0
votes
1answer
40 views

Pair of functions $F(x)$ (transcendental),$A(x)$ (algebraic) with expanded series of positive integer coefficient linked by derivative

$$F(x)=\sum_0^{\infty}b_k x^k,b_k\in \mathcal{N} \bigcup 0,\exists M \space b_k \leq M^k$$. $$A(x)=\sum_0^{\infty}a_k x^k,a_k\in \mathcal{N} \bigcup 0,\exists M \space a_k \leq L^k$$ where $F(x)$ is ...
1
vote
1answer
74 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
2
votes
0answers
31 views

Properties and representations of the the rescaled complementary error function $\mathrm{erfcx}{z}$

Consider the rescaled complementary error function: $$ \mathrm{erfcx}(z) = {e^{z^2}} \left( {1-\mathrm{erf}(z)} \right) $$ $z \in \Bbb{C}$ which also has the following integral representation: $$ ...
0
votes
2answers
72 views

Basic question on complex integration

I have a very basic question on complex integration. How is the definite integral $$ \int_{z_1}^{z_2}{f(z)dz} $$ $z \in \Bbb{C}$ to be interpreted in the absence of a specific path over which ...
10
votes
1answer
143 views

Integral $\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3$

Hi I am trying to prove this below. $$ I:=\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3 $$ where $$ \zeta_3=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I am ...
1
vote
2answers
56 views

Determining if any general funtion u(x,y) makes f(z)=u(x,y)+iv(x,y) analytical

I have a question about Complex Analytical functions. I have some homework that asks: let $f(z) = u(x,y) + iv(x,y)$. Indicate the following functions for which u(x,y) may be analytic: $6(x^2-y^2)$ | ...
1
vote
1answer
73 views

Calculate the residue of $\cot\pi z$ at poles $z=n$

I'm having trouble calculating the residue of $f(z) =\cot\pi z$. The function has a simple pole for every integer n, and i'm, trying to find the residue at n. I know that by the residue theorem: ...
3
votes
1answer
52 views

Is there a simpler form for $\Re \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$?

Is there a simpler (i.e. manifestly real) form for $\Re \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$ or $\Im \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$, or more generally for $\frac{\Gamma(1/2-ia)}{\Gamma(1-ia)}$ with ...
5
votes
2answers
109 views

Integral $\int_0^1 \frac{x\log x+1-x}{x \log^2 x}\log(1+x)\, dx=\log\frac{4}{\pi}$

Hi I am trying to prove this $$ I:=\int_0^1 \frac{x\log x+1-x}{x \log^2 x}\log(1+x)\, dx=\log\frac{4}{\pi}. $$ Thanks. This is just a beautiful integral for many reasons. Logs are everywhere and an ...
7
votes
4answers
161 views

Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$

Hi I am trying to prove$$ I:=\int_0^1 \log\left(\,\Gamma\left(x+\alpha\right)\,\right)\,{\rm d}x =\frac{\log\left(2\pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha\,,\qquad \alpha \geq 0. $$ I am ...
0
votes
1answer
35 views

factorization of an expression involving gamma function

Does the equation $\Gamma(x+1/2)\Gamma(x-1/2)=\Gamma(x+iy)\Gamma(x-iy)$, where $\Gamma(z)$ is the Gamma function and $i=\sqrt{-1}$, have any solution assuming $x,y$ are both real and $x>1/2$? This ...
2
votes
0answers
101 views

An Integral possibly related to Legendre polynomials

Consider the integral $$\int_0^1\frac{(t^2-1)^a}{(t-u)^{b+1}}dz$$ where $b\gg a$, with $a,b$ integers and $u>1$. I know you can write this integral as the sum of two hypergeometric functions but ...
0
votes
1answer
129 views

Evaluation of definite integral using complex analysis

I want to evaluate the following indefinite integral $$ \int_0^{\infty} x^{p - 1} \cos (ax) dx$$ where $0 < p < 1$ and $a > 0$. I was considering the function $f(z) = z^{p - 1} e^{iaz}$ and ...
0
votes
0answers
19 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} ...
6
votes
0answers
226 views

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

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 (yes Beautiful) and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 ...
0
votes
1answer
20 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
162 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: ...
2
votes
0answers
36 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
186 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
70 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 ...
2
votes
1answer
96 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
64 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 ...
2
votes
1answer
248 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 ...
8
votes
1answer
429 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
100 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 ...
19
votes
1answer
358 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
198 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
96 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
80 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
19 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
442 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
50 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
66 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
169 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
182 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
256 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
179 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
85 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
80 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
197 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
90 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
34 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
120 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
127 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
221 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
127 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
102 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
197 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
77 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
70 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 ...