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

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9
votes
3answers
2k views

How to come up with the gamma function?

It always puzzles me, how the Gamma functions's inventor came up with it's definition $$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$ Is there a nice derivation of ...
0
votes
1answer
21 views

Integral over product of two bessel functions and power

I have searched the literature on integrals over bessel functions, but I couldn't find anything. The integral to be evaluated is, $\int_0^a J_{n}(bx)J_{\mu}(cx)xdx =: \mathcal{M}_r(a;n,\mu;b,c)$ ...
1
vote
2answers
244 views

Express complex Bessel function in terms of functions taking real arguements

I want to use the Bessel function in C++. Since this one is not implemented there for complex arguments, I am looking for a way to express the bessel function(first and second kind) as: ...
6
votes
3answers
117 views

How to solve $x^2 = e^x$

The question is to find $x$ in: \begin{equation*} x^2=e^x \end{equation*} I know Newton's method and hence could find the approx as $x\approx -0.7034674225$ from \begin{equation*} ...
1
vote
1answer
393 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 ...
2
votes
1answer
50 views

Closed-form expression for $\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$?

As per the title, I am looking for a closed-form expression for the integral $$\frac{1}{B(\alpha,\beta)}\int_{0}^{1}e^{-ax(1 - bx )}x^{\alpha-1}(1-x)^{\beta - 1}dx$$ where $a,\alpha,\beta>0$ and ...
1
vote
1answer
26 views

Integration of the incomplete beta function

I would like to know if there is a way of computing the following integral analytically ($B_u$ is the incomplete beta function): $$\int B_u(a-1,0)~u^{-a} du$$ Thanks for your ideas.
1
vote
1answer
33 views

Laplace transform involving the gamma function.

Does anyone know how to evaluate the following integral $$ \int_{0}^{\infty} \frac{e^{-qs}\alpha^{s}}{\Gamma(s)\Gamma(s)}\text{d}s $$ where $q,\,\alpha > 0$? I've done some digging in usual ...
4
votes
1answer
147 views

What is the reason to use hypergeometric functions?

I would be grateful if anyone could explain the purpose of using hypergeometric functions. If a function exists in closed form, e.g. $\sum\limits_{k \geq 0}z^k = {}_2 F_1 \bigg[{{1\; 1}\atop{1}} \vert ...
2
votes
2answers
43 views

Why are there four independent solutions of Mathieu equation instead of two?

Consider Mathieu equation: $$\frac{d^2}{d\xi^2}R(\xi)+(a-2q\cos(2\xi))R(\xi)=0.$$ It's a second order ODE, so it should have two linearly independent solutions. One of the choices is to denote one ...
7
votes
1answer
84 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 ...
18
votes
1answer
282 views

Is $K\left(\frac{\sqrt{2-\sqrt3}}2\right)\stackrel?=\frac{\Gamma\left(\frac16\right)\Gamma\left(\frac13\right)}{4\ \sqrt[4]3\ \sqrt\pi}$

Working on this conjecture, I found its corollary, which is also supported by numeric caclulations up to at least $10^5$ decimal digits: ...
0
votes
0answers
53 views

Handling complex arguments of elliptic integrals in Maple

I want to solve the integral \begin{equation} I(y)=\int_0^y\sqrt{\dfrac{x(1+ax)}{(1+ax)^2-b^2}}\,\mathrm{d}x,\qquad a<0,\; b\in(0,1),\; y>0,\; (1+ax)^2-b^2>0, \end{equation} using Maple 18. ...
11
votes
1answer
118 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
4
votes
1answer
64 views

Is the solution of functional equation $x^x=y^y$ when $0\lt x\lt y$ uncountable?

I want to prove that, the set: $S=\{(x,y)\in \mathbb R^2\,\,|\,\,0\lt x\lt y \,\,,\,\,\,\,x^x=y^y \}$ $\,\,$ is uncountable. My idea is the following: Consider the function ...
2
votes
0answers
54 views

A possible dilogarithm identity?

I'm curious to find out if the sum can be expressed in some known constants. What do you think about that? Is it feasible? Have you met it before? $$2 ...
9
votes
1answer
93 views

What is $f_\alpha(x) = \sum\limits_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n$?

I want to understand the function $$f_\alpha(x) = \sum_{n\in \mathbb{N}} \frac{n^\alpha}{n!}x^n, \ \ \ \forall x\in\mathbb{R},$$ for any possible real $\alpha\geq0$. I know that for $\alpha$ integer, ...
1
vote
2answers
66 views

A special modular function: $ j $-invariant.

