# Tagged Questions

63 views

### 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 ...
48 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 ...
77 views

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 ...
156 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 ...
58 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: $6(x^2-y^2)$ | ...
77 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: ...
53 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 ...
127 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 $$I:=\int_{0}^{1} {x\log\left(\,x\,\right) + 1 - x \over x\log^{2}\left(\,x\,\right)}\, \log\left(\,1 + x\,\right)\,{\rm d}x=\log\left(\,4 \over \pi\,\right).$$ Thanks. ...
171 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 ...
38 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 ...
126 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 ...
151 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 ...
19 views

### 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} ...
233 views

191 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: ...
72 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 ...
102 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, ...
72 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 ...
251 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 ...
434 views

50 views

182 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 ...
87 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 ...
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 ...
202 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 ...
### Divergence of $\Gamma$ function for complex values
It is said that $\dfrac{1}{\Gamma(ix)}$ (of purely imaginary part) diverges. But why please?