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Combined display of all available logs of ProofWiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).
- 18:52, 19 April 2024 Robkahn131 talk contribs created page Pi is Irrational/Proof 2/Lemma (Created page with "== Pi is Irrational: Lemma == <onlyinclude> Let $n \in \Z_{> 0}$ be a positive integer. Let it be supposed that $\pi$ is irrational, so that: :$\pi = \dfrac p q$ where $p$ and $q$ are integers and $q \ne 0$. Let $A_n$ be defined as: :$\ds A_n = \frac {q^n} {n!} \int_0^\pi \paren {x \paren {\pi - x} }^n \sin x \rd x$ Then: :$A_n = \paren {4 n - 2} q A_{n - 1} - p^2 A_{n - 2}$...")
- 04:26, 19 April 2024 Robkahn131 talk contribs created page Pi Squared is Irrational/Proof 1/Lemma (Created page with "== Pi Squared is Irrational/Proof 1: Lemma == <onlyinclude> Let $n \in \Z_{> 0}$ be a positive integer. Let $A_n$ be defined as: :$\ds A_n = \frac {q^n} {n!} \int_0^\pi \paren {x \paren {\pi - x} }^n \sin x \rd x$ Let $\pi^2 = \dfrac p q$ where $p$ and $q$ are integers and $q \ne 0$. Note that $\paren {q \pi}^2 = q^2 \paren {\dfrac p q} = p q$ is an integer. Then: :$A_n = \paren {4 n -...")
- 21:39, 16 April 2024 Robkahn131 talk contribs created page Talk:Pi Squared is Irrational/Proof 2 (Created page with "The proof has several errors - one is here: Let us define a polynomial: :$\ds \map f x = \frac {\paren {1 - x^2}^n} {n!} = \sum_{m \mathop = n}^{2 n} \frac {c_m} {n!} x^m$ for $c_m \in \Z$. If $n = 1$ {{begin-eqn}} {{eqn | l = \map f x | r = \frac {\paren {1 - x^2} } {1!} | c = $n = 1$ }} {{eqn | l = \sum_{m \mathop = n}^{2 n} \frac {c_m} {n!} x^m | r = \sum_{m \mathop = 1}^2 \frac {c_m} {1!} x^m | c = }} {{eqn | r = c_1 x + c_2 x^2 | c...")
- 02:34, 30 March 2024 Robkahn131 talk contribs created page Category:Tangent of Sum of Series of Angles (Created page with "Category:Tangent Function")
- 02:33, 30 March 2024 Robkahn131 talk contribs created page Tangent of Sum of Series of Angles/Proof 2 (Created page with "== Theorem == {{:Tangent of Sum of Series of Angles}} == Proof == <onlyinclude> First we note: {{begin-eqn}} {{eqn | l = \cos \sum_j \theta_j + i \sin \sum_j \theta_j | r = \prod_j \paren {\cos \theta_j + i \sin \theta_j} | c = Product of Complex Numbers in Polar Form }} {{eqn | r = \prod_j \cos \theta_j \prod_j \paren {1 + i \tan \theta_j} | c = }} {{eqn | r = \prod_j \cos \theta_j \paren {1 + i \tan \theta_1} \times \paren {1 + i \tan \theta_2}...")
- 02:32, 30 March 2024 Robkahn131 talk contribs created page Tangent of Sum of Series of Angles/Proof 1 (Created page with "== Theorem == {{:Tangent of Sum of Series of Angles}} == Proof == <onlyinclude> First we note: {{begin-eqn}} {{eqn | l = \cos \sum_j \theta_j + i \sin \sum_j \theta_j | r = \prod_j \paren {\cos \theta_j + i \sin \theta_j} | c = Product of Complex Numbers in Polar Form }} {{eqn | r = \prod_j \cos \theta_j \prod_j \paren {1 + i \tan \theta_j} | c = }} {{eqn | r = \prod_j \cos \theta_j \paren {1 + i \tan \theta_1} \times \paren {1 + i \tan \theta_2}...")
- 16:00, 24 March 2024 Robkahn131 talk contribs created page Integral Representation of Bernoulli Number (Created page with "== Theorem == <onlyinclude> Bernoulli numbers can be expressed in integral form as follows: :$\ds \size {B_{2 n} } = 4 n \int_0^\infty \frac {t^{2 n - 1} } {e^t - 1} \rd t$ where: :$B_n$ are the Bernoulli numbers :$n$ is a positive integer. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \map \zeta s \map \Gamma s | r = \int_0^\infty \frac {t^{s - 1} } {e^t -...")
