann Zeta Function"
] | [
"Recurrence Relation for Polygamma Function",
"Polygamma Function in terms of Hurwitz Zeta Function",
"Integral Form of Polygamma Function/Corollary",
"Tonelli's Theorem",
"Power Series Expansion for Logarithm of 1 - x",
"Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x over One... |
proofwiki-21543 | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x over One minus x | :$\ds \int_0^1 \dfrac {\ln x \map \ln {1 - x} } {\paren {1 - x} } \rd x = \map \zeta 3$
where $\map \zeta 3$ is Apéry's constant: the Riemann $\zeta$ function of $3$. | {{begin-eqn}}
{{eqn | l = \int_0^1 \dfrac {\ln x \map \ln {1 - x} } {\paren {1 - x} } \rd x
| r = \int_1^0 \dfrac {\map \ln {1 - u} \ln u } u \paren {-\rd u}
| c = $x \to \paren {1 - u}$ and $\rd x \to -\rd u$
}}
{{eqn | r = \int_0^1 \dfrac {\map \ln {1 - u} \ln u } u \rd u
| c = reversing limits of inte... | :$\ds \int_0^1 \dfrac {\ln x \map \ln {1 - x} } {\paren {1 - x} } \rd x = \map \zeta 3$
where $\map \zeta 3$ is [[Definition:Apéry's Constant|Apéry's constant]]: the [[Definition:Riemann Zeta Function|Riemann $\zeta$ function]] of $3$. | {{begin-eqn}}
{{eqn | l = \int_0^1 \dfrac {\ln x \map \ln {1 - x} } {\paren {1 - x} } \rd x
| r = \int_1^0 \dfrac {\map \ln {1 - u} \ln u } u \paren {-\rd u}
| c = $x \to \paren {1 - u}$ and $\rd x \to -\rd u$
}}
{{eqn | r = \int_0^1 \dfrac {\map \ln {1 - u} \ln u } u \rd u
| c = [[Integration by Substit... | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x over One minus x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_minus_x_over_One_minus_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_minus_x_over_One_minus_x/Proof_1 | [
"Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x over One minus x",
"Apéry's Constant",
"Definite Integrals involving Logarithm Function"
] | [
"Definition:Apéry's Constant",
"Definition:Riemann Zeta Function"
] | [
"Integration by Substitution/Definite Integral",
"Power Series Expansion for Logarithm of 1 - x",
"Fubini's Theorem",
"Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x",
"Gamma Function Extends Factorial"
] |
proofwiki-21544 | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x over One minus x | :$\ds \int_0^1 \dfrac {\ln x \map \ln {1 - x} } {\paren {1 - x} } \rd x = \map \zeta 3$
where $\map \zeta 3$ is Apéry's constant: the Riemann $\zeta$ function of $3$. | With a view to expressing the primitive in the form:
{{begin-eqn}}
{{eqn | l = \int u \frac {\d v} {\d x} \rd x
| r = u v - \int v \frac {\d u} {\d x} \rd x
| c = Integration by Parts
}}
{{end-eqn}}
let:
{{begin-eqn}}
{{eqn | l = u
| r = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d... | :$\ds \int_0^1 \dfrac {\ln x \map \ln {1 - x} } {\paren {1 - x} } \rd x = \map \zeta 3$
where $\map \zeta 3$ is [[Definition:Apéry's Constant|Apéry's constant]]: the [[Definition:Riemann Zeta Function|Riemann $\zeta$ function]] of $3$. | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
{{begin-eqn}}
{{eqn | l = \int u \frac {\d v} {\d x} \rd x
| r = u v - \int v \frac {\d u} {\d x} \rd x
| c = [[Integration by Parts]]
}}
{{end-eqn}}
let:
{{begin-eqn}}
{{eqn | l = u
| r = \ln x
| c =
}}
{{eqn | l... | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x over One minus x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_minus_x_over_One_minus_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_minus_x_over_One_minus_x/Proof_2 | [
"Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x over One minus x",
"Apéry's Constant",
"Definite Integrals involving Logarithm Function"
] | [
"Definition:Apéry's Constant",
"Definition:Riemann Zeta Function"
] | [
"Definition:Primitive",
"Integration by Parts",
"Derivative of Natural Logarithm Function",
"Primitive of Power",
"Integration by Parts",
"Integration by Substitution/Definite Integral",
"Logarithm of Reciprocal",
"Integral Representation of Riemann Zeta Function in terms of Gamma Function/Corollary",... |
proofwiki-21545 | Discriminant of Quadratic Equation | Let $\QQ$ be the quadratic equation:
:$a x^2 + b x + c$
The discriminant of $\QQ$ is given by:
:$\map \Delta \QQ = b^2 - 4 a c$ | Let $\alpha_1$ and $\alpha_2$ be the roots of $\QQ$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Delta \QQ
| r = a^2 \paren {\alpha_1 - \alpha_2}^2
| c = Discriminant Example: Quadratic
}}
{{eqn | r = a^2 \paren { {\alpha_1}^2 - 2 \alpha_1 \alpha_2 + {\alpha_2}^2}
| c = Square of Difference
}}
{{eqn | r ... | Let $\QQ$ be the [[Definition:Quadratic Equation|quadratic equation]]:
:$a x^2 + b x + c$
The [[Definition:Discriminant of Polynomial|discriminant]] of $\QQ$ is given by:
:$\map \Delta \QQ = b^2 - 4 a c$ | Let $\alpha_1$ and $\alpha_2$ be the [[Definition:Root of Equation|roots]] of $\QQ$.
We have:
{{begin-eqn}}
{{eqn | l = \map \Delta \QQ
| r = a^2 \paren {\alpha_1 - \alpha_2}^2
| c = [[Discriminant of Polynomial/Examples/Quadratic|Discriminant Example: Quadratic]]
}}
{{eqn | r = a^2 \paren { {\alpha_1}^2 ... | Discriminant of Quadratic Equation | https://proofwiki.org/wiki/Discriminant_of_Quadratic_Equation | https://proofwiki.org/wiki/Discriminant_of_Quadratic_Equation | [
"Discriminants of Quadratic Equations",
"Quadratic Equations"
] | [
"Definition:Quadratic Equation",
"Definition:Discriminant of Polynomial"
] | [
"Definition:Root of Equation",
"Discriminant of Polynomial/Examples/Quadratic",
"Square of Difference",
"Square of Sum",
"Sum of Roots of Quadratic Equation",
"Product of Roots of Quadratic Equation"
] |
proofwiki-21546 | Digamma Function of One | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \map \psi z
| r = \dfrac {\map {\Gamma'} z} {\map \Gamma z}
| c = {{Defof|Digamma Function}}
}}
{{eqn | r = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }
| c = Reciprocal times Derivative of Gamma Function
}}
{{eqn | ll = \leadsto
| l ... | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \map \psi z
| r = \dfrac {\map {\Gamma'} z} {\map \Gamma z}
| c = {{Defof|Digamma Function}}
}}
{{eqn | r = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }
| c = [[Reciprocal times Derivative of Gamma Function]]
}}
{{eqn | ll = \leadsto
... | Digamma Function of One | https://proofwiki.org/wiki/Digamma_Function_of_One | https://proofwiki.org/wiki/Digamma_Function_of_One | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Reciprocal times Derivative of Gamma Function",
"Definition:Summation",
"Definition:Summation/Vacuous Summation",
"Category:Examples of Digamma Function",
"Category:Euler-Mascheroni Constant"
] |
proofwiki-21547 | Digamma Function of One | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 4}
| r = -\gamma - \ln 8 - \frac \pi 2 \map \cot {\frac 1 4 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {4 / 2} - 1} \map \cos {\frac {2 \pi n} 4} \map \ln {\map \sin {\frac {\pi n} 4} }
| c = Gauss's Digamma Theorem
}}
{{eqn | r = -\gamma - 3 \ln 2 - \frac \pi 2 \ti... | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 4}
| r = -\gamma - \ln 8 - \frac \pi 2 \map \cot {\frac 1 4 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {4 / 2} - 1} \map \cos {\frac {2 \pi n} 4} \map \ln {\map \sin {\frac {\pi n} 4} }
| c = [[Gauss's Digamma Theorem]]
}}
{{eqn | r = -\gamma - 3 \ln 2 - \frac \pi 2... | Digamma Function of One Fourth/Proof 1 | https://proofwiki.org/wiki/Digamma_Function_of_One | https://proofwiki.org/wiki/Digamma_Function_of_One_Fourth/Proof_1 | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Gauss's Digamma Theorem",
"Sum of Logarithms"
] |
proofwiki-21548 | Digamma Function of One | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{4 - 1} \map \psi {\frac k 4}
| r = -\paren {4 - 1} \gamma - 4 \ln 4
| c =
}}... | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{4 - 1} \map \psi {\frac k 4}
| r = -\paren {4 - 1} \gamma - 4 \ln 4
| c =
}}... | Digamma Function of One Fourth/Proof 2 | https://proofwiki.org/wiki/Digamma_Function_of_One | https://proofwiki.org/wiki/Digamma_Function_of_One_Fourth/Proof_2 | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula",
"Digamma Function of One Half",
"Logarithm of Power"
] |
proofwiki-21549 | Digamma Function of One | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 2}
| r = -\gamma - \ln 4 - \frac \pi 2 \map \cot {\frac 1 2 \pi} + 2 \sum_{n \mathop = 1}^0 \map \cos {\frac {2 \pi n} 2} \map \ln {\map \sin {\frac {\pi n} 2} }
| c = Gauss's Digamma Theorem
}}
{{eqn | r = -\gamma - \ln 4
| c = Cotangent of Right Angle, no... | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 2}
| r = -\gamma - \ln 4 - \frac \pi 2 \map \cot {\frac 1 2 \pi} + 2 \sum_{n \mathop = 1}^0 \map \cos {\frac {2 \pi n} 2} \map \ln {\map \sin {\frac {\pi n} 2} }
| c = [[Gauss's Digamma Theorem]]
}}
{{eqn | r = -\gamma - \ln 4
| c = [[Cotangent of Right Ang... | Digamma Function of One Half/Proof 1 | https://proofwiki.org/wiki/Digamma_Function_of_One | https://proofwiki.org/wiki/Digamma_Function_of_One_Half/Proof_1 | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Gauss's Digamma Theorem",
"Cotangent of Right Angle",
"Definition:Summation/Vacuous Summation",
"Logarithm of Power"
] |
proofwiki-21550 | Digamma Function of One | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{6 - 1} \map \psi {\frac k 6}
| r = -\paren {6 - 1} \gamma - 6 \ln 6
| c =
}}... | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{6 - 1} \map \psi {\frac k 6}
| r = -\paren {6 - 1} \gamma - 6 \ln 6
| c =
}}... | Digamma Function of One Sixth/Proof 2 | https://proofwiki.org/wiki/Digamma_Function_of_One | https://proofwiki.org/wiki/Digamma_Function_of_One_Sixth/Proof_2 | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula",
"Sum of Logarithms",
"Digamma Function of One Third",
"Digamma Function of One Half",
"Digamma Function of Two Thirds"
] |
proofwiki-21551 | Digamma Function of One | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 3}
| r = -\gamma - \ln 6 - \frac \pi 2 \map \cot {\frac 1 3 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {3 / 2} - 1} \map \cos {\frac {2 \pi n} 3} \map \ln {\map \sin {\frac {\pi n} 3} }
| c = Gauss's Digamma Theorem
}}
{{eqn | r = -\gamma - \ln 2 - \ln 3 - \frac \pi... | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 3}
| r = -\gamma - \ln 6 - \frac \pi 2 \map \cot {\frac 1 3 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {3 / 2} - 1} \map \cos {\frac {2 \pi n} 3} \map \ln {\map \sin {\frac {\pi n} 3} }
| c = [[Gauss's Digamma Theorem]]
}}
{{eqn | r = -\gamma - \ln 2 - \ln 3 - \frac... | Digamma Function of One Third/Proof 1 | https://proofwiki.org/wiki/Digamma_Function_of_One | https://proofwiki.org/wiki/Digamma_Function_of_One_Third/Proof_1 | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Gauss's Digamma Theorem",
"Sum of Logarithms",
"Difference of Logarithms",
"Logarithm of Power"
] |
proofwiki-21552 | Digamma Function of One | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{3 - 1} \map \psi {\frac k 3}
| r = -\paren {3 - 1} \gamma - 3 \ln 3
| c =
}}... | :$\map \psi 1 = -\gamma$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{3 - 1} \map \psi {\frac k 3}
| r = -\paren {3 - 1} \gamma - 3 \ln 3
| c =
}}... | Digamma Function of One Third/Proof 2 | https://proofwiki.org/wiki/Digamma_Function_of_One | https://proofwiki.org/wiki/Digamma_Function_of_One_Third/Proof_2 | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula"
] |
proofwiki-21553 | Gamma Function of 3 | :$\map \Gamma 3 = 2$ | {{begin-eqn}}
{{eqn | l = \map \Gamma 3
| r = \map \Gamma {2 + 1}
| c =
}}
{{eqn | r = 2 \map \Gamma 2
| c = Gamma Difference Equation
}}
{{eqn | r = 2 \times 1
| c = Gamma Function of 2
}}
{{end-eqn}}
{{qed}}
Category:Examples of Gamma Function Values
gcvkn2iren6z3839y7j5v340mdy1ef0 | :$\map \Gamma 3 = 2$ | {{begin-eqn}}
{{eqn | l = \map \Gamma 3
| r = \map \Gamma {2 + 1}
| c =
}}
{{eqn | r = 2 \map \Gamma 2
| c = [[Gamma Difference Equation]]
}}
{{eqn | r = 2 \times 1
| c = [[Gamma Function of 2]]
}}
{{end-eqn}}
{{qed}}
[[Category:Examples of Gamma Function Values]]
gcvkn2iren6z3839y7j5v340mdy1e... | Gamma Function of 3 | https://proofwiki.org/wiki/Gamma_Function_of_3 | https://proofwiki.org/wiki/Gamma_Function_of_3 | [
"Examples of Gamma Function Values",
"Examples of Gamma Function Values"
] | [] | [
"Gamma Difference Equation",
"Gamma Function of 2",
"Category:Examples of Gamma Function Values"
] |
proofwiki-21554 | Gamma Function of 2 | :$\map \Gamma 2 = 1$ | {{begin-eqn}}
{{eqn | l = \map \Gamma 2
| r = \map \Gamma {1 + 1}
| c =
}}
{{eqn | r = 1 \map \Gamma 1
| c = Gamma Difference Equation
}}
{{eqn | r = 1 \times 0!
