0
votes
0answers
15 views

Calculate $\lim_{z\rightarrow -n} \frac{\Gamma'(iz)}{\Gamma^2(iz)}$

We know that: \begin{equation} \lim_{z\rightarrow -n} \frac{\Gamma'(z)}{\Gamma^2(z)}=(-1)^{n+1} n! \end{equation} What if there is $iz$ instead of $z$? i.e. \begin{equation} \lim_{z\rightarrow -n} ...
4
votes
0answers
97 views

Beautiful Closed form $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\ln^3 2 ...
0
votes
1answer
18 views

Beta function identity for $B(z,z)$

I would like to derive the identity $B(z,z)=2^{1-2z}B(z,\frac{1}{2})$ somehow. The Beta function is defined as $B(p,q)=\int_0^1 t^{p-1}(1-t)^{q-1}dt$ where $Re(p), Re(q)>0$ I used the ...
10
votes
1answer
147 views

How to prove $\int_1^\infty\frac{K(x)^2}x dx=\frac{i\,\pi^3}8$?

How can I prove the following identity? $$\int_1^\infty\frac{K(x)^2}x dx\stackrel{\color{#B0B0B0}?}=\frac{i\,\pi^3}8,\tag1$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind: ...
1
vote
0answers
26 views

Real-valued Irreducible Representations of Lie Groups

I'm interested in the real-valued irreducible representations of a number of Lie groups. For concreteness I'll use the group $M(2)$ of Euclidean motions, which can be parameterized as follows: $$ g(t, ...
13
votes
1answer
170 views

Closed form for $_2F_1\left(\frac12,\frac23;\,\frac32;\,\frac{8\,\sqrt{11}\,i-5}{27}\right)$

I'm trying to find a closed form (in terms of simpler functions) for the following hypergeometric function with a complex argument: ...
0
votes
1answer
52 views

Bessel functions: proof that $J_0(z)=\frac{1}{\pi}\int_0^\pi e^{i z \cos(\theta)}d \theta$.

I encountered the above when dealing with the Bessel functions of the first kind, $J_n(z)$, specifically $n=0$. Using the differential-equation definition of the Bessel function, I obtained the above ...
0
votes
0answers
93 views

Hankel curve formula for gamma function proof

I recently read a proof for the following formula which I don't understand completely. For $Re(z)>0$: $\Gamma(z)=\frac{1}{e^{2\pi iz}-1}\int_{C_\delta}e^{-\zeta}\zeta^{z-1}d\zeta$ , where ...
2
votes
1answer
61 views

Integral representation of the Bessel function (J)

Laurent series expansion of the generator function gives, $g(z,t) = e^{z/2(t - 1/t ) } = \sum J_n (z) t^n. $ The term $t^n$ suggests that this expansion is performed around the origin, so we have, ...
1
vote
1answer
46 views

Laurent expansion of digamma function around $x=0$

I want to check the validity for such a method Define the digamma function as $$\psi_0(x)=\frac{d}{dx}\left( \log \Gamma(x)\right)$$ $$\tag{1}\psi_0(x)=\frac{\Gamma'(x)}{\Gamma(x)}$$ It has the ...
1
vote
1answer
191 views

Evaluate the Bessel Function $J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$

I need to evaluate the following definite integral: $$J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x - \sin x)}\, dx$$ I have attempted basic variable substitution and expanding the cosine term, but I have ...
6
votes
1answer
348 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
1
vote
1answer
77 views

Bessel function with complex argument

So I understand that the bessel functions of the first kind are the ones that satisfy this equation: $$x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-\alpha^2)y = 0$$ and the result is a linear ...
18
votes
1answer
288 views

Derivative of the Meijer G-function with respect to one of its parameters

Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in ...
7
votes
1answer
165 views

Simplification of $G_{2,4}^{4,2}\left(\frac18,\frac12\middle|\begin{array}{c}\frac12,\frac12\\0,0,\frac12,\frac12\\\end{array}\right)$

In this post Cleo gives a misterious result containing the following generalized Meijer G-function: ...
7
votes
1answer
82 views

Calculating $\text{erf}^{-1}(z)$ for $z\in\mathbb{C}$

All the information I found about inverse error function $\text{erf}^{-1}(z)$ was about $z\in\mathbb{R}$. Also I found some Taylor expansions for it, but as the function is unbounded near $z=\pm1$, ...
13
votes
1answer
77 views

Simplification of a trilogarithm of a complex argument

Is it possible to simplify the following expression? $$\large\Im\,\operatorname{Li}_3\left(-e^{\xi\,\left(\sqrt3-\sqrt{-1}\right)-\frac{\pi^2}{12\,\xi}\left(\sqrt3+\sqrt{-1}\right)}\right)$$ where ...
1
vote
0answers
17 views