- 02:03, 20 March 2024 Robkahn131 talk contribs created page Binet's Formula for Logarithm of Gamma Function/Formulation 1/Corollary (Created page with "== Corollary to Binet's Formula for Logarithm of Gamma Function/Formulation 1 == <onlyinclude> Let $z$ be a complex number with a positive real part. Then: :$\ds \lim_{z \mathop \to \infty} \size {\Ln \map \Gamma z - \paren {z - \frac 1 2} \Ln z + z - \frac 1 2 \ln 2 \pi } \to 0$ where: :$\Gamma$ is the Gamma function :$\Ln$ is the Definiti...")
- 22:03, 16 March 2024 Robkahn131 talk contribs created page Abel-Plana Formula (Created page with "== Theorem == Let $\map f z$ be analytic for real part of $\map \Re z \ge 0$ and suppose that either :$\ds \sum_{n \mathop = 0}^{\infty} \map f n$ converges or $\ds \int_0^{\infty} \map f x \rd x $ converges. Assume further that :$\ds \lim_{y \mathop \to \infty} \size {\map f {x \pm i y} } e^{-2 \pi y} = 0$ uniformly in x on every Definitio...")
- 15:46, 16 March 2024 Robkahn131 talk contribs created page Dirichlet's Integral Form of Digamma Function (Created page with "== Theorem == Let $z$ be a complex number with a positive real part. Then: :$\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac 1 {t \paren {1 + t}^z } } \rd t$ where $\psi$ is the digamma function. == Proof == We have: {{begin-eqn}} {{eqn | l = \map \psi z | r = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z...")
- 01:01, 7 March 2024 Robkahn131 talk contribs deleted page There are 77 Minimal Primes in Base 10 if Single-Digit-Primes Subsequences are Allowed (super long and messy page with no redeeming value)
- 01:35, 6 March 2024 Robkahn131 talk contribs created page Sum of Every Nth Term of a Series/Also known as (Created page with "== Sum of Every Nth Term of a Series: Also known as == <onlyinclude> '''Sum of Every Nth Term of a Series''' is also known as '''Simpson's Dissection'''. </onlyinclude> == Sources == * {{citation|author = Thomas Simpson|date = 1757|title = The invention of a general method for determining the sum of every 2d, 3d, 4th, or 5th, &c. term of a series, taken in order; the sum of the whole series being known|journal = Philosophical Transactions of the Royal Society|abbr...")
- 01:31, 6 March 2024 Robkahn131 talk contribs created page Sum of Every Nth Term of a Series (Created page with "== Theorem == <onlyinclude> Let $\omega = e^{\dfrac {2 i \pi} q}$ be a primitive qth root of unity. If $\ds \map f x = \sum_{n \mathop = 0}^\infty a_n x^n$ and $p \not \equiv 0 \pmod q$ Then: :$\ds \sum_{n \mathop = 0}^\infty a_{n q + p} x^{n q + p} = \dfrac 1 q \sum_{j \mathop = 0}^{q - 1} \omega^{- j p} \map f {\omega^j x}$ </onlyinclude> == Proof == Expanding the sum on the {{RHS}}, we obtain: {{begin-eqn}} {{eqn |...")
- 01:10, 25 February 2024 Robkahn131 talk contribs deleted page Definition:Complete Quotient (content was: "{{delete|Not sourced, not referenced, not even defined properly}} == Definition == <onlyinclude> Let $x$ be an irrational number. The '''sequence of complete quotients''' is defined recursively by: :$\alpha_0 = x$ :$\alpha_{n + 1} = \dfrac 1 {\fractpart {\alpha_n} }$ where $\fractpart {\, \cdot \,}$ denotes fractional part. </onlyinclude> {{definition wanted|for a general continued fraction}} == Also see == *...")
- 22:35, 23 February 2024 Robkahn131 talk contribs created page Category:Examples of Dilogarithm Function (Created page with "{{ExampleCategory|def = Dilogarithm Function}} Category:Spence's Function")
- 22:33, 23 February 2024 Robkahn131 talk contribs created page Dilogarithm Function/Examples (Created page with " == Examples of Dilogarithm Function == <onlyinclude> === Example: $\map {\Li_2} {-\phi}$ === {{:Dilogarithm of Minus Golden Mean}} === Example: $\map {\Li_2} {-1}$ === {{:Dilogarithm of Minus One}} === Example: $\map {\Li_2} {-\dfrac 1 \phi}$ === {{:Dilogarithm of Minus Reciprocal of Golden Mean}} === Dilogarithm...")