It is known that j invariant $$j(\tau)= 1728 \frac{g_2^3(\tau)}{\Delta(\tau)} $$ $\tau \in \mathbb{H}$ attains every complex value , Can someone guide me its proof.?? where $L(\tau ) = \{\tau m ...
5
votes
0answers
62 views

Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)-4\text{Li}_2(\alpha^3)-3\text{Li}_2(\alpha^2)+6\text{Li}_2(\alpha)-\tfrac{7}{5}\zeta(2)=0\tag1$$ ...
2
votes
0answers
206 views

Complete Elliptic Integral of the 3rd Kind - Residual Computation

Let us consider the following function $f(a,k)$ in the interval $a,k\in (0,1]$ : $$f(a,k)=\frac{2 \sqrt{1-a^2} \sqrt{a^2-k^2}}{\sqrt{a^2}}\Pi\left(a^2,k^2\right)$$ where $\Pi\left(a^2,k^2\right)$ is ...
0
votes
0answers
16 views

transformation involving elliptic integrals

I have two expressions which I know are equivalent but I just can't see how to go from one to another. I'm sure it involves properties of elliptic integrals however I am not very familiar with the ...
0
votes
0answers
76 views

Value of an elliptic integral of the first kind

The elliptic integral of the first kind $$ \int_0^{\pi/2}{\frac{du}{\sqrt{1-k^2\sin^2{u}}}} $$ cannot be expressed in terms of standard functions. But in the following context from The Pendulum by ...
0
votes
2answers
48 views

How to evaluate the length of the perimeter of a low eccentricity ellipse?

Given that $ e= \frac{a^2-b^2}{b^2} $ , and $L$ is the length of the perimeter, which equals $4aE(e, \pi/2)$, find the length of the perimeter up to $e^2$ in terms of $a$ and $b$. How does one begin ...
0
votes
0answers
23 views

Proof of identity with Hermite polynomials

Let's have Hermite polynomial, $H_{n} = e^{\frac{x^{2}}{2}}\frac{d^{n}}{dx^{n}}e^{-\frac{x^{2}}{2}}$. How to prove the identity $$ \tag 1 \sum_{n = 0}^{\infty}H_{n}(x)H_{n}(y)\frac{t^{n}}{n!} = (1 - ...
0
votes
0answers
40 views

Closed form expression for 3F2 with positive unit argument

Is there any closed form expression for the Hypergeometric function ${}_3F_2(-n,-n,c;-d/2-n,-d/2-n;1)$ for $n>0$ and $d>0$. The parameter $c$ can be both positive and negative.
0
votes
0answers
40 views

Euler type superdivergent

Could you explain where this came from: $$\sum _{k=0}^{\infty } (k!)^2 (-y)^k=\frac{G_{1,3}^{3,1}\left(\frac{1}{y}\mid{{0}\atop{0,0,0}}\right)+2 \left(\log \left(\frac{1}{y}\right)+\log (y)\right) ...
0
votes
0answers
38 views

q-theta function and their properties

I want to compute the residue integral for the $q$-theta function, and derive its properties. First, I'll briefly explain the definition \begin{align} & ...
0
votes
0answers
19 views

Integrals involving Marcum Q and Gaussian functions

I want to evaluate the following double integral. Does any closed form solution exist? $$I(x)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-y^2}e^{-z^2} Q_1 ...
4
votes
1answer
171 views

Integral Representation of Bessel Function (K)

There is an integral representation for the modified Bessel function of the second (or third depending on who you talk to) kind (denoted $K_\nu$) that says: $$K_\nu(z) = ...
0
votes
1answer
190 views

Is there a name for the normal CDF function $\Phi(\cdot)$?

I can't seem to find a plain English name for the CDF of the normal distribution $\Phi(x)$. However, I am aware of several other related functions that have a name, so I feel like this one should as ...
6
votes
0answers
92 views

Weierstrass product expression for Klein's j-invariant

The first sentence of @ccorn's answer to a previous question of mine was: “Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the ...
0
votes
2answers
41 views

Writing $f(x)$ in terms of the heaviside function

I have $f(x,t) = 0$ when $t \le 0$ and $f(x,t) = \sin(-x + t)$ when $t > 0$. I have been told this can be written more concisely in terms of the heavisdie function $u(t - a)$ as $f(x,t) = \sin(-x ...
0
votes
1answer
33 views

Why it is necessary for copula functions to be grounded?

