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

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0answers
69 views

The integral of the product of three Meijer-G-functions

Is there any expression for the integral of the product of three Meijer-G-functions, where the domain of integration is $[0,1]$?
2
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1answer
43 views

Product of two hypergeometric functions

For $\Re a, \Re b, \Re c, \Re a', \Re b', \Re c'>0$, I would calculate the following product $$ {}_2 F_1(a, b; c; x^{-1}) \times \, {}_2 F_1(a', b'; c'; 1-\frac{x}{y}) $$ For all $y>x>1$. ...
2
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0answers
44 views

Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity ...
3
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1answer
105 views

Special functions related to $\sum _{n=1}^{\infty } \frac{x^n \log (n!)}{n!}$

While doing some caculation related to von Neumann entropy, I encountered this kind of convergent series. $$\text{Exl}(x) \equiv \sum _{n=1}^{\infty } \frac{x^n \log (n!)}{n!}$$ In my calculation, ...
4
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2answers
540 views

Digamma function integral

Does anyone how to get a finite value to this integral ? $ \int\nolimits_{0}^{\infty} dx \frac{ \Psi (1/4+ix/2) +\Psi (1/4-ix/2)}{x^{2}+1/4} $ i have tried residue theorem but i got nonsenses :( can ...
3
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1answer
91 views

Integrating a differential equation?

How does $(xJ_0'(x))'+xJ_0(x)=0\implies\int\nolimits_0^1 x J_0(ax)J_0(bx) dx={bJ_0(a)J_0'(b)-aJ_0(b)J_0'(a)\over{a^2-b^2}}?$ Thanks. Perhaps int by parts? But how do I get the RHS form?
12
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3answers
339 views

How this integral $ \int_0^z\frac{1-e^x}{x} dx$ is connected to the Gamma function and Euler constant?

This is my first question in this forum; I hope it is an appropriate question. The Wolframalpha website tells me that $$ \int\nolimits_0^z\frac{1-e^x}{x} dx = \log (-z)+\Gamma(0, -z)+\gamma\quad ...
6
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2answers
292 views

Integral form of $\Gamma (x)$

While trying to represent the poles and analytic continuation of $\Gamma (x)$ , the author uses the following equality: ...
3
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1answer
118 views

Simplifying an Integral

I'm looking to simplify the integral $$\int\nolimits_0^{\infty}\dfrac{(t^b+1)^n}{(1+t)^{nb+2}} dt$$. (This arises out of the sum of a bunch of Beta functions, ie $\displaystyle\sum_{i=0}^{n} ...
10
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2answers
2k views

Integral of product of two error functions (erf)

In the course of my research I came across the following integral: $$\int\nolimits_{-\infty}^{\infty}\operatorname{erf}(a+x)\operatorname{erf}(a-x)dx$$ where ...
0
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1answer
212 views

Coordinate scaling in incomplete gamma function integral

I'm faced with the integral $$\mathcal{I} = \int\nolimits_0^\infty \mathrm d x \; e^{-\beta \, e^x - \mu x} \;,\quad \Re(\beta) > 0 \;.$$ The solution can be looked up. It reads $$\mathcal{I} = ...
3
votes
1answer
421 views

How do I find the inverse Hankel transform of $k^2e^{-k^2}$?

I am trying to solve: $$f_l(r)=\int_0^{\infty}e^{-k^2}k^4j_l(kr)dk,$$ where $j_l$ is the spherical Bessel function of the first kind, for any integer l >= 0. Thanks in advance for any answers!
5
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1answer
190 views

Is this a known special function?

