# 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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### Simplifying a certain polylogarithmic sum in two variables

This question is related to my previous question here. While tinkering around for a solution I found that the integral there can be reduced to the problem of solving the following basic logarithmic ...
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### sum of integral parts of real number and fraction

For any real x and positive integer n ,show that [x] + [x +1/n] + [x + 2/n] + .... + [x + n-1/n] = [nx] I have used the fact that x-1 < [x] <= x,for all terms and added,but not able to get ...
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### How can the Bessel function of the second kind be in the radial eigenfunction?

Let $0<a<b<\infty$. Consider the heat equations or wave equations on the annulus or the spherical layers $$\Omega:=\{x\in\mathbb{R^d}\mid a<\|x\|_2<b\},$$ ...
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### Infinite sum of Hermite polynomials with same order, but different argument

I am looking for any possible simplification of the following sum for positive reals $\alpha,\beta$ and positive integer $n$: $$\sum_{t=-\infty}^{\infty}e^{-\beta(t+\alpha)^{2}}H_{n}(t+\alpha)$$ ...
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### How to proof a basis ${\psi_a}$ is complete?

Why $$\int\text{d}a\psi^*_a(y)\psi_a(x)=\delta(y-x)$$ shows the basis is complete? Even, how is $\delta$ defined? I mean, the most consistent definition. I really dislike the definition by 0 and ...
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### efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
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### Another beta integral due to Cauchy.

I have the following identity which I want to prove: $$C(x,y):= \int_{-\infty}^{\infty} \frac{dt}{(1+it)^x(1-it)^y} = \frac{\pi \cdot 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ where ...
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### Domain of the Gamma function

I need to find the domain of the Gamma function, that is to say all $z \in \mathbb{C}$, for which the integral: $$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \mathrm dt$$ converges. I started by ...
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### Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
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### Laplace transform of the square wave to solve PDE

Solve $$y'' + 3y' +2y = r(t)$$ given $y(0)=0$ and $y'(0) = 0$ where $r(t)$ is the square wave, $$r(t) = u(t-1) - u(t-2)$$ I'm just going to type out the answer as I read it and tell you which ...
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### An expository reference on spherical harmonics on $S^n$.

I am looking for a thorough reference which explains how to compute the spherical harmonics on $S^n$ and how to upper and lower-bound their values. About the first part of my query, I am not so much ...
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### Fourier transform of Si[$x^2 + y^2$]; Energy integrals involving sin integral functions

Problem Statement I'm trying to prove( or disprove ) the following identity \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\Big[\text{Si}[x_1^2 + y^2]- \text{Si}[x_2^2 + ...
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### Understanding a text in a book about the estimation

Now $$e_n-e_0=\sum_{k=0}^{n-1}\left [ -\frac{1}{12(k+x)^2}+\mathcal{O}\left ( \frac{1}{(k+x)^3} \right ) \right ]; \tag{*}$$ therefore, $\lim_{n\to\infty}e_n-e_0=K_1(x)$ exists. Set ...
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### Double integral of symmetric polylogarithmic function over rectangular region

This question was inspired by M.N.C.E.'s wonderful response here. While exploring the possibility of generalizing his result, I found that a significant part of the problem reduced to evaluating the ...
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### A generalization of additive function over $\mathbb R$

Let $f:\mathbb R\to\mathbb R$ be a continuous function and $r\ge0$ a fixed value such that for all $x,y\in\mathbb R$ $$|f(x)+f(y)-f(x+y)|\le r$$ Show there exist $a\in\mathbb R$ and a function ...
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### $n$-th derivative of Beta function

We pretty much know nothing about the high order derivatives of the Beta function. Well, we known for the example some recursive formulae for $\Gamma^{(n)}(1)$ as well as ...
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### Implementation of Jacobi theta functions in Matlab

I need Jacobi theta functions for my Matlab program. The functions are not included in the predefined Matlab functions. Doing a simple Google search, I found a package developed by Moiseev I. in 2008. ...
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### A theta function around its natural boundary

Let $q = e^{2\pi i\tau}$, if $$\psi(q^2)=\sum_{n=0}^{\infty} q^{n(n+1)}$$ is one of ramanujan theta functions,is it possible to evaluate the limit $$\lim_{q\rightarrow 1} (1-q){\psi^2(q^2)}$$ In fact ...
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### Solve equation with lower gamma function: $A \gamma(2;x/B)=x$ for $x$

