# Tagged Questions

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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### Elliptical Integrals and graphing plot

I'm trying to computer plot the graphs of sn(u), cn(u) and dn(u) for k = 1/4, 1/2, 3/4, 0.9 and 0.99 And I am trying to plot 3D graphs of sn, cn and dn as functions of u and k. Here's what I have: ...
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### Why do Bessel functions of the first kind come up in the following 2-dimensional Fourier transform?

The following equation is given for a function $\gamma$: $\gamma = \pm \delta \left[\int \frac{d^2 p}{(2\pi)^2}qp(1-\hat{q}\cdot \hat{p})^2 \pi a^2 e^{-|q-p|^2 a^2/4}\right]^{1/2}$ where q and p are ...
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### Complete Elliptic Integral of the 3rd Kind - Residual Computation

Let us consider the following function $f(a,k)$ in the interval $a,k\in (0,1]$ : $$f(a,k)=\frac{2 \sqrt{1-a^2} \sqrt{a^2-k^2}}{\sqrt{a^2}}\Pi\left(a^2,k^2\right)$$ where $\Pi\left(a^2,k^2\right)$ is ...
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### Compute $\sum_{m>n=1}^{\infty} \frac{1}{m!n!}$

Compute the series $$\sum_{m>n=1}^{\infty} \frac{1}{m!n!}$$
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### Double Integral involving modifed bessel function

I'm try to derive a closed form of the following double integral: $\int\limits_0^x {\int\limits_0^x {{e^{ - {K_1}uv}}{I_0}\left( {2{K_1}\sqrt {uv} } \right)du} dv}$; where $K_1$ is a constant. Do you ...
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### Can this function be expressed in terms of other well-known functions?

Consider the function $$f(a) = \int^1_0 \frac {t-1}{t^a-1}dt$$ Can this function be expressed in terms of 'well-known' functions for integer values of $a$? I know that it can be relatively simply ...
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### Evaluation of an integral involving the Lambert W function

Wikipedia claims that $$\int_0^\infty W\left(\frac{1}{x^2}\right) \,\text dx=\sqrt{2\pi}$$ and a numerical computation seems to confirm this. How can this result be proven?
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### 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: $$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$ ...
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### Why is this function homogenous to the specified degree?

I have this function $$w(q) = (1 - \alpha)q^nBk^\alpha + c$$ The paper I'm reading says that w is homogenous of degree $$n/(1-\alpha)$$ and so small differences in q cause large differences ...
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### 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 ...
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### A Thue-Morse Zeta function ( Generalized Riemann Zeta function and new GRH )

Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ a complex number. Odiuos Number Now consider the sequence of functions below $f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$ ...
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### Prove that $\int_0^1 \psi{(x) \sin(2 n \pi x)} \space\mathrm{dx}=-\frac{\pi}{2}$

Prove that $$\int_0^1 \psi{(x) \sin(2 n \pi x)} \space\mathrm{dx}=-\frac{\pi}{2}, \space n\ge1$$ where $\psi(x)$ - digamma function
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### error function (erf) with better precision

Currently I'm using this C++ routine to approximate the error function ...
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### Solving the integral of a Modified Bessel function of the second kind

I would like to find the answer for the following integral $$\int x\ln(x)K_0(x) dx$$ where $K_0(x)$ is the modified Bessel function of the second kind and $\ln(x)$ is the natural-log. Do you have ...
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### Finding x in $\frac{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,1-x)}{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,x)} = \sqrt{n}$

I was trying to find a closed-form for $0<x<1$ in, $$\frac{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,1-x)}{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,x)} = \sqrt{n}$$ where $\,_2F_1(a,b,c,z)$ ...
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### 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 $J_{iu} (ia)$ behave?...
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### Showing integrability (Riemann)

I was trying to show whether or not the function: $f: [0,1 ] \rightarrow \mathbb{R}$ $f(x)= \frac {1}{n}$ for $x = \frac {1}{n}$ $(n \in \mathbb{N})$ and $f(x) = 1$ if the condition isn't ...
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### What is the “name” of this function?

There is a function I met in complex analysis. $$f(\lambda) = \int \limits_{-\infty}^{\infty}\frac{e^{i\lambda x}}{\sqrt{1 + x^{2n}}}dx$$
Prove that $$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
### Identity concerning $e^{ia\sin{x}}$ as a series of bessel functions
Prove the following identity: $$e^{ia\sin{x}}=\sum_{-\infty}^{+\infty}J_k(a) e^{ikx},$$ where $a$ is a real constant and $J_k$ is the Bessel function of the first type of ...