- 02:20, 23 February 2024 Robkahn131 talk contribs created page Dilogarithm of Minus Golden Mean (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} {-\phi} = -\dfrac 3 5 \map \zeta 2 - \paren {\map \ln \phi}^2$ </onlyinclude> where: :$\map {\Li_2} x$ is the dilogarithm function of $x$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$ :$\phi$ denotes the golden mean. == Proof == We now note: {{begin-eqn}} {{eqn | l = \map {\Li_2} {-z} + \map {\Li_2} {-\dfrac 1 z} |...")
- 02:09, 23 February 2024 Robkahn131 talk contribs created page Dilogarithm of Reciprocal of Golden Mean (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} {\dfrac 1 \phi} = \dfrac 3 5 \map \zeta 2 - \paren {\map \ln \phi}^2$ </onlyinclude> where: :$\map {\Li_2} x$ is the dilogarithm function of $x$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$ :$\phi$ denotes the golden mean. == Proof == We first note the following: {{begin-eqn}} {{eqn | l = -\frac 1 \phi | r = 1 -...")
- 02:08, 23 February 2024 Robkahn131 talk contribs created page Dilogarithm of Minus Reciprocal of Golden Mean (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} {-\dfrac 1 \phi} = -\dfrac 2 5 \map \zeta 2 + \dfrac 1 2 \paren {\map \ln \phi}^2$ </onlyinclude> where: :$\map {\Li_2} x$ is the dilogarithm function of $x$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$ :$\phi$ denotes the golden mean. == Proof == We now note: {{begin-eqn}} {{eqn | l = \map {\Li_2} {1 - z} + \map {\Li_2...")
- 02:08, 23 February 2024 Robkahn131 talk contribs created page Dilogarithm of One Minus Reciprocal of Golden Mean (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} {1 - \dfrac 1 \phi} = \dfrac 2 5 \map \zeta 2 - \paren {\map \ln \phi}^2$ </onlyinclude> where: :$\map {\Li_2} x$ is the dilogarithm function of $x$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$ :$\phi$ denotes the golden mean. == Proof == We first note the following: {{begin-eqn}} {{eqn | n = 1 | l = -\frac 1 \ph...")
- 02:14, 19 February 2024 Robkahn131 talk contribs created page Dilogarithm of One Minus Z Plus Dilogarithm of One Minus Reciprocal of Z (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} {1 - z} + \map {\Li_2} {1 - \dfrac 1 z} = -\dfrac 1 2 \map {\ln^2} z $ </onlyinclude> where: :$\map {\Li_2} z$ is the Dilogarithm function of $z$ :$z \in \C$ and $z < 1$ == Proof == From the definition of the dilogarithm function: :$\ds \map {\Li_2} z = -\int_0^z \dfrac {\map \ln {1 - x} } x \rd x$ Taking the derivative of both side...")
- 22:11, 17 February 2024 Robkahn131 talk contribs created page Dilogarithm of Zero (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} 0 = 0$ </onlyinclude> where: :$\map {\Li_2} x$ is the dilogarithm function of $x$ == Proof == {{begin-eqn}} {{eqn | l = \map {\Li_2} z | r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2} | c = Power Series Expansion for Spence's Function }} {{eqn | r = \sum_{n \mathop = 1}^\infty \frac {0^n} {n^2} | c = $z := 0$ }} {{eqn | r = 0 | c = }} {{end-eqn}} {{qed}} Cat...")
- 17:57, 17 February 2024 Robkahn131 talk contribs created page Dilogarithm of Minus Z Plus Dilogarithm of Minus Reciprocal of Z (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} {-z} + \map {\Li_2} {-\dfrac 1 z} = -\map \zeta 2 - \dfrac 1 2 \map {\ln^2} z $ </onlyinclude> where: :$\map {\Li_2} z$ is the Dilogarithm function of $z$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$. == Proof == From the definition of the dilogarithm function: :$\ds \map {\Li_2} z = -\int_0^z \dfrac {\map \ln {...")
- 18:59, 16 February 2024 Robkahn131 talk contribs created page Dilogarithm of One Half (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} {\dfrac 1 2} = \dfrac 1 2 \paren {\map \zeta 2 - \paren {\map \ln 2}^2}$ </onlyinclude> where: :$\map {\Li_2} x$ is the dilogarithm function of $x$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$. == Proof == {{begin-eqn}} {{eqn | l = \map {\Li_2} z + \map {\Li_2} {1 - z} | r = \map \zeta 2 - \map \ln z \map \ln {1 - z} | c = Dilogarit...")