| c = Gamma Function Extends Factorial
}}
{{eqn | r = 1
| c = {{Defof|Factorial}}: $0! = 1$
}}
{{end-eqn}}
{{qed}}
Category:Exa... | :$\map \Gamma 2 = 1$ | {{begin-eqn}}
{{eqn | l = \map \Gamma 2
| r = \map \Gamma {1 + 1}
| c =
}}
{{eqn | r = 1 \map \Gamma 1
| c = [[Gamma Difference Equation]]
}}
{{eqn | r = 1 \times 0!
| c = [[Gamma Function Extends Factorial]]
}}
{{eqn | r = 1
| c = {{Defof|Factorial}}: $0! = 1$
}}
{{end-eqn}}
{{qed}}
[[C... | Gamma Function of 2 | https://proofwiki.org/wiki/Gamma_Function_of_2 | https://proofwiki.org/wiki/Gamma_Function_of_2 | [
"Examples of Gamma Function Values"
] | [] | [
"Gamma Difference Equation",
"Gamma Function Extends Factorial",
"Category:Examples of Gamma Function Values"
] |
proofwiki-21555 | Definite Integral from 0 to 1 of Zeta of 2 minus Dilogarithm of x over One minus x | :$\ds \int_{\to 0}^{\to 1} \dfrac {\paren {\map \zeta 2 - \map {\Li_2} x } } {1 - x} \rd x = 2 \map \zeta 3$
where:
:$\map {\Li_2} x$ is the Dilogarithm function of $x$
:$\map \zeta 2$ is the Riemann $\zeta$ function of $2$
:$\map \zeta 3$ is Apéry's constant: the Riemann $\zeta$ function of $3$. | With a view to expressing the primitive in the form:
{{begin-eqn}}
{{eqn | l = \int u \frac {\d v} {\d x} \rd x
| r = u v - \int v \frac {\d u} {\d x} \rd x
| c = Integration by Parts
}}
{{end-eqn}}
let:
{{begin-eqn}}
{{eqn | l = u
| r = \map \zeta 2 - \map {\Li_2} x
| c =
}}
{{eqn | ll= \leads... | :$\ds \int_{\to 0}^{\to 1} \dfrac {\paren {\map \zeta 2 - \map {\Li_2} x } } {1 - x} \rd x = 2 \map \zeta 3$
where:
:$\map {\Li_2} x$ is the [[Definition:Spence's Function|Dilogarithm function]] of $x$
:$\map \zeta 2$ is the [[Definition:Riemann Zeta Function|Riemann $\zeta$ function]] of $2$
:$\map \zeta 3$ is [[Defi... | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
{{begin-eqn}}
{{eqn | l = \int u \frac {\d v} {\d x} \rd x
| r = u v - \int v \frac {\d u} {\d x} \rd x
| c = [[Integration by Parts]]
}}
{{end-eqn}}
let:
{{begin-eqn}}
{{eqn | l = u
| r = \map \zeta 2 - \map {\Li_2} x
... | Definite Integral from 0 to 1 of Zeta of 2 minus Dilogarithm of x over One minus x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Zeta_of_2_minus_Dilogarithm_of_x_over_One_minus_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Zeta_of_2_minus_Dilogarithm_of_x_over_One_minus_x | [
"Spence's Function",
"Apéry's Constant",
"Definite Integrals involving Logarithm Function"
] | [
"Definition:Spence's Function",
"Definition:Riemann Zeta Function",
"Definition:Apéry's Constant",
"Definition:Riemann Zeta Function"
] | [
"Definition:Primitive",
"Integration by Parts",
"Primitive of Reciprocal of a x + b",
"Integration by Parts",
"Integration by Substitution/Definite Integral",
"Logarithm of Reciprocal",
"Integral Representation of Riemann Zeta Function in terms of Gamma Function/Corollary",
"Gamma Function of 3",
"C... |
proofwiki-21556 | Sum of Digamma of n plus 2 over n plus 2 squared | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \dfrac {\map \psi {n + 2} } {\paren {n + 2}^2 }
| r = \dfrac {\map \psi 2} {2^2} + \dfrac {\map \psi 3} {3^2} + \dfrac {\map \psi 4} {4^2} + \dfrac {\map \psi 5} {5^2} + \cdots
| c =
}}
{{eqn | r = \gamma \cdot \paren {1 - \map \zeta 2} + \map \zeta 3
... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \dfrac {\map \psi {n + 2} } {\paren {n + 2}^2}
| r = \sum_{n \mathop = 1}^\infty \dfrac {\map \psi n} {n^2} - \map \psi 1
| c = $n \to n - 2$
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \dfrac {\paren {\map \psi {n + 1} - \dfrac 1 n} } {n^2} - \map \psi 1... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \dfrac {\map \psi {n + 2} } {\paren {n + 2}^2 }
| r = \dfrac {\map \psi 2} {2^2} + \dfrac {\map \psi 3} {3^2} + \dfrac {\map \psi 4} {4^2} + \dfrac {\map \psi 5} {5^2} + \cdots
| c =
}}
{{eqn | r = \gamma \cdot \paren {1 - \map \zeta 2} + \map \zeta 3
... | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 0}^\infty \dfrac {\map \psi {n + 2} } {\paren {n + 2}^2}
| r = \sum_{n \mathop = 1}^\infty \dfrac {\map \psi n} {n^2} - \map \psi 1
| c = $n \to n - 2$
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \dfrac {\paren {\map \psi {n + 1} - \dfrac 1 n} } {n^2} - \map \psi 1... | Sum of Digamma of n plus 2 over n plus 2 squared | https://proofwiki.org/wiki/Sum_of_Digamma_of_n_plus_2_over_n_plus_2_squared | https://proofwiki.org/wiki/Sum_of_Digamma_of_n_plus_2_over_n_plus_2_squared | [
"Riemann Zeta Function",
"Digamma Function",
"Apéry's Constant"
] | [
"Definition:Digamma Function",
"Definition:Riemann Zeta Function",
"Definition:Apéry's Constant",
"Definition:Riemann Zeta Function",
"Definition:Multiplication/Real Numbers"
] | [
"Recurrence Relation for Digamma Function",
"Linear Combination of Convergent Series",
"Digamma Function of One",
"Linear Combination of Convergent Series",
"Tonelli's Theorem",
"Power Series Expansion for Spence's Function",
"Definite Integral from 0 to 1 of Zeta of 2 minus Dilogarithm of x over One mi... |
proofwiki-21557 | Cumulative Distribution Function for Discrete Distribution | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.
Then the cumulative distribution function of $X$ is given by:
:$\map F x = \ds \sum_{x_i \mathop \le x} \map \Pr {X = x_i}$ | {{ProofWanted|Not sure whether this really ought to be a theorem or a definition}} | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Then the [[Definition:Cumulative Distribution Function|cumulative distribution function]] of $X$ is given by:
... | {{ProofWanted|Not sure whether this really ought to be a theorem or a definition}} | Cumulative Distribution Function for Discrete Distribution | https://proofwiki.org/wiki/Cumulative_Distribution_Function_for_Discrete_Distribution | https://proofwiki.org/wiki/Cumulative_Distribution_Function_for_Discrete_Distribution | [
"Cumulative Distribution Functions",
"Discrete Probability Distributions"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Discrete",
"Definition:Cumulative Distribution Function"
] | [] |
proofwiki-21558 | Symmetric Integral of Even Function over One Plus Even Function to Power of Odd Function | <onlyinclude>
:$\ds \int_{-a}^a \frac {\map e x} {1 + \map t x^{\map o x} } \rd x = \int_0^a \map e x \rd x$
where:
:$a$ is a positive real number
:$\map e x$ and $\map t x$ are arbitrary even functions
:$\map o x$ is an arbitrary odd function.
<onlyinclude> | Let the integrand be denoted by:
:$\map f x = \dfrac {\map e x} {1 + \map t x^{\map o x} }$
Using Real Function is Expressible as Sum of Even Function and Odd Function, let us express $\map f x$ as:
:$(1): \quad \map f x = \map E x + \map O x$
where:
:$\map E x$ is an even function
:$\map O x$ is an odd function
define... | <onlyinclude>
:$\ds \int_{-a}^a \frac {\map e x} {1 + \map t x^{\map o x} } \rd x = \int_0^a \map e x \rd x$
where:
:$a$ is a [[Definition:Positive Real Number|positive real number]]
:$\map e x$ and $\map t x$ are arbitrary [[Definition:Even Function|even functions]]
:$\map o x$ is an arbitrary [[Definition:Odd Functi... | Let the [[Definition:Integrand|integrand]] be denoted by:
:$\map f x = \dfrac {\map e x} {1 + \map t x^{\map o x} }$
Using [[Real Function is Expressible as Sum of Even Function and Odd Function]], let us express $\map f x$ as:
:$(1): \quad \map f x = \map E x + \map O x$
where:
:$\map E x$ is an [[Definition:Even F... | Symmetric Integral of Even Function over One Plus Even Function to Power of Odd Function | https://proofwiki.org/wiki/Symmetric_Integral_of_Even_Function_over_One_Plus_Even_Function_to_Power_of_Odd_Function | https://proofwiki.org/wiki/Symmetric_Integral_of_Even_Function_over_One_Plus_Even_Function_to_Power_of_Odd_Function | [
"Odd Functions",
"Even Functions",
"Definite Integrals"
] | [
"Definition:Positive/Real Number",
"Definition:Even Function",
"Definition:Odd Function"
] | [
"Definition:Integration/Integrand",
"Real Function is Expressible as Sum of Even Function and Odd Function",
"Definition:Even Function",
"Definition:Odd Function",
"Linear Combination of Integrals/Definite",
"Definite Integral of Odd Function",
"Definite Integral of Even Function",
"Definition:Fractio... |
proofwiki-21559 | Divisibility by 6 | An integer $N$ expressed in decimal notation is divisible by $6$ {{iff}}
:$N$ is even
and:
:$N$ is divisible by $3$. | We have that:
:$6 \divides N \iff 2 \divides N \land 3 \divides N$
The result follows by definition of even integer.
{{qed}} | An [[Definition:Integer|integer]] $N$ expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $6$ {{iff}}
:$N$ is [[Definition:Even Integer|even]]
and:
:$N$ is [[Definition:Divisor of Integer|divisible]] by $3$. | We have that:
:$6 \divides N \iff 2 \divides N \land 3 \divides N$
The result follows by definition of [[Definition:Even Integer|even integer]].