Is this a known special function: $$\int\nolimits_0^1 a^p(1-a)^{1-p}\\,b^{1-p}\\,(1-b)^p dp\qquad ?$$ I am really only interested in maximizing this over $(a,b)$ in $[0,1] \times [0,1]$, so a ...
4
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1answer
43 views

Fundamental solution of a shifted operator

what is the fundamental solution of the shifted operator $ \Delta + \lambda^2 $, i.e, what the function $f$ satisfying the following equation $$ (\Delta + \lambda^2 )f(x) = \delta(x),$$ where $ \Delta ...
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0answers
36 views

Existence of solution for equation with erfc

I'm looking for the self-consistent (e.g. input needs to be the same as output) solution of $r$ in $$ r = \frac{1}{2}\operatorname{erfc}(z(r)) $$ where $\operatorname{erfc}$ is complimentary error ...
0
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0answers
39 views

Relation between hypergeometric functions $_2F_1$ of $z$ and $\frac{1}{1+z}$.

What are the relation between hypergeometric functions $_2F_1$ of $z$ and $\frac{1}{1+z}$. Specifically, I need a transformation that transforms: $_2F_1\left(a,b;c; -\sinh^2(x)\right)$ to ...
0
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0answers
26 views

Seeking a closed form solution to an ODE, if any.

I am try to solve the following ODE by the method of separation of variables. $$\frac{dF}{dx}=\frac{b}{x\left\{\ln^3 x+\ln x[c+d(ax+b)][2\ln x+c+d(ax+b)] \right\}},$$ where $a,b,c,d$ are constants. I ...
8
votes
1answer
110 views

How do I develop numerical routines for the evaluation of my own special functions?

This question has been cross-posted to ComputationalScience.SE here. When performing computational work, I often come across a univariate function, defined in terms of an integral or differential ...
0
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0answers
8 views

Resolvent kernel of Hyperbolic space

The expression of the spherical functions $\varphi_\lambda(x)$ and the resolvent kernel $r_\lambda(x)$ on Hyperbolic space are known. The spherical functions are the Jacobi functions ...
1
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1answer
75 views

meijer g function explicit form

Can the following case of the Meijer G-function $$ G_{2,3}^{3,1}\left(z\left|\begin{smallmatrix}0,1\\ 0,0,0\end{smallmatrix}\right.\right) $$ be expressed more explicitly (in terms of other special ...
4
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3answers
55 views

Alternative definition of Gamma function. Show that $ \lim_{n \to \infty} \frac{n! \; n^m}{m \times (m+1) \times \dots \times (m+n)} = (m-1)!$

Alternative definition of Gamma function on Wikipedia has it defined as a limit. $$ \Gamma(t) = \lim_{n \to \infty} \frac{n! \; n^t}{t \times (t+1) \times \dots \times (t+n)}$$ How do we recover ...
16
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1answer
197 views

Is there any proof that the Riemann Zeta function is not elementary?

I'm just curious, has anyone ever proved that the Riemann Zeta function is not an elementary function? Here I am using the term "elementary" in the sense of Liouville or as defined in this paper. ...
2
votes
1answer
583 views

The approximation of first-ordered modified Bessel function of the second kind

After analysing the outage probability of a single relay selection system, I got to the following form: $P = 1 + \sum\limits_{k = 1}^K {\left( \begin{array}{l} K\\ k \end{array} \right){{\left( { - 1} ...
4
votes
3answers
209 views

Zeta function for negative integers

I already proved that $\zeta(z)=\frac{1}{\Gamma(z)}\int_0^\infty\frac{t^{z-1}}{e^t-1}dt=\frac{\Gamma(z-1)}{2\pi i}\int_{-\infty}^0\frac{t^{z-1}}{e^{-t}-1}dt$ Now the Benoulli numbers are defined by ...
6
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1answer
123 views

How to prove that only the sine waves keep their shape when they are added together and have the same period?

If $f(t)$ is periodic and $f(t) + C \cdot f(t + t_1)$ has the same shape of $f(t)$ for each value of $C$ and $t_1$, then $f(t)$ has the shape of a sine wave. Is there a simple proof? Is there an ...
9
votes
1answer
181 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
2
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2answers
35 views

Integration of hypergeometric functions?