- 18:39, 16 February 2024 Robkahn131 talk contribs created page Dilogarithm of Minus One (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} {-1} = -\dfrac 1 2 \map \zeta 2$ </onlyinclude> where: :$\map {\Li_2} x$ is the dilogarithm function of $x$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$. == Proof == {{begin-eqn}} {{eqn | l = \map {\Li_2} z | r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2} | c = Power Series Expansion for Spence's Function }} {{eqn | r = \sum_{n...")
- 05:38, 16 February 2024 Robkahn131 talk contribs deleted page Recurrence Relation for Dilogarithm (Author request: content was: "== Theorem == <onlyinclude> :$\map {\Li_2} z + \map {\Li_2} {1 - z} = \map \zeta 2 - \map \ln z \map \ln {1 - z}$ </onlyinclude> where: :$\map {\Li_2} z$ is the Dilogarithm function of $z$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$. == Proof == From the {{Defof|Dilogarithm Function}},...", and the only contributor was "Robkahn131" ([[User talk:...)
- 05:37, 16 February 2024 Robkahn131 talk contribs created page Dilogarithm Reflection Formula (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} z + \map {\Li_2} {1 - z} = \map \zeta 2 - \map \ln z \map \ln {1 - z}$ </onlyinclude> where: :$\map {\Li_2} z$ is the Dilogarithm function of $z$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$. == Proof == From the {{Defof|Dilogarithm Function}}, we have: :$\ds \map {\Li_2} z = -\int_0^z \dfrac {\map \ln {1 - x} } x \rd x$ With a view to expressing...")
- 05:22, 16 February 2024 Robkahn131 talk contribs created page Recurrence Relation for Dilogarithm (Created page with "== Theorem == <onlyinclude> :$\map {\Li_2} z + \map {\Li_2} {1 - z} = \map \zeta 2 - \map \ln z \map \ln {1 - z}$ </onlyinclude> where: :$\map {\Li_2} z$ is the Dilogarithm function of $z$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$. == Proof == From the {{Defof|Dilogarithm Function}}, we have: :$\ds \map {\Li_2} z = -\int_0^z \dfrac {\map \ln {1 - x} } x \rd x$ With a view to expressing...")
- 05:20, 16 February 2024 Robkahn131 talk contribs created page Dilogarithm of One (Created page with "== Theorem == <onlyinclude> :$\ds \map {\Li_2} 1 = \map \zeta 2$ </onlyinclude> where: :$\map {\Li_2} x$ is the Dilogarithm function of $x$ :$\map \zeta 2$ is the Riemann $\zeta$ function of $2$ == Proof == {{begin-eqn}} {{eqn | l = \map {\Li_2} z | r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2} | c = Power Series Expansion for Spence's Function }} {{eqn | r = \sum_{n \mathop = 1}...")
- 00:15, 13 February 2024 Robkahn131 talk contribs created page Recurrence Relation for Polylogarithms (Created page with "== Theorem == <onlyinclude> :$\ds \map {\Li_{s + 1} } z = \int_0^z \dfrac {\map {\Li_s } t } t \rd t$ </onlyinclude> where: :$\map {\Li_s} z$ denotes the polylogarithm. == Proof == {{begin-eqn}} {{eqn | l = \int_0^z \dfrac {\map {\Li_s } t } t \rd t | r = \int_0^z \frac 1 t \times \sum_{n \mathop = 1}^\infty \frac {t^n} {n^s} \rd t | c = {{Defof|Polylogarithm}} }} {{eqn | r = \int_0^z \sum_{n \mathop = 1}^\infty \frac {t^{n - 1} }...")
- 23:28, 12 February 2024 Robkahn131 talk contribs created page Polylogarithm of Square (Created page with "== Theorem == <onlyinclude> :$\map {\Li_s} z + \map {\Li_s} {-z} = 2^{1 - s} \map {\Li_s} {z^2}$ </onlyinclude> where $\Li_s$ denotes the polylogarithm. == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \map {\Li_s} z + \map {\Li_s} {-z} | r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^s} + \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} {n^s} | c = {{Defof|Polylogarithm}} }} {{eqn | r = \paren {z + \frac {z^2} {2^s} + \frac...")