{{qed}} | Divisibility by 6 | https://proofwiki.org/wiki/Divisibility_by_6 | https://proofwiki.org/wiki/Divisibility_by_6 | [
"Divisibility Tests",
"6"
] | [
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Divisor (Algebra)/Integer",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Even Integer"
] |
proofwiki-21560 | Definite Integral over Unit Square of Logarithm of x minus Logarithm of y over x minus y | :$\ds \int_0^1 \int_0^1 \frac {\ln x - \ln y} {x - y} \rd x \rd y = 2 \map \zeta 2$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \int_0^1 \dfrac {\ln x - \ln y} {x - y} \rd x \rd y
| r = \int_0^1 \int_0^1 \dfrac {\map \ln {\dfrac x y} } {x - y} \rd x \rd y
| c = Difference of Logarithms
}}
{{eqn | r = 2 \int_0^1 \int_0^x \dfrac {\map \ln {\dfrac x y} } {x - y} \rd x \rd y
| c = symmetry about ... | :$\ds \int_0^1 \int_0^1 \frac {\ln x - \ln y} {x - y} \rd x \rd y = 2 \map \zeta 2$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \int_0^1 \dfrac {\ln x - \ln y} {x - y} \rd x \rd y
| r = \int_0^1 \int_0^1 \dfrac {\map \ln {\dfrac x y} } {x - y} \rd x \rd y
| c = [[Difference of Logarithms]]
}}
{{eqn | r = 2 \int_0^1 \int_0^x \dfrac {\map \ln {\dfrac x y} } {x - y} \rd x \rd y
| c = symmetry ab... | Definite Integral over Unit Square of Logarithm of x minus Logarithm of y over x minus y | https://proofwiki.org/wiki/Definite_Integral_over_Unit_Square_of_Logarithm_of_x_minus_Logarithm_of_y_over_x_minus_y | https://proofwiki.org/wiki/Definite_Integral_over_Unit_Square_of_Logarithm_of_x_minus_Logarithm_of_y_over_x_minus_y | [
"Definite Integrals involving Logarithm Function",
"Riemann Zeta Function",
"Spence's Function"
] | [] | [
"Difference of Logarithms",
"Logarithm of Reciprocal",
"Integration by Substitution/Definite Integral",
"Power Series Expansion for Spence's Function"
] |
proofwiki-21561 | Power Series Expansion for Harmonic Numbers | The harmonic numbers have the power series expansion:
{{begin-eqn}}
{{eqn | l = \map H x
| r = \sum_{k \mathop = 2}^{\infty} \paren {-1}^k \map \zeta k x^{k - 1}
| c =
}}
{{eqn | r = \map \zeta 2 x - \map \zeta 3 x^2 + \map \zeta 4 x^3 - \map \zeta 5 x^4 + \cdots
| c =
}}
{{end-eqn}}
where $\zeta$ d... | {{begin-eqn}}
{{eqn | l = \map H x
| r = \gamma + \frac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }
| c = Extension of Harmonic Number to Non-Integer Argument
}}
{{eqn | r = \gamma - \gamma + \sum_{m \mathop = 1}^\infty \paren {\frac 1 m - \frac 1 {x + m} }
| c = Reciprocal times Derivative of Ga... | The [[Definition:Harmonic Numbers|harmonic numbers]] have the [[Definition:Power Series|power series expansion]]:
{{begin-eqn}}
{{eqn | l = \map H x
| r = \sum_{k \mathop = 2}^{\infty} \paren {-1}^k \map \zeta k x^{k - 1}
| c =
}}
{{eqn | r = \map \zeta 2 x - \map \zeta 3 x^2 + \map \zeta 4 x^3 - \map \ze... | {{begin-eqn}}
{{eqn | l = \map H x
| r = \gamma + \frac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }
| c = [[Extension of Harmonic Number to Non-Integer Argument]]
}}
{{eqn | r = \gamma - \gamma + \sum_{m \mathop = 1}^\infty \paren {\frac 1 m - \frac 1 {x + m} }
| c = [[Reciprocal times Derivative... | Power Series Expansion for Harmonic Numbers | https://proofwiki.org/wiki/Power_Series_Expansion_for_Harmonic_Numbers | https://proofwiki.org/wiki/Power_Series_Expansion_for_Harmonic_Numbers | [
"Harmonic Numbers",
"Examples of Power Series",
"Riemann Zeta Function",
"Taylor Series"
] | [
"Definition:Harmonic Numbers",
"Definition:Power Series",
"Definition:Riemann Zeta Function"
] | [
"Extension of Harmonic Number to Non-Integer Argument",
"Reciprocal times Derivative of Gamma Function",
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Sum of Infinite Geometric Sequence",
"Definition:Complex Modulus",
"Definition:Grandi's Series",
"Definition:Divergent Series"
] |
proofwiki-21562 | Power Series Expansion for Logarithm of Gamma Function | The logarithm of the Gamma function has the power series expansion:
{{begin-eqn}}
{{eqn | l = \map \ln {\map \Gamma {x + 1} }
| r = -\gamma x + \sum_{k \mathop = 2}^\infty \dfrac {\map \zeta k \paren {-x}^k} k
| c =
}}
{{eqn | r = -\gamma x + \dfrac {\map \zeta 2 x^2} 2 - \dfrac {\map \zeta 3 x^3} 3 + \dfr... | {{begin-eqn}}
{{eqn | l = \gamma + \frac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }
| r = \map H x
| c = Extension of Harmonic Number to Non-Integer Argument
}}
{{eqn | ll= \leadsto
| l = \frac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }
| r = -\gamma + \map H x
| c = subtracti... | The [[Definition:Complex Natural Logarithm|logarithm]] of the [[Definition:Gamma Function|Gamma function]] has the [[Definition:Power Series|power series expansion]]:
{{begin-eqn}}
{{eqn | l = \map \ln {\map \Gamma {x + 1} }
| r = -\gamma x + \sum_{k \mathop = 2}^\infty \dfrac {\map \zeta k \paren {-x}^k} k
... | {{begin-eqn}}
{{eqn | l = \gamma + \frac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }
| r = \map H x
| c = [[Extension of Harmonic Number to Non-Integer Argument]]
}}
{{eqn | ll= \leadsto
| l = \frac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }
| r = -\gamma + \map H x
| c = subtr... | Power Series Expansion for Logarithm of Gamma Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_Gamma_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_Gamma_Function | [
"Euler-Mascheroni Constant",
"Examples of Power Series",
"Riemann Zeta Function"
] | [
"Definition:Natural Logarithm/Complex",
"Definition:Gamma Function",
"Definition:Power Series"
] | [
"Extension of Harmonic Number to Non-Integer Argument",
"Fundamental Theorem of Calculus",
"Power Series Expansion for Harmonic Numbers",
"Primitive of Power",
"Primitive of Constant",
"Definition:Decreasing/Sequence",
"Definition:Positive/Real Number",
"Definition:Term of Sequence",
"Definition:Con... |
proofwiki-21563 | Upper Bound for Abundancy Index | Let $n \in \N_{>0}$ be a non-zero natural number.
Let $\map h n$ be the abundancy index of $n$.
Then:
:$\ds \map h n \le \prod_{p \mathop \divides n} \dfrac p {p - 1}$
such that equality holds {{iff}} $n = 1$. | First we note that $\map h 1 = 1$ demonstrating that equality holds if $n = 1$.
Let $n$ be expressed as its prime decomposition:
{{begin-eqn}}
{{eqn | l = n
| r = \prod_{p_k \mathop \divides n} {p_k}^{a_k}
| c =
}}
{{eqn | r = {p_1}^{a_1} {p_2}^{a_2} \cdots {p_k}^{a_k}
| c =
}}
{{end-eqn}}
where $p_... | Let $n \in \N_{>0}$ be a [[Definition:Non-Zero Natural Numbers|non-zero natural number]].
Let $\map h n$ be the [[Definition:Abundancy Index|abundancy index]] of $n$.
Then:
:$\ds \map h n \le \prod_{p \mathop \divides n} \dfrac p {p - 1}$
such that equality holds {{iff}} $n = 1$. | First we note that $\map h 1 = 1$ demonstrating that equality holds if $n = 1$.
Let $n$ be expressed as its [[Definition:Prime Decomposition|prime decomposition]]:
{{begin-eqn}}
{{eqn | l = n
| r = \prod_{p_k \mathop \divides n} {p_k}^{a_k}
| c =
}}
{{eqn | r = {p_1}^{a_1} {p_2}^{a_2} \cdots {p_k}^{a_k}... | Upper Bound for Abundancy Index | https://proofwiki.org/wiki/Upper_Bound_for_Abundancy_Index | https://proofwiki.org/wiki/Upper_Bound_for_Abundancy_Index | [
"Abundancy"
] | [
"Definition:Natural Numbers/Non-Zero",
"Definition:Abundancy Index"
] | [
"Definition:Prime Decomposition",
"Definition:Distinct",
"Definition:Prime Number",
"Definition:Abundancy Index",
"Definition:Divisor Sum Function",
"Divisor Sum Function is Multiplicative",
"Definition:Prime Number",
"Divisor Sum of Power of Prime",
"Category:Abundancy"
] |
proofwiki-21564 | Perfect Number has at least Two Distinct Prime Factors | Let $n \in \N$ be a perfect number.
Then $n$ has at least two distinct prime factors. | {{AimForCont}} the contrary: that $n$ is a perfect number with exactly $1$ prime factor.
Hence let $n = p^k$ where $p$ is prime.
By definition of perfect number:
:$\map {\sigma_1} n = 2 n$
where $\map {\sigma_1} n$ denotes the divisor sum of $n$.
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} n
| r = 2 n
... | Let $n \in \N$ be a [[Definition:Perfect Number|perfect number]].
Then $n$ has at least two [[Definition:Distinct Elements|distinct]] [[Definition:Prime Factor|prime factors]]. | {{AimForCont}} the [[Definition:Contrary Statements|contrary]]: that $n$ is a [[Definition:Perfect Number|perfect number]] with exactly $1$ [[Definition:Prime Factor|prime factor]].
Hence let $n = p^k$ where $p$ is [[Definition:Prime Number|prime]].
By definition of [[Definition:Perfect Number|perfect number]]:
:$\m... | Perfect Number has at least Two Distinct Prime Factors/Proof 1 | https://proofwiki.org/wiki/Perfect_Number_has_at_least_Two_Distinct_Prime_Factors | https://proofwiki.org/wiki/Perfect_Number_has_at_least_Two_Distinct_Prime_Factors/Proof_1 | [
"Perfect Number has at least Two Distinct Prime Factors",
"Perfect Numbers"
] | [
"Definition:Perfect Number",
"Definition:Distinct/Plural",
"Definition:Prime Factor"
] | [
"Definition:Contrary Statements",
"Definition:Perfect Number",
"Definition:Prime Factor",
"Definition:Prime Number",
"Definition:Perfect Number",
"Definition:Divisor Sum Function",
"Divisor Sum of Power of Prime",
"Sum of Geometric Sequence",
"Definition:Divisor (Algebra)/Integer",
"Definition:Con... |
proofwiki-21565 | Perfect Number has at least Two Distinct Prime Factors | Let $n \in \N$ be a perfect number.
Then $n$ has at least two distinct prime factors. | {{AimForCont}} the contrary: that $n$ is a perfect number with exactly $1$ prime factor.
Hence let $n = p^k$ where $p$ is prime.
By definition of perfect number:
:$\map {\sigma_1} n = 2 n$
where $\map {\sigma_1} n$ denotes the divisor sum of $n$.
Hence:
{{begin-eqn}}
{{eqn | l = \map {\sigma_1} n
| r = 2 n
... | Let $n \in \N$ be a [[Definition:Perfect Number|perfect number]].
Then $n$ has at least two [[Definition:Distinct Elements|distinct]] [[Definition:Prime Factor|prime factors]]. | {{AimForCont}} the [[Definition:Contrary Statements|contrary]]: that $n$ is a [[Definition:Perfect Number|perfect number]] with exactly $1$ [[Definition:Prime Factor|prime factor]].
Hence let $n = p^k$ where $p$ is [[Definition:Prime Number|prime]].