I know what the properties "grounded" and "2-increasing" means in copula functions definition but actually I can't understand the reason behind these two! I mean why it is necessary for copulas to be ...
9
votes
0answers
320 views

Mixed Bessel Function integral $\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\mathrm{d}z$

A tricky integral I have been working on, and probably doesn't have a solution in terms of known functions, is: $$\int_{0}^{\infty} e^{- \lambda ...
2
votes
1answer
24 views

Is this function defined in terms of elliptic $\mathrm{K}$ integrals even?

Let $R,z > 0$ be positive real constants, and consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by $$ f(v) = \frac{1}{\sqrt{(R+v)^2+z^2}}\ \mathrm{K}\!\left( \frac{4 R v}{(R+v)^2+z^2} ...
0
votes
1answer
22 views

In what ways can I extend the error function to accept complex arguments?

What are the different approaches to extending the error function to accept complex arguments? When should I favor using one approach over another?
1
vote
0answers
27 views

Proof for the Rodrigues formula for Neumann's Spherical functions.

I've been trying to prove the Rodrigues formula for Neumann's Spherical functions. The Neumann's Spherical functions are: $$N_n(x))=-(-x)^n\left[\frac{1}{x}\frac{d}{dx}\right]^n\frac{\cos{(x)}}{x}$$ ...
2
votes
2answers
56 views

Evaluate $-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$ in $\gamma$.

Evaluate $\gamma$ expressed, involving Lambert function, by $$-\gamma-W_{-1}\left(-\gamma e^{-\gamma}\right)=\frac{\pi}{4}$$ where $\gamma<1$. I doubt that it is possible to find a value for ...
5
votes
2answers
229 views

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ ...
0
votes
0answers
13 views

$K$ which is of second category in itself.let $H = K \cap ( - K)$. Why $H$ is non empty interior

Let $X$ be topological vector space.Let $K$ be closed, convex, dense subset of $X$ and $K$ which is of second category in itself. Put $H = K \cap ( - K)$. Why does $H$ is nonempty interior?
5
votes
1answer
139 views

Infinite series of Hypergeometric function

Any ideas how to find a closed form for the sum given by: $$ \sum^\infty_{n=0} \frac{1}{n!} \frac{a^n b^{n+m}}{(m+n)^2 \Gamma(m+n)} {}_2F_2 \left(m+n,m+n;m+n+1,m+n+1;-b\right) $$ Given that both $a$ ...
2
votes
0answers
28 views

Barnes' double gamma function versus q-gamma function

According to wikipedia, the q-analog of the gamma function is closely related to a multiple gamma function defined by Barnes. Besides the fact that they are both generalizations of the Gamma function, ...
1
vote
1answer
44 views

Heaviside function & Integral Limits

When considering integration, how does one use the Heaviside function in order to alter the limits of integration. For example If i have $$ \int_a^b f(x) dx $$ But want to change this integral to be ...
1
vote
1answer
67 views

How to prove this problem about supermodularity function?

The problem is as follows, and I have solved the subproblem (a), but haven't solved (b) yet. And for (b) the method I think about is proof by contradiction, but I get stuck before I could solve this. ...
1
vote
2answers
69 views

Eliminate the arbitrary funcion - PDE first order

I'm heading the book Elements Of Partial Differential Equations -Sneddon 1957. At chapter two exists this exercise "Eliminate the arbitrary function $f$ fron the equation $$ z= ...
0
votes
1answer
21 views

Difference Between Lyapunov and Strong Lyapunov Function.

Good Day everyone. I was assigned to show that given an autonomous system of Differential Equations and a function $V$, I need to show that $V$ is Lyapunov function. To show that $V$ is Lyapunov. I ...
2
votes
1answer
73 views

Name of function $(1+x)^n-1$

Is there any name for this formula $$(1+x)^n-1$$ When working with floating point numbers this can be calculated with much better precision for very small $|x|<1$ values using Taylor series ...
3
votes
2answers
54 views

Derivative of the Gamma function

How do you prove that $$ \Gamma'(1)=-\gamma, $$ where $\gamma$ is the Euler-Mascheroni constant?
1
vote
2answers
33 views

Integrals involving exponential functions and the gamma function [duplicate]

I'm having trouble evaluating this integral $$\int_0^\infty {e^{-ax^2}} \,dx $$ My guess is that it would evaluate into something like $$\int_0^\infty \frac 12e^{-s}s^{\frac 12} \ldots \,dx = \frac ...
0
votes
1answer
50 views

Best aproximation to an numerical solution using two aproximated functions

I want to find the best aproximation to a numerical solution. For that I want to use two aproximated functions (that I already know). If I plot them I see that one of them underestimates the original ...