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

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5
votes
3answers
178 views

How could one solve $\int_{0}^{\infty} \frac{1}{1-t^4}dt$ with special functions?

How could one solve $$\int_0^\infty \frac{1}{1-t^4} \, dt\,?$$ I have to apply special functions, so I thought that I have to use the change variable $$u=t^4,$$ but $$du=4t^3\,dt$$ and when $$t\...
1
vote
0answers
27 views

Definite Integration of Hypergeometric function combined with two algebraic function?

Please help me to solve the integral: $$\int_0^{-w}\dfrac{{_2F_1(1,k,3/2;\phi/B)}}{\sqrt{\phi}\sqrt{-\phi-\omega}}d\phi$$ I have solved this in Mathematica.but I am not able to way a general result ...
32
votes
1answer
799 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 ...
3
votes
2answers
420 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
vote
0answers
29 views

Combinatorial formula for Legendre Polynomials

Using the recursion formula for the solution of the Legendre equation: $$(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x) = 0$$ With solution $P_n(x)$ such that $P_n(1) = 1$, show that $$P_n(x) = \sum_{k = 0}^{n}\...
1
vote
0answers
22 views

Expressing spherical harmonics as a combination of other spherical harmonics

Spherical harmonics are a useful tool in physics, particularly in classic electrostatics and electrodynamics. Given an integer $l$, the spherical harmonic $Y_{l,m}$, where $-l\leq m\leq l$, solves the ...
1
vote
1answer
489 views

Solution to equations involving Plasma dispersion function

I am trying to solve an equation involving a complex argument for the plasma dispersion function as: $z = x + \iota y$, $ x = \omega / \sqrt2 k v_{Ti} $ $ y = \nu_i /\sqrt{2} k v_{Ti} $ $S[z] = -\...
2
votes
0answers
31 views

Spherical harmonics: how's Laplace's equation related to spheres?

Many spherical harmonics derivations start from finding a solution to Laplace's equation and the results are in fact what are called spherical harmonics. However, how's Laplace's equation really ...
3
votes
1answer
294 views

Deriving the Airy functions from first principles

I have just started reading about the Airy functions and am stuck on a particular step of their derivation. But first here is some background information to give this question some meaning, more ...
8
votes
4answers
281 views

Verify $y=x^aZ_p\left(bx^c\right)$ is a solution to $y''+\left(\frac{1-2a}{x}\right)y'+\left[(bcx^{c-1})^2+\frac{a^2-p^2c^2}{x^2}\right]y=0$

In order for the question that I have to make any sense I must first include some background information as given in my textbook: The standard form of Bessel's differential equation is $$x^2y^{\...
1
vote
1answer
19 views

Derivative of Incomplete Gamma Function

For the following incomplete Gamma function: \begin{equation} Γ(1+d,A-c \ln x)=\int_{A-c\ln x}^{\infty}t^{(1+d)-1}e^{-t}dt \end{equation} I am trying to calculate the derivative of $Γ$ with respect ...
0
votes
0answers
13 views

Solving the definite integral $\int_{-\psi/2}^{\psi/2}\,\exp{(A\sin(\theta+\phi))}d\theta$

I need to solve this definite integral: $$\int_{-\psi/2}^{\psi/2}\,\exp{(A\sin(\theta+\phi))}d\theta$$ where $A$ is a real positive constant and $\psi\in[0,2\pi]$. I know that for $\psi=2\pi$ the ...
5
votes
1answer
102 views

finding the series $\sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$

My goal is to solve this series $$S(x) = \sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$$ I did took the derivative first w.r.t $x$ $$S'(x) = \sum_{n=1}^\infty \frac{x^{n-1}}{n!}$$ which I ...
3
votes
2answers
150 views

Application of Bessel Function

I have read number of books and online literature on Bessel function. Theoretically, I have known about Bessel function. What is practical significance of Bessel function? How can Bessel function ...
1
vote
1answer
33 views

Another Laplace transform of a function with square roots.

This question is very much related to this (one). Let $F\colon \mathbb{C}\to \mathbb{C}$ be defined as $$F(s) = \frac{1}{4+3s+\sqrt{s(4+s)}}.$$ My question is what is the inverse Laplace transform ...
10
votes
3answers
5k views

How to come up with the gamma function?