By definition of [[Definition:Perfect Number|perfect number]]:
:$\m... | Perfect Number has at least Two Distinct Prime Factors/Proof 2 | https://proofwiki.org/wiki/Perfect_Number_has_at_least_Two_Distinct_Prime_Factors | https://proofwiki.org/wiki/Perfect_Number_has_at_least_Two_Distinct_Prime_Factors/Proof_2 | [
"Perfect Number has at least Two Distinct Prime Factors",
"Perfect Numbers"
] | [
"Definition:Perfect Number",
"Definition:Distinct/Plural",
"Definition:Prime Factor"
] | [
"Definition:Contrary Statements",
"Definition:Perfect Number",
"Definition:Prime Factor",
"Definition:Prime Number",
"Definition:Perfect Number",
"Definition:Divisor Sum Function",
"Divisor Sum of Power of Prime",
"Sum of Geometric Sequence",
"Definition:Abundancy Index",
"Upper Bound for Abundanc... |
proofwiki-21566 | Definite Integral from 0 to 1 of Arcsine of x by Arccosine of x over x | :$\ds \int_0^1 \frac {\arcsin x \arccos x } x \rd x = \dfrac 7 8 \map \zeta 3$
where $\map \zeta 3$ is Apéry's constant: the Riemann $\zeta$ function of $3$. | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\arcsin x \arccos x} x \rd x
| r = \int_0^1 \frac {\arcsin x \paren {\dfrac \pi 2 - \arcsin x} } x \rd x
| c = Sum of Arcsine and Arccosine: $\arcsin x + \arccos x = \dfrac \pi 2$
}}
{{eqn | r = \frac \pi 2 \int_0^1 \frac {\arcsin x} x \rd x - \int_0^1 \frac {\paren {... | :$\ds \int_0^1 \frac {\arcsin x \arccos x } x \rd x = \dfrac 7 8 \map \zeta 3$
where $\map \zeta 3$ is [[Definition:Apéry's Constant|Apéry's constant]]: the [[Definition:Riemann Zeta Function|Riemann $\zeta$ function]] of $3$. | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\arcsin x \arccos x} x \rd x
| r = \int_0^1 \frac {\arcsin x \paren {\dfrac \pi 2 - \arcsin x} } x \rd x
| c = [[Sum of Arcsine and Arccosine]]: $\arcsin x + \arccos x = \dfrac \pi 2$
}}
{{eqn | r = \frac \pi 2 \int_0^1 \frac {\arcsin x} x \rd x - \int_0^1 \frac {\par... | Definite Integral from 0 to 1 of Arcsine of x by Arccosine of x over x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Arcsine_of_x_by_Arccosine_of_x_over_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Arcsine_of_x_by_Arccosine_of_x_over_x | [
"Apéry's Constant",
"Arcsine Function",
"Arccosine Function",
"Dirichlet Eta Function",
"Riemann Zeta Function"
] | [
"Definition:Apéry's Constant",
"Definition:Riemann Zeta Function"
] | [
"Sum of Arcsine and Arccosine",
"Linear Combination of Integrals",
"Definition:Primitive",
"Integration by Parts",
"Power Rule for Derivatives",
"Primitive of Cotangent Function",
"Integration by Parts",
"Fourier Series for Logarithm of Sine of x over 0 to Pi",
"Linear Combination of Integrals",
"... |
proofwiki-21567 | Left Zero Divisor of Commutative Ring is Right Zero Divisor | Let $\struct {R, +, \circ}$ be a commutative ring.
Let $x \in R$ be a left zero divisor of $R$.
Then $x$ is also a right zero divisor of $R$. | Let $x \in R$ be a left zero divisor of $R$.
Let $0_R$ denote the zero of $R$.
Then:
{{begin-eqn}}
{{eqn | q = \exists y \in R^*
| l = x \circ y
| r = 0_R
| c = {{Defof|Left Zero Divisor}}
}}
{{eqn | ll= \leadsto
| l = y \circ x
| r = 0_R
| c = {{Defof|Commutative Ring}}
}}
{{end-eqn... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $x \in R$ be a [[Definition:Left Zero Divisor|left zero divisor]] of $R$.
Then $x$ is also a [[Definition:Right Zero Divisor|right zero divisor]] of $R$. | Let $x \in R$ be a [[Definition:Left Zero Divisor|left zero divisor]] of $R$.
Let $0_R$ denote the [[Definition:Ring Zero|zero]] of $R$.
Then:
{{begin-eqn}}
{{eqn | q = \exists y \in R^*
| l = x \circ y
| r = 0_R
| c = {{Defof|Left Zero Divisor}}
}}
{{eqn | ll= \leadsto
| l = y \circ x
... | Left Zero Divisor of Commutative Ring is Right Zero Divisor | https://proofwiki.org/wiki/Left_Zero_Divisor_of_Commutative_Ring_is_Right_Zero_Divisor | https://proofwiki.org/wiki/Left_Zero_Divisor_of_Commutative_Ring_is_Right_Zero_Divisor | [
"Zero Divisors"
] | [
"Definition:Commutative Ring",
"Definition:Left Zero Divisor",
"Definition:Right Zero Divisor"
] | [
"Definition:Left Zero Divisor",
"Definition:Ring Zero",
"Definition:Right Zero Divisor"
] |
proofwiki-21568 | Right Zero Divisor of Commutative Ring is Left Zero Divisor | Let $\struct {R, +, \circ}$ be a commutative ring.
Let $x \in R$ be a right zero divisor of $R$.
Then $x$ is also a left zero divisor of $R$. | Let $x \in R$ be a right zero divisor of $R$.
Let $0_R$ denote the zero of $R$.
Then:
{{begin-eqn}}
{{eqn | q = \exists y \in R^*
| l = x \circ y
| r = 0_R
| c = {{Defof|Left Zero Divisor}}
}}
{{eqn | ll= \leadsto
| l = y \circ x
| r = 0_R
| c = {{Defof|Commutative Ring}}
}}
{{end-eq... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $x \in R$ be a [[Definition:Right Zero Divisor|right zero divisor]] of $R$.
Then $x$ is also a [[Definition:Left Zero Divisor|left zero divisor]] of $R$. | Let $x \in R$ be a [[Definition:Right Zero Divisor|right zero divisor]] of $R$.
Let $0_R$ denote the [[Definition:Ring Zero|zero]] of $R$.
Then:
{{begin-eqn}}
{{eqn | q = \exists y \in R^*
| l = x \circ y
| r = 0_R
| c = {{Defof|Left Zero Divisor}}
}}
{{eqn | ll= \leadsto
| l = y \circ x
... | Right Zero Divisor of Commutative Ring is Left Zero Divisor | https://proofwiki.org/wiki/Right_Zero_Divisor_of_Commutative_Ring_is_Left_Zero_Divisor | https://proofwiki.org/wiki/Right_Zero_Divisor_of_Commutative_Ring_is_Left_Zero_Divisor | [
"Zero Divisors"
] | [
"Definition:Commutative Ring",
"Definition:Right Zero Divisor",
"Definition:Left Zero Divisor"
] | [
"Definition:Right Zero Divisor",
"Definition:Ring Zero",
"Definition:Left Zero Divisor"
] |
proofwiki-21569 | Odd Perfect Number has at least Three Distinct Prime Factors | Let $n \in \N$ be an odd perfect number.
Then $n$ has at least $3$ distinct prime factors. | {{AimForCont}} the contrary: that $n$ is an odd perfect number with at most $2$ distinct prime factors.
By Perfect Number has at least Two Distinct Prime Factors, $n$ must have exactly $2$ distinct prime factors.
Hence let $n = p^a q^b$ where $p$ and $q$ are distinct primes and where $a, b \in \N_{>0}$.
As $n$ is odd, ... | Let $n \in \N$ be an [[Definition:Odd Integer|odd]] [[Definition:Perfect Number|perfect number]].
Then $n$ has at least $3$ distinct [[Definition:Prime Factor|prime factors]]. | {{AimForCont}} the [[Definition:Contrary Statements|contrary]]: that $n$ is an [[Definition:Odd Integer|odd]] [[Definition:Perfect Number|perfect number]] with at most $2$ distinct [[Definition:Prime Factor|prime factors]].
By [[Perfect Number has at least Two Distinct Prime Factors]], $n$ must have exactly $2$ distin... | Odd Perfect Number has at least Three Distinct Prime Factors | https://proofwiki.org/wiki/Odd_Perfect_Number_has_at_least_Three_Distinct_Prime_Factors | https://proofwiki.org/wiki/Odd_Perfect_Number_has_at_least_Three_Distinct_Prime_Factors | [
"Perfect Numbers"
] | [
"Definition:Odd Integer",
"Definition:Perfect Number",
"Definition:Prime Factor"
] | [
"Definition:Contrary Statements",
"Definition:Odd Integer",
"Definition:Perfect Number",
"Definition:Prime Factor",
"Perfect Number has at least Two Distinct Prime Factors",
"Definition:Prime Factor",
"Definition:Prime Number",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Perfec... |
proofwiki-21570 | Arctangent of One | :$\map \arctan 1 = \dfrac \pi 4$ | By definition, $\arctan$ is the inverse of the tangent function's restriction to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
By Tangent of $45 \degrees$:
:$\tan \dfrac \pi 4 = 1$.
As $\dfrac \pi 4 \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$, we have by the definition of an inverse function:
:$\map \arctan 1 = \dfrac... | :$\map \arctan 1 = \dfrac \pi 4$ | By [[Definition:Real Arctangent|definition]], $\arctan$ is the [[Definition:Inverse of Mapping|inverse]] of the [[Definition:Real Tangent Function|tangent function]]'s [[Definition:Restriction of Mapping|restriction]] to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
By [[Tangent of 45 Degrees|Tangent of $45 \degrees$]]:... | Arctangent of One | https://proofwiki.org/wiki/Arctangent_of_One | https://proofwiki.org/wiki/Arctangent_of_One | [
"Arctangent Function"
] | [] | [
"Definition:Inverse Tangent/Real/Arctangent",
"Definition:Inverse of Mapping",
"Definition:Tangent Function/Real",
"Definition:Restriction/Mapping",
"Tangent of 45 Degrees",
"Definition:Inverse of Mapping",
"Category:Arctangent Function"
] |
proofwiki-21571 | Sum of Arctangents/Corollary | Let $n \in \N_{>0}$ where $\N_{>0}$ denotes the non-zero natural numbers.
Then:
:$\ds \map \arctan {\frac 1 n} + \map \arctan {\frac {n - 1} {n + 1} } = \frac \pi 4$
where $\arctan$ denotes the arctangent. | {{begin-eqn}}
{{eqn | l = \arctan a + \arctan b
| r = \map \arctan {\dfrac {a + b} {1 - a b} }
| c = Sum of Arctangents
}}
{{eqn | ll= \leadsto
| l = \map \arctan {\dfrac 1 n} + \map \arctan {\dfrac {n - 1} {n + 1} }
| r = \map \arctan {\dfrac {\dfrac 1 n + \dfrac {n - 1} {n + 1} } {1 - \paren {... | Let $n \in \N_{>0}$ where $\N_{>0}$ denotes the [[Definition:Non-Zero Natural Numbers|non-zero natural numbers]].
