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

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1
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1answer
16 views

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 ...
3
votes
1answer
65 views

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 $$\Omega:=\{x\in\mathbb{R}^d\mid a<\|x\|_2<b\},$$ ...
1
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1answer
22 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
1answer
31 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
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1answer
32 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 $n$: $$ \sum_{t=-\infty}^{\infty}e^{-\beta(t+\alpha)^{2}}H_{n}(t+\alpha) $$ ...
0
votes
2answers
28 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
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0answers
16 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
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0answers
20 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
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1answer
21 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 ...
3
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0answers
230 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
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1answer
50 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 ...
4
votes
1answer
81 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
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1answer
38 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
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0answers
22 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. ...
1
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0answers
27 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
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1answer
31 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 ...
0
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0answers
15 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 ...
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0answers
34 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 ...
3
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1answer
100 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 ...
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0answers
28 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 $a$, $b$, ...
0
votes
1answer
38 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
1answer
12 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
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1answer
39 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
27 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,&& ...
0
votes
0answers
21 views

Let $\theta(z) = \sum q^{n^2}$, is $\theta(-1/z)$ also a theta function?

I am learning about theta functions. Let $q = e^{2\pi i \, z}$ and $\theta(z) = \sum q^{n^2}$. How does it behave under $\mathrm{SL}_2(\mathbb{Z})$ ? In general we have: $$ \theta\left( - ...
1
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1answer
31 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
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2answers
63 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 ...
1
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0answers
28 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 ...
0
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1answer
40 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 ...
3
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1answer
43 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} ...
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
39 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 ...
1
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0answers
34 views

general procedure for contour integration of $\int_{0}^{\infty} \mathrm{Ai}(x)^{n} dx$

In Richard Crandall's On The Quantum Zeta Function, following eq. 4.11: $$ \int_{0}^{\infty}\mathrm{Ai}(x)^{2}dx=\frac{\Gamma(\tfrac{5}{6})}{2\pi^{5/6}12^{1/6}} $$ “again derivable by contour ...
0
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0answers
12 views

The domain of the Digamma function and its extension

First, we know that $\Gamma(x)>0$, for all $x>0$, so define $\psi(x)=\frac{\mathrm{d} }{\mathrm{d} x}\ln(\Gamma(x))=\Gamma'(x)/\Gamma(x)$, for all $x>0$. This is the Digamma function. It is ...
1
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0answers
80 views

Definite integral $\int_{0}^{\infty} e^{-a t} \log(t)\log(1+t)\,dt$

Is there a closed-form expression (possibly in terms of special functions) for the integral: $$ \int_{0}^{\infty} e^{-a t}\log(t)\log(1+t)\,dt, $$ where $a>0$?
5
votes
1answer
169 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) $$ ...
0
votes
1answer
25 views

Integral of a power of the complementary error function

I would like to know if is it possible to calculate analytically the following integral: $$J=\int_0^{x_0}\operatorname{erfc}(x)^kdx$$ with $k=2,3,4,...N$ where $\operatorname{erfc}(x)$ is the ...
0
votes
2answers
31 views

Write function $F$, based on parameters?

Let $$U=x+y+z\, ,$$ $$V=xy+yz+zx\, ,$$ $$W=xyz\, .$$ and we have $F(U,V,W)=x^4+y^4+z^4$. My question it is, How to write function $F$, based on parameters $U$ , $V$ and $W$?
1
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1answer
37 views

Inequality for Gamma function

Prove that $$0<\frac{\Gamma(x+y)}{\Gamma(xy)-1}\leq3$$ for all $x>0,y>0, xy>2.$ And equality holds $x=y=2.$
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0answers
41 views

Where was the mistake

we know that $$\frac{\pi^2}{6}=\int_{0}^{\infty}\frac{t}{e^t-1}dt$$, we also know that $\frac{t}{e^t-1}$ is the generating function for the Bernoulli numbers i.e $ \frac{t}{e^t-1} ...
0
votes
0answers
13 views

Support width of the difference of two rect functions.

We let $\operatorname{rect}(t)\equiv \chi_{(-\frac 12, \frac 12)}(t) $ when $|t|\ne \frac 12$ and $\operatorname{rect}(t)=\frac 12$ if $|t|=\frac 12$. Let $\operatorname{rect}(t_1,t_2)\equiv 1$ when ...
1
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1answer
25 views

The Wronskian of parabolic cylinder functions

Suppose I have the second order differential equation $$ y''(t) + (k^{2} + m^{2}t^2)y(t) = 0, \quad 0< t_{0} < t < \infty $$ The solution of this equation is parabolic cylinder functions, ...
0
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0answers
30 views

About asymptotic expansion of parabolic cylinder functions

Let's have the parabolic cylinder function $U(a,z)$. I'm interested in its asymptotics for large argument $z$. Here I've found it, but I'm a bit confuzed now because of expressions $(12.9.1)$ and ...
1
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0answers
70 views

Integrating a product of complicated exponential function and error function

I have a problem with the following integral $$\int_0^\infty\dfrac{{\rm e}^{-t-(x^2-a^2)/t}}{t}Erf\left(\frac{a}{\sqrt{t}}\right)\,{\rm d}t, $$ where $0\le a<x$ and Erf stands for the error ...
2
votes
1answer
25 views

$\operatorname{Re}(\operatorname{Li}_3(z))$ for real $z\geq1$ in terms of elementary functions?

According to the article by Wood, D. "The Computation of Polylogarithms. Technical Report 15-92*", listed in the references about the polylogarithm on the Wikipedia, there is a form in terms of ...
2
votes
0answers
52 views

How to evaluate following integral?

Suppose the integral $$ \tag 1 I = \int \limits_{-\pi}^{\pi}dx \int \limits_{-\pi}^{\pi}\frac{dy}{\tau - \cos (2x) -2\cos(x)\cos(y)}, \quad t > 3 $$ How to evaluate it in terms of elliptic ...
14
votes
1answer
195 views

Proving that $\int_0^1 \frac{\log^2(x)\tanh^{-1}(x)}{1+x^2}dx=\beta(4)-\frac{\pi^2}{12}G$

I am trying to prove that $$I=\int_0^1 \frac{\log^2(x)\tanh^{-1}(x)}{1+x^2}dx=\beta(4)-\frac{\pi^2}{12}G$$ where $\beta(s)$ is the Dirichlet Beta function and $G$ is the Catalan's constant. I managed ...
1
vote
2answers
33 views

Functions that satisfy the identity $f\left(\frac{x}{t}\right) f\left(-\frac{y}{r}\right)=f\left(\frac{x-y}{t-r}\right)$

I am looking for function(s) which satisfy the following property: $$f\left(\frac{x}{t}\right) f\left(-\frac{y}{r}\right)=f\left(\frac{x-y}{t-r}\right)$$ I am not sure if there is any function ...
2
votes
1answer
51 views

combinatorial identity involving fraction and product of bionomial coefficients

How can I prove the following identity for $i\geq 1$: $$ \sum_{t=i}^{s-1} \frac{i}{t}\binom{2(s-t-1)}{s-t-1}\binom{2t-i-1}{t-1}= \binom{2s-i-2}{s-1}. $$ Perhaps I'll need to go to hypergeometric ...