It always puzzles me, how the Gamma function's inventor came up with its 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 ...
1
vote
1answer
36 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
votes
2answers
43 views

Problem on series expansion and Bessel functions

One way to define Bessel functions is $$ e^{\frac{x}{2}(t-\frac{1}{t})}=\sum_{n=-\infty}^{+\infty}J_{n}(x)t^n. $$ How do I prove that? I can't see a way of writing the L.H.S. as a geometrical (or ...
3
votes
2answers
106 views

Integral of Bessel function multiplied with sine

I need advice on how to solve the following integral: $$ \int_0^\infty J_0(bx) \sin(ax) dx $$ I've seen it referenced, e.g. here on MathSE, so I know the solution is $(a^2-b^2)^{-1/2}$ for $a>b$ ...
1
vote
0answers
58 views

What can I do to learn special functions? [closed]

I want to learn special functions but I'm finding the book by Ranjan Roy far too advanced for me. Please help.
0
votes
0answers
28 views

Building a good penalizing function

Currently I'm working with the following penalizing function: $$ \psi(x) = \left\{ \begin{array}{lr} 0 & : x < 0 \\ \frac{1}{1+e^{\frac{1}{x-1}+\frac{1}{x}}} = \frac{g(x)}{...
3
votes
1answer
130 views

Series involving the Riemann zeta function

Consider the series: $$\sum_{n=1}^{\infty}\frac{\zeta(2n+1)}{n(2n+1)}$$ We can easily prove that it's a convergent series. My question, is there a way to express this series in terms of zeta ...
4
votes
0answers
31 views

Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$ \theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) , $$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
2
votes
0answers
30 views

Confusion about the convergence of Riemann zeta function in terms of the integral

Titchmarsh wrote that $$\zeta(s)=s\int_{1}^{\infty}\frac{\left \lfloor x \right \rfloor-x+\frac{1}{2}}{x^{s+1}}\,\mathrm{d}x+\frac{1}{s-1}+\frac{1}{2}\tag{2.14}$$ using the Euler-Maclaurin summation, ...
21
votes
2answers
540 views

Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,\mathrm{d}x$ have a closed form?

$$\newcommand{\arctanh}{~\mathrm{arctanh}~}\newcommand{\sech}{~\mathrm{sech}~}$$ $$I=\int_{-1}^1\frac{\arctan x}{\arctanh x}\,\mathrm{d}x$$ Mathematica gives an approximate result of $I=1....
2
votes
1answer
63 views

Series expansion of $\int x^xdx$

The indefinite integral: $$J=\int x^xdx$$ has no known closed form solution. Expanding in series the function $f=x^x$ we get: $$f\simeq\sum_{k=0}^N \dfrac{x^k\ln(x)^k}{k!}$$ So we can write: $$J\...
0
votes
0answers
38 views

What is the minimum growing function here?

What is the minimal growth of $a$ as a function of $N$ for which in $${x}{a^x}>\frac{\log N^{}}{c\log\log N}$$ $x=O(1)$ holds for a fixed $c>0$? Clearly $a=O\big(\big(\frac{\log N}{c\log\log N}\...
0
votes
3answers
38 views

Asymptotics of incomplete Beta function $B_{1/2}(y+1,y)$ when $y\to\infty$

My question concerns the behavior of the incomplete Beta function $$B_{1/2}(y+1,y)=\int_0^{1/2}x^y (1-x)^{y-1}dx$$ in the large $y$ limit. I have been looking everywhere, but I can't find anything. ...
0
votes
0answers
53 views

Complicated identity relating two 2F2 hypergeometric functions

I would like to relate the following hypergeometric 2F2 function: $\,_2F_2 \left(1, n+1-\alpha;n+2,n+2-\alpha-\beta;-x\right)$, where $\alpha,\beta>0$ to another 2F2 function: $\,_2F_2 \left(n+1,...
0
votes
1answer
32 views

Integration of physicists' Hermite polynomial with exponential

I am trying to prove the lhs of the following equation is equal to rhs. \begin{align*} \int_{-\infty}^\infty H_n(x)e^{-x^2/2}\,\mathrm{d}x = \begin{cases} \frac{2\pi n!}{(n/2)!},& \text{if } ...
0
votes
2answers
55 views

Proof of $\Gamma(x)\Gamma\left(\dfrac{1}{x}\right)\gt 1$

Is it possible to prove the following inequality: $$\forall x\gt0,f(x)=\Gamma(x)\Gamma\left(\dfrac{1}{x}\right)\gt 1?$$ $f(x)$ has a minimum where $f(x)'=0$ which means: $$\Psi(x)\Gamma(x)\Gamma(1/x)=\...
0
votes
0answers
5 views

Invertibility of a polylogarithmic map

Consider a map defined on $\Bbb R\times(0,+\infty)$ and given by $$M:(a,b)\to(\rho,E),$$ $$\rho = \int_{\Bbb R^n}\frac{dx}{1+\exp(a+b|x|^2)}\\E=\int_{\Bbb R^n}\frac{|x|^2dx}{1+\exp(a+b|x|^2)}.$$ I ...
5
votes
1answer
102 views

Closed form of a “harmonic” alternating dilogarithm sum

Does the following sum $$ S = \sum_{n\geq 2}(-1)^n \mathrm{Li}_2(2/n) = 1.14434\ 42096\ 91982\ 23727\ 39852\ 45805\ldots $$ have a closed form in terms of known constants? Neither the inverse ...
0
votes
0answers
25 views

How to simplify the following expression involving Jacobian elliptic functions?