Then:
:$\ds \map \arctan {\frac 1 n} + \map \arctan {\frac {n - 1} {n + 1} } = \frac \pi 4$
where $\arctan$ denotes the [[Definition:Real Arctangent|arctangent]]. | {{begin-eqn}}
{{eqn | l = \arctan a + \arctan b
| r = \map \arctan {\dfrac {a + b} {1 - a b} }
| c = [[Sum of Arctangents]]
}}
{{eqn | ll= \leadsto
| l = \map \arctan {\dfrac 1 n} + \map \arctan {\dfrac {n - 1} {n + 1} }
| r = \map \arctan {\dfrac {\dfrac 1 n + \dfrac {n - 1} {n + 1} } {1 - \par... | Sum of Arctangents/Corollary | https://proofwiki.org/wiki/Sum_of_Arctangents/Corollary | https://proofwiki.org/wiki/Sum_of_Arctangents/Corollary | [
"Sum of Arctangents",
"Arctangent Function"
] | [
"Definition:Natural Numbers/Non-Zero",
"Definition:Inverse Tangent/Real/Arctangent"
] | [
"Sum of Arctangents",
"Arctangent of One",
"Category:Sum of Arctangents",
"Category:Arctangent Function"
] |
proofwiki-21572 | Digamma Function of One Third | :$\map \psi {\dfrac 1 3} = -\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 3}
| r = -\gamma - \ln 6 - \frac \pi 2 \map \cot {\frac 1 3 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {3 / 2} - 1} \map \cos {\frac {2 \pi n} 3} \map \ln {\map \sin {\frac {\pi n} 3} }
| c = Gauss's Digamma Theorem
}}
{{eqn | r = -\gamma - \ln 2 - \ln 3 - \frac \pi... | :$\map \psi {\dfrac 1 3} = -\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 3}
| r = -\gamma - \ln 6 - \frac \pi 2 \map \cot {\frac 1 3 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {3 / 2} - 1} \map \cos {\frac {2 \pi n} 3} \map \ln {\map \sin {\frac {\pi n} 3} }
| c = [[Gauss's Digamma Theorem]]
}}
{{eqn | r = -\gamma - \ln 2 - \ln 3 - \frac... | Digamma Function of One Third/Proof 1 | https://proofwiki.org/wiki/Digamma_Function_of_One_Third | https://proofwiki.org/wiki/Digamma_Function_of_One_Third/Proof_1 | [
"Digamma Function of One Third",
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Gauss's Digamma Theorem",
"Sum of Logarithms",
"Difference of Logarithms",
"Logarithm of Power"
] |
proofwiki-21573 | Digamma Function of One Third | :$\map \psi {\dfrac 1 3} = -\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{3 - 1} \map \psi {\frac k 3}
| r = -\paren {3 - 1} \gamma - 3 \ln 3
| c =
}}... | :$\map \psi {\dfrac 1 3} = -\gamma - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{3 - 1} \map \psi {\frac k 3}
| r = -\paren {3 - 1} \gamma - 3 \ln 3
| c =
}}... | Digamma Function of One Third/Proof 2 | https://proofwiki.org/wiki/Digamma_Function_of_One_Third | https://proofwiki.org/wiki/Digamma_Function_of_One_Third/Proof_2 | [
"Digamma Function of One Third",
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula"
] |
proofwiki-21574 | Digamma Function of One Fourth | :$\map \psi {\dfrac 1 4} = -\gamma - 3 \ln 2 - \dfrac \pi 2$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 4}
| r = -\gamma - \ln 8 - \frac \pi 2 \map \cot {\frac 1 4 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {4 / 2} - 1} \map \cos {\frac {2 \pi n} 4} \map \ln {\map \sin {\frac {\pi n} 4} }
| c = Gauss's Digamma Theorem
}}
{{eqn | r = -\gamma - 3 \ln 2 - \frac \pi 2 \ti... | :$\map \psi {\dfrac 1 4} = -\gamma - 3 \ln 2 - \dfrac \pi 2$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 4}
| r = -\gamma - \ln 8 - \frac \pi 2 \map \cot {\frac 1 4 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {4 / 2} - 1} \map \cos {\frac {2 \pi n} 4} \map \ln {\map \sin {\frac {\pi n} 4} }
| c = [[Gauss's Digamma Theorem]]
}}
{{eqn | r = -\gamma - 3 \ln 2 - \frac \pi 2... | Digamma Function of One Fourth/Proof 1 | https://proofwiki.org/wiki/Digamma_Function_of_One_Fourth | https://proofwiki.org/wiki/Digamma_Function_of_One_Fourth/Proof_1 | [
"Digamma Function of One Fourth",
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Gauss's Digamma Theorem",
"Sum of Logarithms"
] |
proofwiki-21575 | Digamma Function of One Fourth | :$\map \psi {\dfrac 1 4} = -\gamma - 3 \ln 2 - \dfrac \pi 2$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{4 - 1} \map \psi {\frac k 4}
| r = -\paren {4 - 1} \gamma - 4 \ln 4
| c =
}}... | :$\map \psi {\dfrac 1 4} = -\gamma - 3 \ln 2 - \dfrac \pi 2$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{4 - 1} \map \psi {\frac k 4}
| r = -\paren {4 - 1} \gamma - 4 \ln 4
| c =
}}... | Digamma Function of One Fourth/Proof 2 | https://proofwiki.org/wiki/Digamma_Function_of_One_Fourth | https://proofwiki.org/wiki/Digamma_Function_of_One_Fourth/Proof_2 | [
"Digamma Function of One Fourth",
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula",
"Digamma Function of One Half",
"Logarithm of Power"
] |
proofwiki-21576 | Digamma Function of Two Thirds | :$\map \psi {\dfrac 2 3} = -\gamma - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 3} - \map \psi {\frac 2 3}
| r = -\pi \map \cot {\frac \pi 3}
| c = Digamma Reflection Formula
}}
{{eqn | ll= \leadsto
| l = \map \psi {\frac 2 3}
| r = \pi \map \cot {\frac \pi 3} + \map \psi {\frac 1 3}
| c = rearranging
}}
{{eqn | r = \dfrac... | :$\map \psi {\dfrac 2 3} = -\gamma - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 3} - \map \psi {\frac 2 3}
| r = -\pi \map \cot {\frac \pi 3}
| c = [[Digamma Reflection Formula]]
}}
{{eqn | ll= \leadsto
| l = \map \psi {\frac 2 3}
| r = \pi \map \cot {\frac \pi 3} + \map \psi {\frac 1 3}
| c = rearranging
}}
{{eqn | r = \d... | Digamma Function of Two Thirds | https://proofwiki.org/wiki/Digamma_Function_of_Two_Thirds | https://proofwiki.org/wiki/Digamma_Function_of_Two_Thirds | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula",
"Digamma Function of One Third",
"Category:Examples of Digamma Function",
"Category:Euler-Mascheroni Constant"
] |
proofwiki-21577 | Digamma Function of Three Fourths | :$\map \psi {\dfrac 3 4} = -\gamma - 3 \ln 2 + \dfrac \pi 2$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 4} - \map \psi {\frac 3 4}
| r = -\pi \map \cot {\frac \pi 4}
| c = Digamma Reflection Formula
}}
{{eqn | ll= \leadsto
| l = \map \psi {\frac 3 4}
| r = \pi \map \cot {\frac \pi 4} + \map \psi {\frac 1 4}
| c = rearranging
}}
{{eqn | r = \pi \t... | :$\map \psi {\dfrac 3 4} = -\gamma - 3 \ln 2 + \dfrac \pi 2$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 4} - \map \psi {\frac 3 4}
| r = -\pi \map \cot {\frac \pi 4}
| c = [[Digamma Reflection Formula]]
}}
{{eqn | ll= \leadsto
| l = \map \psi {\frac 3 4}
| r = \pi \map \cot {\frac \pi 4} + \map \psi {\frac 1 4}
| c = rearranging
}}
{{eqn | r = \p... | Digamma Function of Three Fourths | https://proofwiki.org/wiki/Digamma_Function_of_Three_Fourths | https://proofwiki.org/wiki/Digamma_Function_of_Three_Fourths | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula",
"Digamma Function of One Fourth",
"Category:Examples of Digamma Function",
"Category:Euler-Mascheroni Constant"
] |
proofwiki-21578 | Digamma Function of One Sixth | :$\map \psi {\dfrac 1 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{6 - 1} \map \psi {\frac k 6}
| r = -\paren {6 - 1} \gamma - 6 \ln 6
| c =
}}... | :$\map \psi {\dfrac 1 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 - \dfrac {\pi \sqrt 3} 2$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}
| r = -\paren {n - 1} \gamma - n \ln n
| c = {{Corollary|Digamma Additive Formula}}
}}
{{eqn | ll= \leadsto
| l = \sum_{k \mathop = 1}^{6 - 1} \map \psi {\frac k 6}
| r = -\paren {6 - 1} \gamma - 6 \ln 6
| c =
}}... | Digamma Function of One Sixth/Proof 2 | https://proofwiki.org/wiki/Digamma_Function_of_One_Sixth | https://proofwiki.org/wiki/Digamma_Function_of_One_Sixth/Proof_2 | [
"Digamma Function of One Sixth",
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula",
"Sum of Logarithms",
"Digamma Function of One Third",
"Digamma Function of One Half",
"Digamma Function of Two Thirds"
] |
proofwiki-21579 | Digamma Function of Five Sixths | :$\map \psi {\dfrac 5 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 + \dfrac {\pi \sqrt 3} 2$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 6} - \map \psi {\frac 5 6}
| r = -\pi \map \cot {\frac \pi 6}
| c = Digamma Reflection Formula
}}
{{eqn | ll= \leadsto
| l = \map \psi {\frac 5 6}
| r = \pi \map \cot {\frac \pi 6} + \map \psi {\frac 1 6}
| c = rearranging
}}
{{eqn | r = \pi \t... | :$\map \psi {\dfrac 5 6} = -\gamma - 2 \ln 2 - \dfrac 3 2 \ln 3 + \dfrac {\pi \sqrt 3} 2$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 6} - \map \psi {\frac 5 6}
| r = -\pi \map \cot {\frac \pi 6}
| c = [[Digamma Reflection Formula]]
}}
{{eqn | ll= \leadsto
| l = \map \psi {\frac 5 6}
| r = \pi \map \cot {\frac \pi 6} + \map \psi {\frac 1 6}
| c = rearranging
}}
{{eqn | r = \p... | Digamma Function of Five Sixths | https://proofwiki.org/wiki/Digamma_Function_of_Five_Sixths | https://proofwiki.org/wiki/Digamma_Function_of_Five_Sixths | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula",
"Digamma Function of One Sixth",
"Category:Examples of Digamma Function",
"Category:Euler-Mascheroni Constant"
] |
proofwiki-21580 | Digamma Reflection Formula | :$\map \psi z - \map \psi {1 - z} = -\pi \cot \pi z$ | === Lemma ===
{{:Polygamma Reflection Formula/Lemma}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \map \Gamma z \map \Gamma {1 - z}
| r = \dfrac \pi {\sin \pi z}
| c = Euler's Reflection Formula
}}
{{eqn | ll= \leadsto
| l = \map \ln {\map \Gamma z \map \Gamma {1 - z} }
| r = \map \ln {\dfrac \pi {\s... | :$\map \psi z - \map \psi {1 - z} = -\pi \cot \pi z$ | === [[Polygamma Reflection Formula/Lemma|Lemma]] ===
{{:Polygamma Reflection Formula/Lemma}}{{qed|lemma}}
{{begin-eqn}}
{{eqn | l = \map \Gamma z \map \Gamma {1 - z}
| r = \dfrac \pi {\sin \pi z}
| c = [[Euler's Reflection Formula]]
}}
{{eqn | ll= \leadsto
| l = \map \ln {\map \Gamma z \map \Gamma {... | Digamma Reflection Formula | https://proofwiki.org/wiki/Digamma_Reflection_Formula | https://proofwiki.org/wiki/Digamma_Reflection_Formula | [
"Cotangent Function",
"Digamma Function",
"Reflection Formulas"
] | [] | [
"Polygamma Reflection Formula/Lemma",
"Euler's Reflection Formula",
"Sum of Logarithms",
"Difference of Logarithms",
"Definition:Differentiation",
"Derivative of Natural Logarithm Function",
"Derivative of Sine Function",
"Derivative of Composite Function",
"Derivative of Constant",
"Definition:Do... |
proofwiki-21581 | Recurrence Relation for Digamma Function | :$\map \psi {z + 1} = \map \psi z + \dfrac 1 z$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {z + 1}
| r = z \map \Gamma z
| c = Gamma Difference Equation
}}
{{eqn | ll= \leadsto
| l = \map \ln {\map \Gamma {z + 1} }
| r = \map \ln {z \map \Gamma z}
| c = applying $\ln$ on both sides
}}
{{eqn | r = \ln z + \map \ln {\map \Gamma z}
| c = Sum of L... | :$\map \psi {z + 1} = \map \psi z + \dfrac 1 z$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {z + 1}
| r = z \map \Gamma z
| c = [[Gamma Difference Equation]]
}}
{{eqn | ll= \leadsto
| l = \map \ln {\map \Gamma {z + 1} }
| r = \map \ln {z \map \Gamma z}
| c = applying $\ln$ on both sides
}}
{{eqn | r = \ln z + \map \ln {\map \Gamma z}
| c = [[Su... | Recurrence Relation for Digamma Function | https://proofwiki.org/wiki/Recurrence_Relation_for_Digamma_Function | https://proofwiki.org/wiki/Recurrence_Relation_for_Digamma_Function | [
"Digamma Function",
"Recurrence Relations"
] | [] | [
"Gamma Difference Equation",
"Sum of Logarithms",
"Definition:Differentiation",
"Derivative of Natural Logarithm Function",
"Derivative of Composite Function"
] |
proofwiki-21582 | Lattice Homomorphism is Both Meet and Join Semilattice Homomorphism | Let $L_1 = \struct{S_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct{S_2, \vee_2, \wedge_2, \preceq_2}$ be lattices.