I would calculate the following integral \begin{equation} I_x = \int_{0}^{1} y^{b+\mu-1} (1-y)^{\nu-1}\, _2F_1(a,b+\nu +\mu;c; xy) \, dy. \end{equation} Such that $\quad \Re a,\Re b,\Re \mu, \Re \nu ...
0
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0answers
75 views

How Can I Find A Closed Form For This Double Summation?

I am looking for a closed form for the following summation that resembles the binomial theorem (to some degree): $$ F_n(x,z) = \sum_{k=2}^n \sum_{c=1}^{k-1} \frac{L_c^{(k-2c)}(-fg)}{(k-c)!} ...
9
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2answers
3k views

Integrate $\sqrt{1+9x^4} \, dx$

I have puzzled over this for at least an hour, and have made little progress. I tried letting $x^2 = \frac{1}{3}\tan\theta$, and got into a horrible muddle... Then I tried letting $u = x^2$, but ...
2
votes
1answer
87 views

Show $\int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x} = \frac{\pi z}{sin(\pi z)}$

I need to solve the following integral: $$ I = \int_{0}^{\infty} \frac{x^{-z}}{(1 + x)^{2}} ~ \mathrm{d}{x}. $$ Wolfram Alpha gives the answer as $ \frac{\pi z}{sin(\pi z)}$, or equivalently, $\pi ...
0
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0answers
15 views

Zernike Polynomials in Higher Dimensions

Let $n,l$ be integers with $n-l\geq 0$. Set $\alpha =\frac{n-l}{2}$ and $\beta =\frac{n+l}{2}.\ $Then the radial part of the Zernike polynomials in dimension 4 is given by $\tag 1R_{n}^{(l)}(\rho ...
0
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1answer
44 views

How to solve harmonic oscillator-like equation with $\theta$-function?

Suppose second order linear differential equation $$ \frac{d^2 y(t)}{dt^2} + \omega^2(t)y(t) = 0, \quad \omega^2(t) = q^2 + \theta (t-t_{0})m^2 $$ ($\theta (t)$ denotes Heaviside step-function) with ...
7
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2answers
97 views

A generalization of Bell numbers to arbitrary complex arguments

For $n\in\mathbb N$, the Bell number $B_n$ is a number of ways to partition the integer range $[1,\,n]$ into pairwise disjoint non-empty subsets. E.g. $B_3=5$ because ...
0
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0answers
35 views

When $\frac{e^{-\lambda } \lambda ^x}{x!}$ over positive integers is invertible?

I am curious for what values of $\lambda \in \mathbb{R}^+$, the function $f(x)=\frac{e^{-\lambda } \lambda ^x}{x!}$ defined only on positive integers i-e $x \in \mathbb{Z}^+$ is invertible? When ...
1
vote
1answer
1k views

Associated Legendre polynomials of fractional order

My question concerns the associated or generalized Legendre polynomials. They are labeled by two numbers $m$ and $l$; i.e., $P_l^m(x)$, for $x \in [-1,1]$. Usually one assumes that $m$ and $l$ are ...
3
votes
2answers
290 views

Integration over a combination of confluent hypergeometric, power, and exponential functions

I am trying to work out this integral. If there is no closed form, can you think of any approximations to it? $$\int_0^T e^{a (T-x)} (T-x)^{1+m+n} x^k \, _1F_1\Big(1+n;2+m+n;a (x-T) \Big) \, dx$$ ...
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2answers
212 views
3
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1answer
22 views

Orthogonality of Laguerre polynomials from generating function

I'm trying to show the orthogonality relation for Laguerre Polynomials $L_n(x)$ through their generating function $G(x,t)$. $$G(x,t)=\frac{1}{1-t}e^{\frac{-xt}{1-t}}=\sum_{n=0}^{\infty} L_n(x) t^n$$ ...
3
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0answers
89 views
4
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2answers
127 views