Zeros of hypergeometric function $_2F_1(a,a;2a;z)$ in the unit disc

I have numerically verified that the hypergeometric function $_2F_1(a,a;2a;z)$ has no zeros in the unit disc, for a range of real positive $a$ parameters. Is it possible to prove this for all real ...
20
votes
1answer
405 views

Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $

Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- ...
2
votes
1answer
46 views

decay rate of series involving the confluent hypergeometric function

I have a question concerning the series: $$c_n:= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \left(-a\right)^k \frac{1}{k!} \frac{b^{n-2k}}{\left(n-2k\right)!} ~ ~ ~ ~; ~ ~ ~ ~ a,b>0 ~ ~ ~ ; ~ ~ ...
0
votes
1answer
61 views

improper intgeral with fresnel integrand

I would like to show that $$\int_0^\infty e^{-S(x)}dx$$ is divergent , where $S(x)$ is the Fresnel integral defines as $$S(x)=\int_0^x \sin \frac{\pi s^2}{2}ds.$$ Thank you
2
votes
0answers
159 views

Separate incomplete elliptic integral into real and imaginary parts

I am working in a problem that involves Incomplete Elliptic Integrals of the First and Second kind of the form $F(\sin^{-1}x~|~m)$ and $E(\sin^{-1}x~|~m)$ where the parameters $m$, $x$ are real ...
0
votes
1answer
142 views

Analytic continuation of factorial function

We know that the factorial can be extended to the whole complex plane except at negative integers and $0$ . But are there any theorems that allow us to do so ? . I know we can use the Identity ...
7
votes
2answers
231 views

The Laurent series of the digamma function at the negative integers

To find the Laurent series of $\psi(z)$ at $z= 0$, I would first find the Taylor series of $\psi(z+1)$ at $z=0$ and then use the functional equation of the digamma function. $$\displaystyle\psi(z + ...
2
votes
1answer
165 views

elements of $SL(2,\mathbb{Z})$ which fix roots of Klein's absolute invariant $j(\tau)$

As a followup to this question (resulting video here), I'd like to make a video showing elements of $\mathbf{SL}(2,\mathbb{R})$ which fix roots of Klein's absolute invariant $j(\tau)$, stylized before ...
1
vote
0answers
78 views

Integral of Bessel functions

Does anybody know if there is an analytical solution to the following integral of Bessel functions: $$\int J_m^*(kx) \, J_m(kx) \, x \,dx,$$ where $m$ is integer and the problem is that $k$ may be ...
4
votes
1answer
79 views

Closed form for the product $G(x)G(-x)$ of two Barnes functions

Is there a "closed form" expression for the following product of two Barnes $G$-functions, $$G(x)G(-x),$$ where $x$ is real? Plotting the graph I have noticed that for $-1<x<1$ we have ...
4
votes
1answer
179 views

How to better understand where the circles and lines go under fractional linear transformations?

Today I encountered the transformation $f(z) = \frac{z}{z-1}$. It has the following property: As the point $z$ makes a counter-clockwise revolution around the unit circle beginning at $1$, the point ...
1
vote
1answer
83 views

Divergence of $\Gamma$ function for complex values

It is said that $\dfrac{1}{\Gamma(ix)}$ (of purely imaginary part) diverges. But why please?
1
vote
0answers
30 views

Approximation the function $f(t)=I_0(-rt)e^{-rt}$ with sum of Exponentials.

Consider the function $f(t)=I_0(-rt)e^{-rt}$ where $I_0(t)$ is modified Bessel’s function and $r>0$. I am looking for an approximation for the function with a sum of exponential functions in $t ...
4
votes
1answer
117 views

Integral formula for $\frac{1}{\Gamma(z)}$

Let $c>0$. How to prove that for any complex number $z$, $$\frac{1}{\Gamma(z)}=\frac{1}{2\pi}\int_{-\infty}^\infty (c+it)^{-z}e^{c+it}\,dt?$$ where $\Gamma(z)$ is the Gamma function.
3
votes
1answer
113 views

Riemann zeta function at zero

Can the value of Riemann zeta function at 0, $\zeta(0)=-1/2$, be deduced from the identity $E(z)=E(1-z)$, where $$E(z)=\pi^{-z/2}\Gamma(z/2)\zeta(z)?$$
7
votes
0answers
190 views

Contour integral representation of Confluent Hypergeometric Function

My brain is spinning around in circles trying to reconcile three distinct contour-integral representation of the confluent hypergeometric function $_1F_1(a,b,z)$ for $b \in \mathbb{Z}_+$: From ...
7
votes
1answer
116 views

A question related to Riemann zeta function

Does anyone know why the following statement is correct? Let $f(x)$ be the function whose value on the interval $m\pi<x<(m+1)\pi, m=0,1,2,\cdots$, is $(-1)^m\frac{\pi}{4}$. Let $0<s<1$. ...
3
votes
1answer
84 views

How to prove this identity for ${}_3F_2$ (Generalized Hypergeometric Function)?