Let $\phi: L_1 \to L_2$ be a lattice homomorphsm between $L_1$ and $L_2$.
Then:
:* $\phi: \struct{S_1, \vee_1, \preceq_1} \to \struct{S_2, \vee_2, \preceq_2}$
:* $\phi: \struct{S_1, \wedge_1, \preceq... | This follows immediately from:
:* Definition:Lattice (Order Theory)
:* Definition:Semilattice
:* Definition:Lattice Homomorphism
:* Definition:Semilattice Homomorphism
{{qed}}
Category:Lattice Homomorphisms
bxj5phrwt6qbj1pyl4ltpbnuosvdew4 | Let $L_1 = \struct{S_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct{S_2, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Lattice (Order Theory)|lattices]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Lattice Homomorphism|lattice homomorphsm]] between $L_1$ and $L_2$.
Then:
:* $\phi: \struct{S_1, \vee_1, \preceq_1}... | This follows immediately from:
:* [[Definition:Lattice (Order Theory)]]
:* [[Definition:Semilattice]]
:* [[Definition:Lattice Homomorphism]]
:* [[Definition:Semilattice Homomorphism]]
{{qed}}
[[Category:Lattice Homomorphisms]]
bxj5phrwt6qbj1pyl4ltpbnuosvdew4 | Lattice Homomorphism is Both Meet and Join Semilattice Homomorphism | https://proofwiki.org/wiki/Lattice_Homomorphism_is_Both_Meet_and_Join_Semilattice_Homomorphism | https://proofwiki.org/wiki/Lattice_Homomorphism_is_Both_Meet_and_Join_Semilattice_Homomorphism | [
"Lattice Homomorphisms"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Lattice Homomorphism",
"Definition:Semilattice Homomorphism"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Semilattice",
"Definition:Lattice Homomorphism",
"Definition:Semilattice Homomorphism",
"Category:Lattice Homomorphisms"
] |
proofwiki-21583 | Lattice Homomorphism is Order-Preserving | Let $L_1 = \struct{S_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct{S_2, \vee_2, \wedge_2, \preceq_2}$ be lattices.
Let $\phi: L_1 \to L_2$ be a lattice homomorphsm between $L_1$ and $L_2$.
Then:
:$\phi: \struct{S_1, \preceq_1} \to \struct{S_2, \preceq_2}$ is order-preserving | From Lattice Homomorphism is Both Meet and Join Semilattice Homomorphism:
:$\phi: \struct{S_1, \vee_1, \preceq_1} \to \struct{S_2, \vee_2, \preceq_2}$ is a semilattice homomorphism
From Semilattice Homomorphism is Order-Preserving:
:$\phi: \struct{S_1, \preceq_1} \to \struct{S_2, \preceq_2}$ is order-preserving
{{qed}}... | Let $L_1 = \struct{S_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct{S_2, \vee_2, \wedge_2, \preceq_2}$ be [[Definition:Lattice (Order Theory)|lattices]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Lattice Homomorphism|lattice homomorphsm]] between $L_1$ and $L_2$.
Then:
:$\phi: \struct{S_1, \preceq_1} \to \stru... | From [[Lattice Homomorphism is Both Meet and Join Semilattice Homomorphism]]:
:$\phi: \struct{S_1, \vee_1, \preceq_1} \to \struct{S_2, \vee_2, \preceq_2}$ is a [[Definition:Semilattice Homomorphism|semilattice homomorphism]]
From [[Semilattice Homomorphism is Order-Preserving]]:
:$\phi: \struct{S_1, \preceq_1} \to \st... | Lattice Homomorphism is Order-Preserving | https://proofwiki.org/wiki/Lattice_Homomorphism_is_Order-Preserving | https://proofwiki.org/wiki/Lattice_Homomorphism_is_Order-Preserving | [
"Lattice Homomorphisms"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Lattice Homomorphism",
"Definition:Increasing/Mapping"
] | [
"Lattice Homomorphism is Both Meet and Join Semilattice Homomorphism",
"Definition:Semilattice Homomorphism",
"Semilattice Homomorphism is Order-Preserving",
"Definition:Increasing/Mapping",
"Category:Lattice Homomorphisms"
] |
proofwiki-21584 | Ordered Set has Upper Bound iff has Greatest Element | Let $\struct{S,\preceq}$ be an ordered set.
Then:
:$S$ has an upper bound in $\struct{S,\preceq}$ {{iff}} $\struct{S,\preceq}$ has a greatest element | This follows immediately from:
:* Definition:Upper Bound
:* Definition:Greatest Element
{{qed}}
Category:Greatest Elements
Category:Ordered Sets
57i8my33m9a21nxibyxjkmcdo4hop64 | Let $\struct{S,\preceq}$ be an [[Definition:Ordered Set|ordered set]].
Then:
:$S$ has an [[Definition:Upper Bound|upper bound]] in $\struct{S,\preceq}$ {{iff}} $\struct{S,\preceq}$ has a [[Definition:Greatest Element|greatest element]] | This follows immediately from:
:* [[Definition:Upper Bound]]
:* [[Definition:Greatest Element]]
{{qed}}
[[Category:Greatest Elements]]
[[Category:Ordered Sets]]
57i8my33m9a21nxibyxjkmcdo4hop64 | Ordered Set has Upper Bound iff has Greatest Element | https://proofwiki.org/wiki/Ordered_Set_has_Upper_Bound_iff_has_Greatest_Element | https://proofwiki.org/wiki/Ordered_Set_has_Upper_Bound_iff_has_Greatest_Element | [
"Greatest Elements",
"Ordered Sets"
] | [
"Definition:Ordered Set",
"Definition:Upper Bound",
"Definition:Greatest Element"
] | [
"Definition:Upper Bound",
"Definition:Greatest Element",
"Category:Greatest Elements",
"Category:Ordered Sets"
] |
proofwiki-21585 | Ordered Set has Lower Bound iff has Smallest Element | Let $\struct{S,\preceq}$ be an ordered set.
Then:
:$S$ has a lower bound in $\struct{S,\preceq}$ {{iff}} $\struct{S,\preceq}$ has a smallest element | This follows immediately from:
:* Definition:Lower Bound
:* Definition:Smallest Element
{{qed}}
Category:Smallest Elements
Category:Ordered Sets
q89lwnug5ka01sl6xy8p7kkhkgg8qrq | Let $\struct{S,\preceq}$ be an [[Definition:Ordered Set|ordered set]].
Then:
:$S$ has a [[Definition:Lower Bound|lower bound]] in $\struct{S,\preceq}$ {{iff}} $\struct{S,\preceq}$ has a [[Definition:Smallest Element|smallest element]] | This follows immediately from:
:* [[Definition:Lower Bound]]
:* [[Definition:Smallest Element]]
{{qed}}
[[Category:Smallest Elements]]
[[Category:Ordered Sets]]
q89lwnug5ka01sl6xy8p7kkhkgg8qrq | Ordered Set has Lower Bound iff has Smallest Element | https://proofwiki.org/wiki/Ordered_Set_has_Lower_Bound_iff_has_Smallest_Element | https://proofwiki.org/wiki/Ordered_Set_has_Lower_Bound_iff_has_Smallest_Element | [
"Smallest Elements",
"Ordered Sets"
] | [
"Definition:Ordered Set",
"Definition:Lower Bound",
"Definition:Smallest Element"
] | [
"Definition:Lower Bound",
"Definition:Smallest Element",
"Category:Smallest Elements",
"Category:Ordered Sets"
] |
proofwiki-21586 | Ramanujan Phi Function in terms of Digamma Function | :$\map \psi {\dfrac 1 z} + \map \psi {1 - \dfrac 1 z} = -2 \gamma - z \map \phi z$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 z} + \map \psi {1 - \frac 1 z}
| r = -z + \map \psi {1 + \frac 1 z} + \map \psi {1 - \frac 1 z}
| c = Recurrence Relation for Digamma Function: $\map \psi {z + 1} = \map \psi z + \dfrac 1 z$
}}
{{eqn | r = -z + \paren {-\gamma + \sum_{n \mathop = 1}^\infty \paren... | :$\map \psi {\dfrac 1 z} + \map \psi {1 - \dfrac 1 z} = -2 \gamma - z \map \phi z$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 z} + \map \psi {1 - \frac 1 z}
| r = -z + \map \psi {1 + \frac 1 z} + \map \psi {1 - \frac 1 z}
| c = [[Recurrence Relation for Digamma Function]]: $\map \psi {z + 1} = \map \psi z + \dfrac 1 z$
}}
{{eqn | r = -z + \paren {-\gamma + \sum_{n \mathop = 1}^\infty \p... | Ramanujan Phi Function in terms of Digamma Function | https://proofwiki.org/wiki/Ramanujan_Phi_Function_in_terms_of_Digamma_Function | https://proofwiki.org/wiki/Ramanujan_Phi_Function_in_terms_of_Digamma_Function | [
"Digamma Function",
"Ramanujan Phi Function"
] | [] | [
"Recurrence Relation for Digamma Function",
"Reciprocal times Derivative of Gamma Function",
"Linear Combination of Convergent Series",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares"
] |
proofwiki-21587 | Bounded Lattice has Both Greatest Element and Smallest Element | Let $\struct {S, \vee, \wedge, \preceq}$ be a bounded lattice.
Then:
:$(1): \quad \struct {S, \preceq}$ has a smallest element, namely:
::::$\bot := \sup \O$
:$(2): \quad \struct {S, \preceq}$ has a greatest element, namely:
::::$\top := \inf \O$ | By definition of bounded lattice:
:$\bot = \sup \O$ and $\top = \inf \O$ exist
From Supremum of Empty Set is Smallest Element:
:$\bot = \sup \O$ {{iff}} $\bot$ is the smallest element of $\struct {S, \preceq}$
From Infimum of Empty Set is Greatest Element:
:$\top = \inf \O$ {{iff}} $\top$ is the greatest element of $\s... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Bounded Lattice|bounded lattice]].
Then:
:$(1): \quad \struct {S, \preceq}$ has a [[Definition:Smallest Element|smallest element]], namely:
::::$\bot := \sup \O$
:$(2): \quad \struct {S, \preceq}$ has a [[Definition:Greatest Element|greatest element]], namely... | By definition of [[Definition:Bounded Lattice|bounded lattice]]:
:$\bot = \sup \O$ and $\top = \inf \O$ exist
From [[Supremum of Empty Set is Smallest Element]]:
:$\bot = \sup \O$ {{iff}} $\bot$ is the [[Definition:Smallest Element|smallest element]] of $\struct {S, \preceq}$
From [[Infimum of Empty Set is Greatest E... | Bounded Lattice has Both Greatest Element and Smallest Element | https://proofwiki.org/wiki/Bounded_Lattice_has_Both_Greatest_Element_and_Smallest_Element | https://proofwiki.org/wiki/Bounded_Lattice_has_Both_Greatest_Element_and_Smallest_Element | [
"Bounded Lattices"
] | [
"Definition:Bounded Lattice",
"Definition:Smallest Element",
"Definition:Greatest Element"
] | [
"Definition:Bounded Lattice",
"Supremum of Empty Set is Smallest Element",
"Definition:Smallest Element",
"Infimum of Empty Set is Greatest Element",
"Definition:Greatest Element",
"Category:Bounded Lattices"
] |
proofwiki-21588 | Dirichlet Eta Function at Non-Positive Integers | Let $n \ge 0$ be a integer.
Then:
:$\map \eta {-n} = \dfrac {\paren {-1}^{n + 1} \paren {2^{n + 1 } - 1} B_{n + 1} } {n + 1}$
where:
:$B_n$ is the $n$th Bernoulli number
:$\eta$ is the Dirichlet eta function | {{begin-eqn}}
{{eqn | l = \map \eta s
| r = \paren {1 - 2^{1 - s} } \map \zeta s
| c = Riemann Zeta Function in terms of Dirichlet Eta Function
}}
{{eqn | ll = \leadsto
| l = \map \eta {- n}
| r = \paren {1 - 2^{1 - \paren {-n} } } \map \zeta {-n}
| c = setting $s := - n$
}}
{{eqn | r = \p... | Let $n \ge 0$ be a [[Definition:Integer|integer]].
