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

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-2
votes
0answers
22 views

help with proving question? [on hold]

The following inclusion property holds true for the class $\sum p,q,s(\alpha ;A, B,λ),\alpha 1 > 0$ $ِِِِِِِ\sum p,q,s(\alpha 1 +1;A, B,λ) \subset \sum p,q,s(\alpha 1;A, B,λ)$ when $f (z) =z^{-p} ...
2
votes
1answer
76 views

Solution to a particular Wave Equation

Consider the partial differential equation \begin{align} \frac{1}{c^{2}} \, \frac{ \partial^{2} U}{\partial t^{2}} &= \frac{\partial^{2} U}{\partial x^{2}} + x \, \frac{\partial U}{\partial x} + ...
114
votes
1answer
4k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
15
votes
2answers
495 views
9
votes
1answer
1k views

Connection between Hermite & Legendre polynomials

Prove that $$H_n(x)= 2^{n+1}e^{x^2}\int_x^\infty e^{-t^2}t^{n+1}P_n\left(\frac{x}t\right)dt,$$ where $H_n$ is Hermite polynomial & $P_n$ is Legendre polynomial
1
vote
1answer
17 views

Bessel Function Integral with sin argument

I would like to find if possible a solution (closed form) or approximation for the following integral: $$\int_{\pi/2}^{\pi}\int_{\pi/2}^{\pi}J_{0}\left(\alpha \sin\theta_{k}\right)J_{0}\left(\alpha ...
1
vote
4answers
2k views

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 $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
6
votes
1answer
167 views

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 ...
0
votes
0answers
15 views

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 ...
0
votes
0answers
20 views

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 $n$: $$ \sum_{t=-\infty}^{\infty}e^{-\beta(t+\alpha)^{2}}H_{n}(t+\alpha) $$ ...
4
votes
2answers
990 views

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 ...
30
votes
1answer
767 views

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 ...
0
votes
2answers
20 views

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 ...
0
votes
0answers
13 views

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 ...
0
votes
0answers
14 views

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 \begin{equation} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\Big[\text{Si}[x_1^2 + y^2]- \text{Si}[x_2^2 + ...
0
votes
1answer
20 views

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 ...
4
votes
0answers
82 views

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 ...
3
votes
0answers
54 views

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 ...
4
votes
1answer
48 views

$n$-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 ...
1
vote
0answers
14 views

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. ...
2
votes
2answers
248 views

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 ...
3
votes
2answers
407 views

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 ...
3
votes
0answers
46 views

How to evaluate $\int_{0}^{\infty }\frac{e^{-x^{2}}}{\sqrt{t^{2}+x}}\mathrm{d}x$

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 \{ ...
1
vote
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 ...
1
vote
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!
1
vote
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 ...
1
vote
1answer
28 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 ...
12
votes
1answer
966 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 ...
3
votes
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 ...
0
votes
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 ...
0
votes
1answer
30 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 ...
5
votes
1answer
654 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 ...
1
vote
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 ...
1
vote
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 ...
14
votes
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 ...
3
votes
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 ...
5
votes
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) $$ ...
1
vote
0answers
27 views

Can anybody recognise this equation?

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 $a$, $b$, ...
19
votes
0answers
525 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
0
votes
1answer
36 views

Improper integral involving sinc function and Pochhammer symbol

Can anyone please advise me how to integrate expressions of the form $\text{sinc}\,(x) / (1-x)_n$ along the real axis? Using a CAS, one could suggest that $$ n! \int_{-\infty}^\infty \frac{\sin \pi ...
0
votes
0answers
5 views

Completeness Relation for Tricomi Confluent Hypergeometric Function

Consider the Kummer differential equation $$ \frac{d}{dz}\left[z^be^{-z}\frac{dw}{dz}\right]=az^{b-1}e^{-z}w,\quad z\in\mathbb{R}. $$ It is an eigenvalue problem of Sturm-Liouville type with weight ...
1
vote
1answer
38 views

Asymptotics of a series involving cos integral functions

I'm looking for the asymptotic expansion( or value ) of the following function \begin{equation} F[y,t] = \sideset{}{'}\sum_{n \in \mathbb{Z}}\text{Ci}\big[\frac{n^2}{t}\big] - ...
1
vote
1answer
26 views

Are polylogarithms the simplest functions that decay exponentially in one limit, and grow polynomially in another limit?

I have a function $f(x)$ which is defined as the solution to a certain differential equation. The boundary conditions are that in the $x\rightarrow \infty$ limit, it should be asymptotically ...
0
votes
1answer
18 views

Mixing distribution.

Let $\theta$ in $[0,1]$ and defined for $u,v\in [0,1]$, $$C_\theta (u,v)=\begin{cases}\min(u.v),&&|v-u|\ge \theta\\ \max(u+v-1,0), && |u+v-1|\ge 1-\theta\\ (u+v-\theta)/2,&& ...
1
vote
1answer
27 views

Definition of elementary and special functions

This is a (perhaps) naive question, but one that I have been thinking about lately. Is it a true statement that all functions (elementary or special) can be defined as the solution to a particular ...
1
vote
0answers
26 views

Does asymptotic expansion of Whittaker function $W_{\lambda , \mu}(z)$ exist for $|\lambda| \to 0$?

Suppose Whittaker function $$ \tag 1 W_{\lambda , \mu}(z) $$ Does some asymptotic expansion exist for the case $|\lambda| \to 0$? I'm interested not in the case of $\lambda = 0$, but in the case of ...
2
votes
1answer
38 views

Asymptotics of this HyperGeometric Function

I have a function $$f(x)=x^{2m}\text{ }_2F_1\left(\frac{1}{2},-m;\frac{3}{2};-\frac{1}{x^2}\right)$$ where $x>0$. I am interested in asymptotics in the two extreme limits: $$\lim_{x\rightarrow 0} ...
1
vote
2answers
61 views

Closed-form solution to $\frac{\ln x}{x} = k$

What is the solution in $x$ to $$\frac{\ln x}{x} = k ?$$ I suspect this has something to do with the Lambert W function, since that's used in solutions of the form $x\ln(x) = k$, but the ...
0
votes
0answers
12 views

The Wronskian of parabolic cylinder function and the plane wave

Suppose equation $$ \tag 1 \ddot{y} + (t^2\theta (t - t_{i}) + p^2)y(t) = 0, \quad t \in (t_{0}, \infty) $$ (here $\theta (t - t_{i})$ is the step function) with initial condition $$ \tag 2 y(t \to ...
1
vote
1answer
36 views

The expression of the sum of infinite gaussian functions

Let $f(x|\mu,\sigma^2)$ be the gaussian function (normal distribution): $$f(x|\mu,\sigma^2)=\frac{1}{\sigma\sqrt{2\pi}}e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$$ We know its integral over $\mathbb{R}$ is ...