- 19:11, 11 February 2024 Robkahn131 talk contribs created page Properties of Digamma Function (Created page with "== Theorem == This page gathers together some of the properties of the '''digamma function''': {{:Definition:Digamma Function}} === Digamma Function in terms of General Harmonic Number === {{:Digamma Function in terms of General Harmonic Number}} === Recurrence Relation for Digamma Function === {{:Recurrence Relation for Digamma Function}} === Digamma Reflection Formula === {{:Digamma Reflection Formula}} === Digamma Add...")
- 18:55, 11 February 2024 Robkahn131 talk contribs created page Category:Dilogarithm of Square (Created page with "Category:Spence's Function")
- 18:54, 11 February 2024 Robkahn131 talk contribs created page Dilogarithm of Square/Proof 2 (Created page with "== Theorem == {{:Dilogarithm of Square}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \map {\Li_2} z + \map {\Li_2} {-z} | r = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2} + \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} {n^2} | c = Power Series Expansion for Spence's Function }} {{eqn | r = \paren {z + \frac {z^2} {2^2} + \frac {z^3} {3^2} + \frac {z^4} {4^2} + \frac {z^5} {5^2} + \frac {z^6} {6^2} + \cdots} + \paren {-z + \frac {z^2} {2^2}...")
- 18:54, 11 February 2024 Robkahn131 talk contribs created page Dilogarithm of Square/Proof 1 (Created page with "== Theorem == {{:Dilogarithm of Square}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \map {\Li_2} z + \map {\Li_2} {-z} | r = -\paren {\int_0^z \frac {\map \ln {1 - t} } t \rd t + \int_0^z \frac {\map \ln {1 + t} } t \rd t} | c = {{Defof|Dilogarithm Function}} }} {{eqn | r = -\int_0^z \frac {\map \ln {\paren {1 - t} \paren {1 + t} } } t \rd t | c = Linear Combination of Definite Integrals, Sum of Logarithms }} {{eqn | r = -\int_0^z \fr...")
- 21:19, 10 February 2024 Robkahn131 talk contribs created page Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 1/Proof 1 (Created page with "== Theorem == {{:Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 1}} == Proof == <onlyinclude> {{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 1 | r = \paren {-1}^0 \dfrac {E_0 \pi} {4 \paren 0!} | c = setting $n \gets 0$ }} {{e...")
- 21:19, 10 February 2024 Robkahn131 talk contribs created page Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 1/Proof 2 (Created page with "== Theorem == {{:Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 1}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \frac 1 {1 + x^2} | r = 1 - x^2 + x^4 - x^6 + \cdots | c = Sum of Infinite Geometric Sequence }} {{eqn | l = \int_0^1 \frac 1 {1 + x^2} \rd x | r = \int_0^1 \paren {1 - x^2 + x^4 - x^6 + \cdots } \rd x | c = integrating both sides from $0$ to $1$ }} {{eqn | ll = \leadsto | l = \map...")
- 20:49, 10 February 2024 Robkahn131 talk contribs created page Dirichlet Beta Function of 5 (Redirected page to Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 5) Tag: New redirect
- 20:43, 10 February 2024 Robkahn131 talk contribs created page Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 5 (Created page with "== Theorem == Let $\beta$ denote the Dirichlet beta function. Then: <onlyinclude> :$\map \beta 5 = \dfrac {5 \pi^5} {1536} $ </onlyinclude> == Proof == {{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 5 | r = \paren {-1}^2 \dfrac {...")
- 16:24, 10 February 2024 Robkahn131 talk contribs created page Even Derivatives of Cotangent of Pi Z at One Fourth (Created page with "== Theorem == <onlyinclude> :$ \ds \valueat {\dfrac {\d^{2 n} } {\d z^{2 n} } \cot \pi z} {z \mathop = \frac 1 4} = \paren {-1}^n \paren {2 \pi}^{2 n} E_{2 n}$ where: :$E_n$ denotes the $n$th Euler number :$n$ is a non-negative integer. </onlyinclude> == Proof == === Lemma === {{:Even Derivatives of Cotangent of Pi Z at One Fourth/Lemma}}{{qed...")
- 16:15, 10 February 2024 Robkahn131 talk contribs created page Even Derivatives of Cotangent of Pi Z at One Fourth/Lemma (Created page with "== Even Derivatives of Cotangent of Pi Z at One Fourth: Lemma == <onlyinclude> Let $z \ne \paren {4 n + 1} \dfrac 1 4$ Then: :$\ds \map \tan {\pi z + \dfrac \pi 4} = \map \sec {2 \pi z} + \map \tan {2 \pi z}$ </onlyinclude> where: :$\sec$ and $\tan$ are secant and tangent respectively. == Proof == {{begin-eqn}} {{eqn | l = \map \tan {\pi z + \dfrac \pi 4} | r = \dfrac {\map \tan {\pi z } + \map \tan {...")
