# Tagged Questions

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### Contour integration with branch points inside the contour.

In my scientific research I ran into an unpleasant situation with specific type of contour integrals. Being more specific I have problems not with integrals themselves (I can use various numeric ...
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### Tractable indefinite integral of the exponentiation of some function

Consider the function $z(s)\in\mathbb{C}$ defined as $z(s)=\int_0^s \exp\left[i(q u+\lambda(u))\right]du$ for some $q\in\mathbb{Q}-\mathbb{Z}$ and $\lambda(s)$ a $2\pi$-periodic real differentiable ...
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### A generalization of Bell numbers to arbitrary complex arguments

For $n\in\mathbb N$, the Bell number $B_n$ is a number of ways to partition the integer range $[1,\,n]$ into pairwise disjoint non-empty subsets. E.g. $B_3=5$ because ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: $6(x^2-y^2)$ | ...
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### 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: ...
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### 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 ...
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### 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 $$I:=\int_{0}^{1} {x\log\left(\,x\,\right) + 1 - x \over x\log^{2}\left(\,x\,\right)}\, \log\left(\,1 + x\,\right)\,{\rm d}x=\log\left(\,4 \over \pi\,\right).$$ Thanks. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Calculate $\lim_{z\rightarrow -n} \frac{\Gamma'(iz)}{\Gamma^2(iz)}$

We know that: $$\lim_{z\rightarrow -n} \frac{\Gamma'(z)}{\Gamma^2(z)}=(-1)^{n+1} n!$$ What if there is $iz$ instead of $z$? i.e. \lim_{z\rightarrow -n} ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 $z=\pm1$, ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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### 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 ...
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### 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. Specifically, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### Divergence of $\Gamma$ function for complex values
It is said that $\dfrac{1}{\Gamma(ix)}$ (of purely imaginary part) diverges. But why please?