Calculating in closed form an integral in Airy function

Can we hope for a nice closed form for the integral below? $$\int_0^1 \frac{\displaystyle \text{Ai}\left(-\frac{t}{2^{2/3} \sqrt[3]{3-3 t}}\right)^2+\text{Bi}\left(-\frac{t}{2^{2/3} \sqrt[3]{3-3 ...
2
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0answers
29 views

(n+1)-th derivative of the following function [closed]

I have the Runge function $f(x)=\frac{1}{1+x^2}, x\in [-5,5]$ and the $n$-th derivation of this function: $$f^{(n)}(x)=(-1)^nn!\frac{\sin((n+1)\text{arccot}(x))}{(\sqrt{1+x^2})^{n+1}}.$$ I have to ...
2
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0answers
41 views

What is known about $\sum_{n=0}^{\infty} x^{n^3} $.

$f(x) =\sum_{n=0}^{\infty} x^{n^2}$ and similar "theta-type" functions are extensively studied. They have many properties and occur in number theory , algebra (in particular solving the quintic ...
4
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1answer
128 views

Prove that $\int_{-\infty}^\infty \frac{\operatorname{Ai}^2(x+a_n)}{x^2}dx = 1$

While I've been thinking about this question, I've found that for all $n \geq 1$ integer values, we have $$ \mathcal{I}_n = \int_{-\infty}^\infty \frac{\operatorname{Ai}^2(x+a_n)}{x^2}dx ...
2
votes
1answer
51 views

Zeros of Bessel functions

Let $J_\nu(x):=\displaystyle\sum^\infty_{k=0}\frac{(-1)^k(x/2)^{\nu+2k}}{k!~\Gamma(\nu+k+1)}$ denote a Bessel function. When $\nu\geq0$, let $0<j_{\nu,1}<j_{\nu,2}<\cdots$ denote the positive ...
2
votes
3answers
250 views

Adding imaginary number to exponential of Euler Gamma function

This is gamma function: $\Gamma (n) = \int_0^\infty x^{n-1}e^{-x}\,dx$ What will be Result if I add Imaginary Number to Exponential of Euler Gamma Function? $$? = \int_0^\infty x^{n-1}e^{-ix}\,dx$$ ...
0
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0answers
42 views

How can simplify this expression?

Can we write the following expression $$ c_1 P_{i \lambda - \frac{1}{2}}^{\frac{3}{2}}(\cosh t) + c_2 Q_{i \lambda - \frac{1}{2}}^{\frac{3}{2}}(\cosh t); \quad t>0 \, \mbox{and} \, \lambda \in ...
10
votes
1answer
4k views

Solution of an integral with strange imprecision of gamma functions

Trying to solve the following integral, with $n,m \in \mathbb{Z^+}$, $\alpha>1$, $0 < \epsilon < 1$, and $\Gamma(.)$ and $\Gamma(.,.)$ the gamma and incomplete gamma functions, respectively: ...
1
vote
1answer
62 views

Solving a differential equation of order two

I can not to solve the following equation $$ y^{ ''}(t) + \left( \lambda^{2} - \frac{2}{\sinh^{2}(t)} \right) y(t) = 0, \quad \mbox{with} \, t>0 $$ where $\lambda \in \mathbb C$. Someone can ...
3
votes
1answer
54 views

On what domain is the dilogarithm analytic?

The series $\displaystyle\sum \dfrac{z^n}{n^2}$ converges for $\lvert z\rvert<1$ by the ratio test, meaning that the dilogarithm function $\text{Li}_2(z),$ which is equal to the series ...
1
vote
0answers
44 views

How to solve a homogeneous Fredholm integral equation of the second kind with a symmetric non-seperable kernel?

I have the equation \begin{eqnarray} \lambda L(p)=\int dq\,K(p,q)L(q) \end{eqnarray} Where $L$ is an unknown function, $\lambda$ is some constant, and $K$ is a known function. This is a homogeneous ...