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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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 Wikipedia ...
1
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0answers
31 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
62 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
49 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} ...
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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 t_{...
1
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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
35 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 ...
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0answers
13 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
84 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
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1answer
174 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
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1answer
28 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
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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
43 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} =\sum_{n=1}^{\infty}\...
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0answers
14 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
26 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
40 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 $(12....
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0answers
72 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
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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
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0answers
57 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 integral?...
14
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1answer
197 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
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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 ...
1
vote
1answer
31 views

What is an example of a polynomial of degree as small as possible which meets this condition?

If $f$ is a function, what polynomial is a good approximation of order $n$ for $f$ near $x=0$? Here we say that $P$ is a good approximation of order $n$ for $f$ near $x=0$ when $E(x)$ approaches $0$ ...
1
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0answers
37 views

Derivatives wrt order of MacDonald function

I'm looking for a closed-form expression for $$ \left.\left[\frac{\partial^n}{\partial \nu^n}K_{\nu}(z)\right]\right|_{\nu=\pm\tfrac{1}{2}},\;\;n\ge1 $$ where $K_{\nu}(z)$ denotes the MacDonald ...
5
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0answers
48 views

Find the extended form of the group generated by an operator?

I tried to find the extended form of the group generated by the following operators. (I): The first operator $$A=z\frac{\partial }{\partial z}+1$$ To find the extended form of the group ...
0
votes
1answer
45 views

Check this value of $\int_{0}^{x}\frac{t^m}{(x-t)^\alpha}dt$

I want to prove that: $$\int_{0}^{x}\frac{t^m}{(x-t)^\alpha}dt=\frac{\Gamma(1-\alpha)\Gamma(m+1)}{\Gamma(m-\alpha+2)}x^{m-\alpha+1}$$ where $m$ is a positive integer and $\alpha \in [0,1]$. I ...
1
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0answers
18 views

How can I scale a value when it is within a threshold?

I am not a mathematician so I'm not even sure of the correct language to describe this. I also don't know what appropriate tags are for this question so please amend as necessary. I am looking ...
1
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1answer
48 views

hypergeometric transformation

I came across the following ${}_3F_2$ hypergeometric polynomial: $$ {}_3F_2\left(\left.\begin{array}{c} 1,1,-n\\ 2, -1-2n \end{array}\right| -x\right) $$ for some large $x > 0$. I am wondering ...
4
votes
1answer
266 views

Limit involving the inverse beta regularized function

Let $0<p<\frac{1}{2}$. I am looking for the limit: $$\lim_{t \to \infty} \left(\frac{t}{\frac{t}{I_{2 p}^{-1}\left(\frac{t}{2},\frac{1}{2}\right)}-2 \sqrt{t} \sqrt{\frac{1}{I_{2 p}^{-1}\left(\...
7
votes
1answer
170 views

How can I evaluate $\int_{0}^{1}\frac{(\arctan x)^2}{1+x^{2}}\ln\left ( 1+x^{2} \right )\mathrm{d}x$

How to calculate this relation? $$I=\int_{0}^{1}\frac{(\arctan x)^2}{1+x^{2}}\ln\left ( 1+x^{2} \right )\mathrm{d}x=\frac{\pi^3}{96}\ln{2}-\frac{3\pi\zeta{(3)}}{128}-\frac{\pi^2G}{16}+\frac{\beta{(4)}...
1
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0answers
38 views

How to make analytic continuation and compute imaginary part

Suppose I have the function $$ \tag 0 G(x) = g(x)K\left(k(x)\right), $$ where $$ k(x) = \frac{4\sqrt{x}}{(x-1)^{\frac{3}{2}}(x+3)^{\frac{1}{2}}}, $$ $$ g(x) = \frac{2}{\sqrt{x}}k(x), $$ and $K(x)$ is ...
2
votes
1answer
43 views

Hermite Polynomials: Rodrigues to Integral Representation

I would like to go from this representation of the Hermite polynomials: $$H_n(z)=(-1)^ne^{z^2}\frac{d^n}{dz^n}e^{-z^2} \tag{1}$$ to this representation $$H_n(z)=\frac{2^n}{\sqrt{\pi}}\int_{-\infty}^...
0
votes
0answers
35 views

Integral involving power of incomplete gamma function

I have the following integral that I am trying to solve $$I= \int_0^\infty e^{-\beta x} x^{\mu-1} \tilde{\gamma}(\nu, \alpha x)^\xi dx $$ where $\beta \in \mathbb{R}^+ $, $\nu \in \mathbb{R}^+$, $\xi ...
0
votes
1answer
62 views

Modified Bessel Function Integral representation proof $K_{\nu}(z)=\frac{z^{\nu}}{2^{\nu+1}}\int_{0}^{\infty}t^{-\nu-1}e^{-t-z^{2}/4t}dt $

How do I proof the following integral representation for the Modified Bessel function of the second kind (Macdonald Function). $K_{\nu}(z)=\frac{z^{\nu}}{2^{\nu+1}}\int_{0}^{\infty}t^{-\nu-1}e^{-t-z^{...
3
votes
0answers
42 views

About the domain of the Gamma function

I started to read about the history of the Gamma Function. There are three places I liked most, The early history of the factorial function (p. 239 - 243) Leonhard Euler's Integral: An Historical ...
1
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1answer
39 views

Name of a particular improper integral

I am curious if there is a particular name for this, $\int\limits_{-\infty}^\infty e^{i\xi^2}d\xi$. I think it might be related the Fresnel integral but I cannot see it, any suggestions?
0
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1answer
27 views

Asymptotic limit of the following integral?