This may look like homework, but it is not. I've found this identity (using Mathematica): $$ {}_3F_2 \left( \matrix{1,1,1 \\ 2, e} ; 1 \right) = (e-1) \psi^{\prime}(e-1), $$ valid for $e$ with ...
5
votes
0answers
189 views

Identity involving $\zeta(3)$

This is related to this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that ...
1
vote
0answers
72 views

Gamma Function Problem

Hi is it fair to write $$\Gamma(1+ix)=ix\Gamma(ix) $$ and then to say that $\Gamma(ix)=\exp(i\arg(\Gamma(x)))$. If not can anyone explain what $\arg(\Gamma(x))$ is defined as please, I always get ...
2
votes
0answers
69 views

Closed form formula for a double series related to wave equation

Does anyone have a closed form formula for the double series $$\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)?$$ This is related ...
9
votes
1answer
198 views

The series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}\mbox{sech}\left(\frac{(2n-1)\pi}{2}\right)$

Does anyone have a proof that $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}\mbox{sech}\left(\frac{(2n-1)\pi}{2}\right)=\frac{\pi}{8}$$
4
votes
1answer
310 views

Integral of $\ln |\sin(x)|$

Does anyone have a real formula for the integral $$\int\ln |\sin(x)|\,dx ?$$ Neither Maple nor Mathematica give a real answer. Using integration by parts and the series for $x\cot x$, I get $$x\ln ...
1
vote
0answers
56 views

What is the closed form expression for this?

Let $r_1...r_k$ be the $k$ roots unity or solutions to the expression $x^k = 1$ What is the expression: ...
5
votes
1answer
150 views

Dirichlet L-series and Gamma function question

Could someone help me, please, with this exercise? Consider a sequence of complex numbers $\{a_n\}$ such that $a_n=a_m $ iff $ n\cong m $ mod $q$ for some positive integer $q$. Define the ...
5
votes
1answer
172 views

Prove that $\int_0^{\pi/2} \cos^{p+q-2}(\theta) \cos((p-q)\theta)d\theta = \frac{\pi}{(p+q-1)2^{p+q-1}B(p,q)}$

Does anybody know how to prove this identity? $$\int_0^{\pi/2} \cos^{p+q-2}(\theta) \cos((p-q)\theta)d\theta = \frac{\pi}{(p+q-1)2^{p+q-1}B(p,q)}\quad p+q>1,q<1$$ $B(x,y)$ denotes Beta ...
1
vote
2answers
478 views
4
votes
1answer
101 views

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$

Prove that $\Gamma (-n+x)=\frac{(-1)^n}{n!}\left [ \frac{1}{x}-\gamma +\sum_{k=1}^{n}k^{-1}+O(x) \right ]$ I don't know how to do this ? Note that $\gamma $ is the Euler-Mascheroni constant
1
vote
1answer
157 views

asymptotic behavior of the real part of the Riemann zeta function for $0<\sigma<1$

consider the zeta function $\zeta(\sigma+it)$ for $\sigma>1$ : $$\zeta(\sigma+it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma+it}}$$ And: $$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$ ...
2
votes
1answer
82 views

About the values of the $\Gamma$ function

The $\Gamma$ function is defined by $$\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt$$ where $z$ is a complex number. We know that if $z$ is real then the values of $\Gamma$ are also real. I am ...
2
votes
0answers
66 views

asymptotics of $ J_{iu} (ia)$ for a Bessel function

Let $J_{iu}(ia)$ be the Bessel function of imaginary order. ($a$ is a real number (positive or negative) and $u$ is also real.) In the limit $u \to \infty $ how does the function $J_{iu} (ia)$ ...
2
votes
1answer
76 views

What is the “name” of this function?

There is a function I met in complex analysis. $$f(\lambda) = \int \limits_{-\infty}^{\infty}\frac{e^{i\lambda x}}{\sqrt{1 + x^{2n}}}dx$$
2
votes
1answer
241 views

Conformal mapping from triangle to upper half plane in terms of Weierstrass $\wp$

I'm trying to explicitly compute a conformal map $f:\Delta \rightarrow \mathbb{H}$ where $\Delta$ is a triangle and $\mathbb{H}$ is the upper half plane, in terms of the Weierstrass $\wp$ function. I ...