Then:
:$\map \eta {-n} = \dfrac {\paren {-1}^{n + 1} \paren {2^{n + 1 } - 1} B_{n + 1} } {n + 1}$
where:
:$B_n$ is the [[Definition:Bernoulli Numbers|$n$th Bernoulli number]]
:$\eta$ is the [[Definition:Dirichlet Eta Function|Dirichlet eta function]] | {{begin-eqn}}
{{eqn | l = \map \eta s
| r = \paren {1 - 2^{1 - s} } \map \zeta s
| c = [[Riemann Zeta Function in terms of Dirichlet Eta Function]]
}}
{{eqn | ll = \leadsto
| l = \map \eta {- n}
| r = \paren {1 - 2^{1 - \paren {-n} } } \map \zeta {-n}
| c = setting $s := - n$
}}
{{eqn | r ... | Dirichlet Eta Function at Non-Positive Integers | https://proofwiki.org/wiki/Dirichlet_Eta_Function_at_Non-Positive_Integers | https://proofwiki.org/wiki/Dirichlet_Eta_Function_at_Non-Positive_Integers | [
"Dirichlet Eta Function"
] | [
"Definition:Integer",
"Definition:Bernoulli Numbers",
"Definition:Dirichlet Eta Function"
] | [
"Riemann Zeta Function in terms of Dirichlet Eta Function",
"Riemann Zeta Function at Non-Positive Integers"
] |
proofwiki-21589 | Integral Form of Ramanujan Phi Function | :$\ds \map \phi x = \int_{\to 0}^{\to 1} \paren {\dfrac {u^{x - 2} \paren {1 - u}^2} {1 - u^x} } \rd u + 1$
where:
:$\map \phi x$ denotes the Ramanujan phi function
:$x \in \C$
:$\map \Re x > 1$ | {{begin-eqn}}
{{eqn | l = \int_{\to 0}^{\to 1} \paren {\frac {u^{x - 2} \paren {1 - u}^2} {1 - u^x} } \rd u + 1
| r = 1 + \int_{\to 0}^{\to 1} \paren {\frac {u^x} {1 - u^x} } \times \paren {u^{-2} \paren {1 - 2 u + u^2} } \rd u
| c = Square of Sum
}}
{{eqn | r = 1 + \int_{\to 0}^{\to 1} \paren {\frac {u^x} ... | :$\ds \map \phi x = \int_{\to 0}^{\to 1} \paren {\dfrac {u^{x - 2} \paren {1 - u}^2} {1 - u^x} } \rd u + 1$
where:
:$\map \phi x$ denotes the [[Definition:Ramanujan Phi Function of One Argument|Ramanujan phi function]]
:$x \in \C$
:$\map \Re x > 1$ | {{begin-eqn}}
{{eqn | l = \int_{\to 0}^{\to 1} \paren {\frac {u^{x - 2} \paren {1 - u}^2} {1 - u^x} } \rd u + 1
| r = 1 + \int_{\to 0}^{\to 1} \paren {\frac {u^x} {1 - u^x} } \times \paren {u^{-2} \paren {1 - 2 u + u^2} } \rd u
| c = [[Square of Sum]]
}}
{{eqn | r = 1 + \int_{\to 0}^{\to 1} \paren {\frac {u... | Integral Form of Ramanujan Phi Function | https://proofwiki.org/wiki/Integral_Form_of_Ramanujan_Phi_Function | https://proofwiki.org/wiki/Integral_Form_of_Ramanujan_Phi_Function | [
"Ramanujan Phi Function",
"Definite Integrals"
] | [
"Definition:Ramanujan Phi Function/One Argument"
] | [
"Square of Sum",
"Sum of Infinite Geometric Sequence",
"Linear Combination of Integrals",
"Tonelli's Theorem",
"Integral of Power",
"Definition:Common Denominator",
"Difference of Two Squares"
] |
proofwiki-21590 | Complete Lattice has Both Greatest Element and Smallest Element | Let $\struct{S, \vee, \wedge, \preceq}$ be a complete lattice.
Then:
:$(\text{1}) \quad \struct{S, \preceq}$ has a smallest element, namely:
::$\quad \bot := \sup \O$
:$(\text{2}) \quad \struct{S, \preceq}$ has a greatest element, namely:
::$\quad \top := \inf \O$ | From Complete Lattice is Bounded:
:$\struct{S, \vee, \wedge, \preceq}$ is a bounded lattice
From Bounded Lattice has Both Greatest Element and Smallest Element:
:$(\text{1}) \quad \struct{S, \preceq}$ has a smallest element, namely:
::$\quad \bot := \sup \O$
:$(\text{2}) \quad \struct{S, \preceq}$ has a greatest elemen... | Let $\struct{S, \vee, \wedge, \preceq}$ be a [[Definition:Complete Lattice|complete lattice]].
Then:
:$(\text{1}) \quad \struct{S, \preceq}$ has a [[Definition:Smallest Element|smallest element]], namely:
::$\quad \bot := \sup \O$
:$(\text{2}) \quad \struct{S, \preceq}$ has a [[Definition:Greatest Element|greatest el... | From [[Complete Lattice is Bounded]]:
:$\struct{S, \vee, \wedge, \preceq}$ is a [[Definition:Bounded Lattice|bounded lattice]]
From [[Bounded Lattice has Both Greatest Element and Smallest Element]]:
:$(\text{1}) \quad \struct{S, \preceq}$ has a [[Definition:Smallest Element|smallest element]], namely:
::$\quad \bot :... | Complete Lattice has Both Greatest Element and Smallest Element | https://proofwiki.org/wiki/Complete_Lattice_has_Both_Greatest_Element_and_Smallest_Element | https://proofwiki.org/wiki/Complete_Lattice_has_Both_Greatest_Element_and_Smallest_Element | [
"Complete Lattices"
] | [
"Definition:Complete Lattice",
"Definition:Smallest Element",
"Definition:Greatest Element"
] | [
"Complete Lattice is Bounded",
"Definition:Bounded Lattice",
"Bounded Lattice has Both Greatest Element and Smallest Element",
"Definition:Smallest Element",
"Definition:Greatest Element",
"Category:Complete Lattices"
] |
proofwiki-21591 | Equivalence of Complete Semilattice and Complete Lattice | Let $\struct {S, \preceq}$ be an ordered set.
{{TFAE}}:
:$(\text 1) \quad \struct {S, \preceq}$ is a complete join semilattice
:$(\text 2) \quad \struct {S, \preceq}$ is a complete meet semilattice
:$(\text 3) \quad \struct {S, \preceq}$ is a complete lattice | === Statement $(\text 1)$ implies Statement $(\text 2)$ ===
Let $\struct {S, \preceq}$ be a complete join semilattice.
Let $A \subseteq S$.
Let $T = \leftset{s \in S : s}$ is a lower bound for $\rightset{A}$
By definition of complete join semilattice:
:$\sup T$ exists in $\struct {S, \preceq}$
Let $a = \sup T$.
We have... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
{{TFAE}}:
:$(\text 1) \quad \struct {S, \preceq}$ is a [[Definition:Complete Join Semilattice|complete join semilattice]]
:$(\text 2) \quad \struct {S, \preceq}$ is a [[Definition:Complete Meet Semilattice|complete meet semilattice]]
:$(\text 3)... | === Statement $(\text 1)$ implies Statement $(\text 2)$ ===
Let $\struct {S, \preceq}$ be a [[Definition:Complete Join Semilattice|complete join semilattice]].
Let $A \subseteq S$.
Let $T = \leftset{s \in S : s}$ is a [[Definition:Lower Bound|lower bound]] for $\rightset{A}$
By definition of [[Definition:Complete... | Equivalence of Complete Semilattice and Complete Lattice | https://proofwiki.org/wiki/Equivalence_of_Complete_Semilattice_and_Complete_Lattice | https://proofwiki.org/wiki/Equivalence_of_Complete_Semilattice_and_Complete_Lattice | [
"Complete Lattices"
] | [
"Definition:Ordered Set",
"Definition:Complete Join Semilattice",
"Definition:Complete Meet Semilattice",
"Definition:Complete Lattice"
] | [
"Definition:Complete Join Semilattice",
"Definition:Lower Bound",
"Definition:Complete Join Semilattice",
"Definition:Upper Bound",
"Definition:Supremum of Set",
"Definition:Lower Bound",
"Definition:Lower Bound",
"Definition:Supremum of Set",
"Definition:Lower Bound",
"Definition:Complete Meet Se... |
proofwiki-21592 | Complete Join Semilattice is Dual to Complete Meet Semilattice | Let $\struct {S, \preceq}$ be an ordered set.
The following are dual statements:
:$\struct{S, \preceq}$ is a complete join semilattice
:$\struct{S, \preceq}$ is a complete meet semilattice | By definition of complete join semilattice:
:$\struct{S, \preceq}$ is a complete join semilattice
{{iff}}:
:$\forall S' \subseteq S : \sup S' \in S$, where $\sup S'$ is the supremum of $S'$
The dual of this statement is:
:$\forall S' \subseteq S : \inf S' \in S$, where $\inf S'$ is the infimum of $S'$ by Dual Pair... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
The following are [[Definition:Dual Statement (Order Theory)|dual statements]]:
:$\struct{S, \preceq}$ is a [[Definition:Complete Join Semilattice|complete join semilattice]]
:$\struct{S, \preceq}$ is a [[Definition:Complete Meet Semilattice|co... | By definition of [[Definition:Complete Join Semilattice|complete join semilattice]]:
:$\struct{S, \preceq}$ is a [[Definition:Complete Join Semilattice|complete join semilattice]]
{{iff}}:
:$\forall S' \subseteq S : \sup S' \in S$, where $\sup S'$ is the [[Definition:Supremum of Set|supremum]] of $S'$
The [[Defi... | Complete Join Semilattice is Dual to Complete Meet Semilattice | https://proofwiki.org/wiki/Complete_Join_Semilattice_is_Dual_to_Complete_Meet_Semilattice | https://proofwiki.org/wiki/Complete_Join_Semilattice_is_Dual_to_Complete_Meet_Semilattice | [
"Complete Lattices",
"Dual Pairs (Order Theory)"
] | [
"Definition:Ordered Set",
"Definition:Dual Statement (Order Theory)",
"Definition:Complete Join Semilattice",
"Definition:Complete Meet Semilattice"
] | [
"Definition:Complete Join Semilattice",
"Definition:Complete Join Semilattice",
"Definition:Supremum of Set",
"Definition:Dual Statement (Order Theory)",
"Definition:Infimum of Set",
"Dual Pairs (Order Theory)",
"Definition:Complete Meet Semilattice",
"Definition:Complete Meet Semilattice"
] |
proofwiki-21593 | Axes of Symmetry for Ellipse | Let $K$ be an ellipse.
Then $K$ has exactly $2$ axis of bilateral symmetry:
:the major axis of $K$
:the minor axis of $K$. | From:
:Major Axis of Ellipse is Axis of Symmetry
:Minor Axis of Ellipse is Axis of Symmetry
we have that these diameters of $K$ are in fact axis of bilateral symmetry.
It remains to be shown that there are no more.
{{ProofWanted}} | Let $K$ be an [[Definition:Ellipse|ellipse]].
Then $K$ has exactly $2$ [[Definition:Axis of Bilateral Symmetry|axis of bilateral symmetry]]:
:the [[Definition:Major Axis of Ellipse|major axis]] of $K$
:the [[Definition:Minor Axis of Ellipse|minor axis]] of $K$. | From:
:[[Major Axis of Ellipse is Axis of Symmetry]]
:[[Minor Axis of Ellipse is Axis of Symmetry]]
we have that these [[Definition:Diameter of Ellipse|diameters]] of $K$ are in fact [[Definition:Axis of Bilateral Symmetry|axis of bilateral symmetry]].
It remains to be shown that there are no more.