I am interested in the asymptotic limit of the following integral for $a\rightarrow\infty$, $$\int_0^1\mathop{\mathrm{d}x}J_2(ax)x^n,$$ where $n>-1$ and $J_2(x)$ is the Bessel function of first ...
2
votes
1answer
35 views

Legendre Polynomial definite integral identity

I'm doing a problem involving legendre polynomials and I got stuck in this integral: $$I_k=\int_{-1}^{1} x P_{2k+1}(x)dx $$ Update: Note that the function in the integral is even If $k=0$, then ...
1
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0answers
26 views

Differential equation with variable coefficients

I saw this differential equation somewhere $$y''+xy=0$$ it was solved using the substitution $$y=x^\alpha u$$ where $\alpha$, is a constant. My question is how can we substitute for $y$ with an ...
0
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1answer
18 views

What is the gamma term in the modified Bessel function of second kind of Order Zero?

In my version of Schaums mathematical handbook (4th edition page 156), it gives the following for $K_0$ $$ K_0= -\big(ln(x/2)+\gamma\Big) I_0(x)+\frac{x^2}{2}+\frac{x^4}{2^2\cdot4^2}\left(1+\frac{1}{...
0
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0answers
30 views

Infinite sum of a product of hyperbolic functions, help!!

Let $g_{a,b}=\mathrm{csch}(n(a-b))$ when $a$ is different from $b$ and $0$ if $a=b$. $n$ is a positive real. I am trying to compute the following sum \begin{equation} \sum_{k=0}^{\infty}(2k-a)g_{0,k}...
0
votes
1answer
52 views

Deriving Hermite polynomial derivative recurrence relation straight from differential equation.

I want to derive the derivative recurrence relation for the Hermite polynomials straight from the Hermite differential equation. That is, I want to go from left to right in the following sequence ...
0
votes
0answers
16 views

On Hyper-geometric function differential equation

The hypergeometric function $$_2F_1(a,b;c+n:z) = \sum_{m=0}^\infty \frac{(a)_m(b)_m}{(c+n)_m}\frac{z^m}{m!}$$ should satisfy the differential equation $$z(1-z)\frac{d^2u}{dz^2} + [c+n-(a+b+1)]\frac{...
1
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2answers
44 views

Does anyone know a function that can describe a harmonic series?

I want to find a function that satisfies the following functional equation: $F(z+1)=1/z+F(z)$ This is a generalization of harmonic series 1 + 1/2 + 1/3 + 1/4 + ...,...
1
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0answers
30 views

On Hyper-geometric Functions and its recurrence relation

I research in generating functions of Hyper-geometric functions $_2F_1(a+n,b;c+n;x)$ using Lie group theoretic method and so the recurrence relation is important in this method. I want recurrence ...
1
vote
1answer
34 views

Want to check that $\sum_{j=0}^{k-1}w^{ jm}=0$, $m\not\equiv 0 \pmod{k}$ where $w=e^{2\pi i/k}$

If $f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$, then $$ \sum_{n=0}^{\infty}a_{kn+m}x^{kn+m}=\frac{1}{k}\sum_{j=0}^{k-1}w^{-jm}f(w^j x) \tag{1},$$ where $w=e^{2\pi i/k}$ is a primitive $k$th root of ...
0
votes
2answers
43 views

How to prove this gamma identity?

How to prove this? $$2^n \ \Gamma(n+\frac{1}{2})\ =\ 1.3.5...(2n-1)\ \sqrt{\pi}$$ I tried rewriting the right-hand side as $$\frac{(2n-1)!}{2(n-1/2)}\ \sqrt{\pi}=\frac{\Gamma(2n)}{2\Gamma(n+1/2)}\sqrt{...
1
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1answer
74 views

Evaluating an integral by substitution and special functions [duplicate]

How can I evaluate this integral? $$\int_{0}^{1} \frac{dx}{\sqrt{{1+x^4} }}$$ I tried using the substitution $x=\mathrm{e}^{-u}$ but I got nowhere.
2
votes
1answer
83 views

If $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$ holds for $0<z<1$, then also for $0<\operatorname{Re}(z)<1$

In Special Functions p. 10, it has proven that $$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)},$$ for $0<z<1$. Then it says that this equality implies for $0<\operatorname{Re}(z)<1$. I do ...