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

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0answers
3 views

Tools for constructing real functions with derivatives constraints

I was playing around with stuff and I realized I'm incredibly bad and unaware of tools to construct real functions which satisfy things. For example: I want to construct a function that is infinitely ...
4
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1answer
60 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 \{ ...
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0answers
19 views

The Cauchy product $\sum_{n=1}^\infty \frac{\log n}{e^n}= \left( 1-\frac{1}{e} \right)\sum_{n=1}^\infty\frac{\log n!}{e^n} $

I know that the Cauchy product is defined $$\left(\sum_{n=1}^\infty\frac{\log n}{e^n}\right)\left( \sum_{n=1}^\infty\frac{1}{e^n} \right)= \sum_{n=1}^\infty\sum_{k=1}^n\frac{\log k}{e^{k+n-k+1}},$$ ...
3
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1answer
49 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 $$\Omega:=\{x\in\mathbb{R}^d\mid a<\|x\|_2<b\},$$ ...
2
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1answer
1k views

What is the sigmoid *squashing* function?

I've just read the following The basic unit ("neuron" i) performs the following computation to update its state $y_i$: it computes a weighted sum $v_i$ of its inputs $x:j$ which is passed ...
0
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0answers
17 views

Definite integral involving algebraic, exponential, and product of two Meijer's G function

I am having trouble with calculating the following integral: \begin{equation} I = \int_{0}^{\infty}x\exp({-\beta x})\large{G}_{2,2}^{1,2}\left( x \left| \begin{array}{cc} 1,1 \\ 1,0 \end{array} ...
2
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1answer
142 views

Sigmoid Function in Neural Network

I am studying a doctoral thesis on control-theory and have trouble understanding the notions and the notation introduced there. I am doing this out of interest on the subject, so I haven't had a ...
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0answers
34 views

Integrals with erf^N

Can anyone help with integral of type. In general, what to do if erf is in power higher than 1? $$g(S|S<L)=\frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{+\infty} \left [ \frac{1}{\sqrt{2 \pi ...
3
votes
3answers
90 views

Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)}dx =\ln(\pi)-\gamma $

I have to prove that given $\gamma=0.577216\ldots$, the Euler-Mascheroni constant, and $\pi=3.14159\ldots$, we have: $$2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)}dx =\ln(\pi)-\gamma $$
17
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1answer
387 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
6
votes
1answer
126 views

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 ...
1
vote
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 ...
0
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1answer
31 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 $n$: $$ \sum_{t=-\infty}^{\infty}e^{-\beta(t+\alpha)^{2}}H_{n}(t+\alpha) $$ ...
1
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1answer
29 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 ...
4
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1answer
198 views

An infinite series of a product of three logarithms

I was told this interesting question today, but I haven't managed to get very far: Evaluate $$\sum_{n=1}^\infty \log \left(1+\frac{1}{n}\right)\log \left(1+\frac{1}{2n}\right)\log ...
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0answers
24 views

help with proving question? [closed]

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
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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 ...
9
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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
20 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 ...
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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
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1answer
179 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 ...
4
votes
2answers
993 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
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1answer
769 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
23 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
15 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 ...
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0answers
15 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 + ...
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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
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0answers
84 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
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0answers
55 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
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 ...
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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
409 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 ...
1
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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 ...
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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
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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
30 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
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1answer
970 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 ...
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0answers
11 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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1answer
32 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
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1answer
661 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 ...
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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 ...
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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
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0answers
445 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
680 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
168 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 $a$, $b$, ...