{{ProofWanted}} | Axes of Symmetry for Ellipse | https://proofwiki.org/wiki/Axes_of_Symmetry_for_Ellipse | https://proofwiki.org/wiki/Axes_of_Symmetry_for_Ellipse | [
"Axes of Symmetry for Ellipse",
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Bilateral Symmetry/Axis",
"Definition:Ellipse/Major Axis",
"Definition:Ellipse/Minor Axis"
] | [
"Major Axis of Ellipse is Axis of Symmetry",
"Minor Axis of Ellipse is Axis of Symmetry",
"Definition:Diameter of Ellipse",
"Definition:Bilateral Symmetry/Axis"
] |
proofwiki-21594 | Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 3 | :$\map \beta 3 = \dfrac {\pi^3} {32} $ | {{begin-eqn}}
{{eqn | l = \map \beta {2 n + 1}
| r = \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}
| c = Dirichlet Beta Function at Odd Positive Integers
}}
{{eqn | ll = \leadsto
| l = \map \beta 3
| r = \paren {-1}^1 \dfrac {E_2 \pi^3 } {4^2 \paren {2}!}
| c = se... | :$\map \beta 3 = \dfrac {\pi^3} {32} $ | {{begin-eqn}}
{{eqn | l = \map \beta {2 n + 1}
| r = \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}
| c = [[Dirichlet Beta Function at Odd Positive Integers]]
}}
{{eqn | ll = \leadsto
| l = \map \beta 3
| r = \paren {-1}^1 \dfrac {E_2 \pi^3 } {4^2 \paren {2}!}
| c ... | Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 3 | https://proofwiki.org/wiki/Dirichlet_Beta_Function_at_Odd_Positive_Integers/Examples/Dirichlet_Beta_Function_of_3 | https://proofwiki.org/wiki/Dirichlet_Beta_Function_at_Odd_Positive_Integers/Examples/Dirichlet_Beta_Function_of_3 | [
"Dirichlet Beta Function at Odd Positive Integers",
"Examples of Dirichlet Beta Function Values"
] | [] | [
"Dirichlet Beta Function at Odd Positive Integers",
"Definition:Euler Numbers",
"Category:Dirichlet Beta Function at Odd Positive Integers",
"Category:Examples of Dirichlet Beta Function Values"
] |
proofwiki-21595 | Eccentricity of Ellipse is Linear Eccentricity over Semimajor Axis | Let $K$ be a ellipse such that:
:$a$ denotes the length of semimajor axis of $K$
:$c$ denotes the linear eccentricity of $K$
:$e$ denotes the eccentricity of $K$.
Then:
:$e = \dfrac c a$ | {{Recall|Focus-Directrix Property of Ellipse|focus-directrix property of ellipse}}
{{:Definition:Focus-Directrix Property of Ellipse}}
{{ProofWanted|straightforward but I no longer have the patience}} | Let $K$ be a [[Definition:Ellipse|ellipse]] such that:
:$a$ denotes the [[Definition:Length (Linear Measure)|length]] of [[Definition:Semimajor Axis of Ellipse|semimajor axis]] of $K$
:$c$ denotes the [[Definition:Linear Eccentricity|linear eccentricity]] of $K$
:$e$ denotes the [[Definition:Eccentricity of Ellipse|ecc... | {{Recall|Focus-Directrix Property of Ellipse|focus-directrix property of ellipse}}
{{:Definition:Focus-Directrix Property of Ellipse}}
{{ProofWanted|straightforward but I no longer have the patience}} | Eccentricity of Ellipse is Linear Eccentricity over Semimajor Axis | https://proofwiki.org/wiki/Eccentricity_of_Ellipse_is_Linear_Eccentricity_over_Semimajor_Axis | https://proofwiki.org/wiki/Eccentricity_of_Ellipse_is_Linear_Eccentricity_over_Semimajor_Axis | [
"Eccentricity of Ellipse is Linear Eccentricity over Semimajor Axis",
"Eccentricity of Ellipse",
"Linear Eccentricity",
"Semimajor Axis of Ellipse",
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Linear Measure/Length",
"Definition:Ellipse/Major Axis/Semimajor Axis",
"Definition:Linear Eccentricity",
"Definition:Ellipse/Eccentricity"
] | [] |
proofwiki-21596 | Eccentricity of Ellipse in terms of Semimajor and Semiminor Axes | Let $K$ be a ellipse such that:
:$a$ denotes the length of the semimajor axis of $K$
:$b$ denotes the length of the semiminor axis of $K$
:$e$ denotes the eccentricity of $K$.
Then:
:$e^2 = 1 - \dfrac {b^2} {a^2}$ | Let $c$ denote the linear eccentricity of $K$.
{{begin-eqn}}
{{eqn | l = a^2
| r = b^2 + c^2
| c = Linear Eccentricity of Ellipse from Major and Minor Axis
}}
{{eqn | ll = \leadsto
| l = c^2
| r = a^2 - b^2
}}
{{eqn | l = e
| r = \frac c a
| c = Eccentricity of Ellipse is Linear Ecce... | Let $K$ be a [[Definition:Ellipse|ellipse]] such that:
:$a$ denotes the [[Definition:Length (Linear Measure)|length]] of the [[Definition:Semimajor Axis of Ellipse|semimajor axis]] of $K$
:$b$ denotes the [[Definition:Length (Linear Measure)|length]] of the [[Definition:Semiminor Axis of Ellipse|semiminor axis]] of $K$... | Let $c$ denote the [[Definition:Linear Eccentricity|linear eccentricity]] of $K$.
{{begin-eqn}}
{{eqn | l = a^2
| r = b^2 + c^2
| c = [[Linear Eccentricity of Ellipse from Major and Minor Axis]]
}}
{{eqn | ll = \leadsto
| l = c^2
| r = a^2 - b^2
}}
{{eqn | l = e
| r = \frac c a
| c ... | Eccentricity of Ellipse in terms of Semimajor and Semiminor Axes | https://proofwiki.org/wiki/Eccentricity_of_Ellipse_in_terms_of_Semimajor_and_Semiminor_Axes | https://proofwiki.org/wiki/Eccentricity_of_Ellipse_in_terms_of_Semimajor_and_Semiminor_Axes | [
"Eccentricity of Ellipse in terms of Semimajor and Semiminor Axes",
"Eccentricity of Ellipse",
"Semimajor Axis of Ellipse",
"Semiminor Axis of Ellipse",
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Linear Measure/Length",
"Definition:Ellipse/Major Axis/Semimajor Axis",
"Definition:Linear Measure/Length",
"Definition:Ellipse/Minor Axis/Semiminor Axis",
"Definition:Ellipse/Eccentricity"
] | [
"Definition:Linear Eccentricity",
"Linear Eccentricity of Ellipse from Major and Minor Axis",
"Eccentricity of Ellipse is Linear Eccentricity over Semimajor Axis"
] |
proofwiki-21597 | Axes of Symmetry for Ellipsoid | Let $\EE$ be an ellipsoid.
Then $\EE$ has exactly $3$ axes of symmetry:
:the major axis of $\EE$
:the mean axis of $\EE$
:the minor axis of $\EE$. | From:
:Major Axis of Ellipsoid is Axis of Symmetry
:Mean Axis of Ellipsoid is Axis of Symmetry
:Minor Axis of Ellipsoid is Axis of Symmetry
we have that these diameters of $\EE$ are in fact axes of symmetry.
It remains to be shown that there are no more.
{{ProofWanted}} | Let $\EE$ be an [[Definition:Ellipsoid|ellipsoid]].
Then $\EE$ has exactly $3$ [[Definition:Axis of Symmetry|axes of symmetry]]:
:the [[Definition:Major Axis of Ellipsoid|major axis]] of $\EE$
:the [[Definition:Mean Axis of Ellipsoid|mean axis]] of $\EE$
:the [[Definition:Minor Axis of Ellipsoid|minor axis]] of $\EE$. | From:
:[[Major Axis of Ellipsoid is Axis of Symmetry]]
:[[Mean Axis of Ellipsoid is Axis of Symmetry]]
:[[Minor Axis of Ellipsoid is Axis of Symmetry]]
we have that these [[Definition:Diameter of Ellipsoid|diameters]] of $\EE$ are in fact [[Definition:Axis of Symmetry|axes of symmetry]].
It remains to be shown that t... | Axes of Symmetry for Ellipsoid | https://proofwiki.org/wiki/Axes_of_Symmetry_for_Ellipsoid | https://proofwiki.org/wiki/Axes_of_Symmetry_for_Ellipsoid | [
"Axes of Symmetry for Ellipsoid",
"Ellipsoids"
] | [
"Definition:Ellipsoid",
"Definition:Axis of Symmetry",
"Definition:Ellipsoid/Major Axis",
"Definition:Mean Axis of Ellipsoid",
"Definition:Minor Axis of Ellipsoid"
] | [
"Major Axis of Ellipsoid is Axis of Symmetry",
"Mean Axis of Ellipsoid is Axis of Symmetry",
"Minor Axis of Ellipsoid is Axis of Symmetry",
"Definition:Ellipsoid/Diameter",
"Definition:Axis of Symmetry"
] |
proofwiki-21598 | Minor Axis of Ellipsoid is Axis of Symmetry | Let $\EE$ be an ellipsoid.
The minor axis of $\EE$ is an axis of symmetry of $\EE$. | Let us arrange $\EE$ such that the minor axis is aligned with the $x$-axis, say.
By Standard Equation of Ellipsoid, $\EE$ can be expressed by the equation:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} + \dfrac {z^2} {c^2} = 1$
where $2 a$ is the length of the minor axis.
Let $R$ denote the space rotation of $180 \degrees$... | Let $\EE$ be an [[Definition:Ellipsoid|ellipsoid]].
The [[Definition:Minor Axis of Ellipsoid|minor axis]] of $\EE$ is an [[Definition:Axis of Symmetry|axis of symmetry]] of $\EE$. | Let us arrange $\EE$ such that the [[Definition:Minor Axis of Ellipsoid|minor axis]] is aligned with the [[Definition:X-Axis|$x$-axis]], say.
By [[Standard Equation of Ellipsoid]], $\EE$ can be expressed by the [[Definition:Equation|equation]]:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} + \dfrac {z^2} {c^2} = 1$
where... | Minor Axis of Ellipsoid is Axis of Symmetry | https://proofwiki.org/wiki/Minor_Axis_of_Ellipsoid_is_Axis_of_Symmetry | https://proofwiki.org/wiki/Minor_Axis_of_Ellipsoid_is_Axis_of_Symmetry | [
"Minor Axis of Ellipsoid",
"Symmetries (Geometry)"
] | [
"Definition:Ellipsoid",
"Definition:Minor Axis of Ellipsoid",
"Definition:Axis of Symmetry"
] | [
"Definition:Minor Axis of Ellipsoid",
"Definition:Axis/X-Axis",
"Standard Equation of Ellipsoid",
"Definition:Equation",
"Definition:Linear Measure/Length",
"Definition:Minor Axis of Ellipsoid",
"Definition:Rotation (Geometry)/Space",
"Definition:Axis/X-Axis",
"Definition:Minor Axis of Ellipsoid",
... |
proofwiki-21599 | General Harmonic Numbers in terms of Riemann Zeta and Hurwitz Zeta Functions | :$\harm r x = \map \zeta r - \map \zeta {r, x + 1}$ | {{begin-eqn}}
{{eqn | l = \harm r x
| r = \sum_{k \mathop = 1}^\infty \paren {\frac 1 {k^r} - \frac 1 {\paren {k + x}^r} }
| c = {{Defof|General Harmonic Numbers}}
}}
{{eqn | r = \sum_{k \mathop = 1}^\infty \frac 1 {k^r} - \sum_{k \mathop = 1}^{\infty} \frac 1 {\paren {k + x}^r}
| c =
}}
{{eqn | r = ... | :$\harm r x = \map \zeta r - \map \zeta {r, x + 1}$ | {{begin-eqn}}
{{eqn | l = \harm r x
| r = \sum_{k \mathop = 1}^\infty \paren {\frac 1 {k^r} - \frac 1 {\paren {k + x}^r} }
| c = {{Defof|General Harmonic Numbers}}
}}
{{eqn | r = \sum_{k \mathop = 1}^\infty \frac 1 {k^r} - \sum_{k \mathop = 1}^{\infty} \frac 1 {\paren {k + x}^r}
| c =
}}
{{eqn | r = ... | General Harmonic Numbers in terms of Riemann Zeta and Hurwitz Zeta Functions | https://proofwiki.org/wiki/General_Harmonic_Numbers_in_terms_of_Riemann_Zeta_and_Hurwitz_Zeta_Functions | https://proofwiki.org/wiki/General_Harmonic_Numbers_in_terms_of_Riemann_Zeta_and_Hurwitz_Zeta_Functions | [
"General Harmonic Numbers",
"Riemann Zeta Function",
"Hurwitz Zeta Function"
] | [] | [] |
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