I need to find an expression for $x$ given: $A \gamma(2;x/B)=x$ where $\gamma(a,x)=\int\limits_0^x t^{a-1} e^{-t} \mathrm{d}t$ is the lower incomplete gamma function. $A$ and $B$ are real, positive ...
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How to evaluate the integral below $$\int_{0}^{\infty }\frac{e^{-x^{2}}}{\sqrt{t^{2}+x}}\mathrm{d}x~~~~~~(t＞0)$$ The WolframAlpha gave me a horrible answer $$\frac{t}{2}e^{-\frac{t^{4}}{2}}\left \{ ... 1answer 36 views ### Maximum/minimum of a special function I was given a function f(x)=\mbox{Li}_{-n}(x), where Li is the polylogarithm of order -n (n>0\in\mathbb{N}) and x\in(-\infty,0). The function in this domain is bounded and has some ... 0answers 74 views ### What does subcopula mean? In copula concept, what does "subcopula" exactly mean? Does it mean a subset of copula? Would you please explain a little bit in details? Thanks in advance! 0answers 25 views ### Fourier series in spherical coordinates? I'm reading an article and he just state: let f\left(\theta,\varphi\right) be of this form$$f\left(\theta,\varphi\right)={\sum}g_{m}\left(\theta\right)e^{im\varphi},I'm on the unitary ... 1answer 29 views ### How does one simplify this series expression? I am trying to prove the Rodrigues formula for the Legendre polynomials from the power series recursion relation (obtained through the Frobenius method). On page 3 of this article, I can follow the ... 1answer 967 views ### a conjectured continued-fraction for \cot\left(\frac{z\pi}{4z+2n}\right) that leads to a new limit for \pi Given a complex number \begin{aligned}\frac{z}{n}=x+iy\end{aligned} and a gamma function \Gamma(z) with x\gt0, it is conjectured that the following continued fraction for ... 1answer 99 views ### Find monotonic functions going from 0 to +\infty for x \in (-\infty,+\infty) (similar to e^x) How can we find functions on \mathbb{R} with exponential-like properties, namely: f(x) is infinitely differentiable; f(x) and all its derivatives are monotonic; f(x) and all its derivatives ... 0answers 10 views ### Exist a quasi periodic function whose derivative is not almost periodic? I learned from "A Note On Almost Periodic Variational Equations" by P. Giesl and M. Rasmussen that there exist almost periodic functions (in fact limit periodic) such that their (strong) derivative is ... 1answer 31 views ### Almost periodic function vs quasi periodic function I am doing some work regarding quasi periodic function but I am not able to figure out the difference between almost periodic and quasi periodic functions.Can anyone let me know about it? Thanks ... 1answer 655 views ### Inverse of elliptic integral of second kind The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ... 0answers 30 views ### Definite integral with exponential and algebraic functions I came across definte integral: I(a, b) = \int_{a-b}^{a+b} \frac{1}{e^x -1} \frac{1}{\sqrt{1-(x-a)^2/b^2}} ~\mathrm{d}x  Mathematica was not able to guide a closed form solution, but I am hoping ... 1answer 24 views ### How can I build a C^\infty-function like this on a cube of \mathbb{R}^m Build a function f: \mathbb{R}^m \to \mathbb{R}, f\in C^\infty(\mathbb{R}^m), such that for a cube C^m_\epsilon(0):=\{x|\space\epsilon>=|x_i|\}\subseteq\mathbb{R^m}. \forall x \in ... 0answers 443 views ### a conjectured continued fraction for \displaystyle\tan\left(\frac{z\pi}{4z+2n}\right) Given a complex number \begin{aligned}\frac{z}{n}=x+iy\end{aligned} and a gamma function \Gamma(z) with x\gt0, it is conjectured that the following continued fraction for ... 1answer 677 views ### 2-increasing functions I'm trying to learn about copulas. Two definitions I've come across are the H-volume of a rectangle, which is defined as V_H=H(x_2,y_2)-H(x_2,y_1)-H(x_1,y_2)+H(x_1,y_1). The function H, whose ... 1answer 167 views ### Why does the asymptotic expansion of the real-valued Kummer function contain complex terms? Working on a problem in spectral theory, I need to study the asymptotics of a confluent hypergeometric function (here (a)_0=1 and (a)_s=a(a+1)\cdots(a+s-1) denote the Pochhammer symbol) ...
I just wonder if the following equation is a known special function? $$\left(u(1-u^2)\frac{d^2}{du^2}-(u^2+1)\frac{d}{du}-\frac{au}{(1-u^2)}-\frac{bu^3}{(1-u^2)}+c\right)G(u,u')=0,$$ where $a$, $b$, ...