- 21:10, 8 February 2024 Robkahn131 talk contribs created page Cotangent of Complement equals Tangent/Corollary 1 (Created page with "== Corollary to Cotangent of Complement equals Tangent == <onlyinclude> Let $x \ne \paren {4 n + 1} \dfrac \pi 4$ Then: :$\ds \map \cot {\dfrac \pi 4 + x } = \map \tan {\dfrac \pi 4 - x}$ </onlyinclude> where: :$\cot$ and $\tan$ are cotangent and tangent respectively. == Proof == {{begin-eqn}} {{eqn | l = \map \cot {\dfrac \pi 2 - \theta} | r = \tan \theta | c = Cotangent of Complement equa...")
- 21:09, 8 February 2024 Robkahn131 talk contribs created page Cotangent of Complement equals Tangent/Corollary 2 (Created page with "== Corollary to Cotangent of Complement equals Tangent == <onlyinclude> Let $x \ne \paren {4 n + 1} \dfrac \pi 4$ Then: :$\ds \map \cot {\dfrac \pi 4 - x } = \map \tan {\dfrac \pi 4 + x}$ </onlyinclude> where: :$\cot$ and $\tan$ are cotangent and tangent respectively. == Proof == {{begin-eqn}} {{eqn | l = \map \cot {\dfrac \pi 2 - \theta} | r = \tan \theta | c = Cotangent of Complement equa...")
- 01:01, 7 February 2024 Robkahn131 talk contribs created page Properties of General Harmonic Numbers (Created page with "== Theorem == From the {{Defof|General Harmonic Numbers}}, we have: :Let $\ds r \in \R_{>0}$. :For $z \in \C \setminus \Z_{< 0}$ the '''harmonic numbers order $r$''' can be extended to the complex plane as: :$\ds \harm r z = \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + z}^r} }$ This page gathers together some of the properties of Defi...")
- 00:33, 7 February 2024 Robkahn131 talk contribs created page Category:Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function (Created page with "Category:Jacobi Theta Functions Category:Riemann Zeta Function")
- 22:55, 6 February 2024 Robkahn131 talk contribs created page Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function/Lemma 3 (Created page with "== Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function: Lemma 3 == <onlyinclude> :$\ds \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x} = \dfrac 1 2 \paren {\map {\vartheta_3} {\pi t, e^{-\pi x } } - 1}$ </onlyinclude> where: :$\ds \map {\vartheta_3} {\pi t, e^{-\pi x } }$ is the Third Type of Jacobi Theta Function :$t \in \Z$. == Proof == {{begin-eqn}} {{eqn | l = \map {\vartheta_3} {z, q}...")
- 22:55, 6 February 2024 Robkahn131 talk contribs created page Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function/Lemma 2 (Created page with "== Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function: Lemma 2 == <onlyinclude> :$\ds \int_0^1 x^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x } \rd x = -\frac 1 {s \paren {1 - s} } + \int_1^\infty x^{-\paren {s + 1} / 2} \sum_{n \mathop = 1}^\infty e^{-\paren {\pi n^2 x} } \rd x$ </onlyinclude> where: :$s \in \C$ is a complex number with real part $\sigma>1$. == Proof =...")
- 22:54, 6 February 2024 Robkahn131 talk contribs created page Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function/Lemma 1 (Created page with "== Integral Representation of Riemann Zeta Function in terms of Jacobi Theta Function: Lemma 1 == <onlyinclude> :$\ds \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s = \int_0^1 x^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x } \rd x + \int_1^\infty x^{s / 2 - 1} \sum_{n \mathop = 1}^\infty e^{-\pi n^2 x } \rd x$ </onlyinclude> where: :$\map \Gamma s$ is the gamma function :$\map \zeta s$ is the Definition:Riemann Zeta Func...")
- 01:57, 31 January 2024 Robkahn131 talk contribs created page Category:Reflection Formulas (Created page with " Category:Digamma Function Category:Gamma Function Category:Polygamma Function Category:General Harmonic Numbers")
- 01:42, 31 January 2024 Robkahn131 talk contribs created page Category:General Harmonic Number Reflection Formula (Created page with "{{SubjectCategory|result = General Harmonic Number Reflection Formula}} Category:General Harmonic Numbers")