I would like to show that a certain elliptic function $F(x)$ (that is periodic, say with some period $h$) has exactly two zeroes in $[0,h)$. Let us recall some notation. Given a parameter $m \in [0,1]$...
3
votes
1answer
66 views

Prove $\int_0^{\pi/2} J_0 (\cos x) dx=\frac{\pi}{2} \left(J_0 \left(\frac{1}{2} \right)\right)^2$

I got curious about this integral because we have the following identity: $$\frac{2}{\pi}\int_0^{\pi/2} \cos (x\cos t) dt=J_0(x)$$ So we have an interesting (if useless) symmetry: $$\int_0^{\pi/2} ...
3
votes
0answers
63 views

Integration of Laguerre polynomial $\int_{0}^{x}u^{p-1}(1-u)^{q-1}e^{-\theta u}L_n^{(m)}(\theta u)\mathrm du$

It's been several days that I'm confronted to this integral, without much success in its resolution. To give you more details, in my case: $n$ is an integer $>1$ $m=n-2$ $p,q \in \{n-1, n\}$ $x ...
8
votes
1answer
224 views

the solution for an integral including exponential integral function

I have the following integral $$\int_c^\infty{x^{a-1} e^{\ p \ x} \ \mathrm{Ei}(-p\ x) \ \mathrm{d}x}.$$ I'd like you to help me to evaluate it or giving me a hint to proceed.
1
vote
1answer
566 views

Integral of Meijer G-function

I am trying to integrate this: $$\int_0^\infty \log(1+x^r)x^{a-1}e^{-\beta x}I_v(kx) \ \mathrm dx$$ where $r, a, \beta, v, k$ are arbitrary constants, $v$ is the order of the modified bessel function ...
3
votes
1answer
437 views

A definite integral in terms of Meijer G-function

I am trying to find some relevant functional identities involving Meijer G-functions in order to prove $$ \int_0^\infty\frac{\log(x+1)}{x}\mathrm{e}^{-zx}\,\mathrm{d}x = G^{3,1}_{2,3}\left(z \middle| ...
2
votes
1answer
41 views

Integral representation of a Meijer G-function

How to prove that, the integral $$I_{a,b}:=\int_{1}^{+\infty}e^{-at}(1-t^{-1})^b\,dt ; \, a,b>0$$ is given by $\Gamma(b+1)$ times a Meijer G-function, i.e., $$I_{a,b}:=\Gamma(b+1) \times G^{m,...
1
vote
1answer
92 views

Conversion of Hypergeometric functions from Maple to Mathematica

After a calculation I found in Maple an expression of the kind: $$f(x)=hypergeom\left(\left[\dfrac{a}{b},1\right],\left[c,d\right],h\right)$$ What is the equivalent notation in Mathematica which I ...
0
votes
1answer
39 views

Proof of a formula containing double factorial

How can I prove the formula: $$\sum_{k=0}^\infty\dfrac{x^k}{k!!}=\dfrac{1}{2}e^{\dfrac{x^2}{2}}\left[2+\sqrt{2\pi}erf\left(\dfrac{x}{\sqrt\pi}\right)\right]?$$ Thanks
8
votes
3answers
392 views

How to find integral of $\int_0^\infty \frac{\ln ^2z} {1+z^2}\mathrm{d}z$?

How do I find the value of $$\int_{0}^{\infty} \frac{(\ln z)^2}{1+z^2}\mathrm{d}z$$ without using contour integration, - using the usual special functions, e.g., zeta/gamma/beta/etc. Thank you,
4
votes
1answer
939 views

What is a cardinal basis spline?

Wikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as "Bk". Meanwhile, cnx.org Signal Reconstruction says: The basis splines Bn are shown ... ...
0
votes
0answers
27 views

How to get analytical summation of this series?

How to get the analytical summation of this series? $$\sum\limits_{n = 2}^{ + \infty } {{\varepsilon ^{n - 1}}\frac{1}{{{n^3}}}\frac{{{d^2}P_n^2\left( {\cos \theta } \right)}}{{d{\theta ^2}}}} = ?$$ ...
1
vote
2answers
71 views

Limit of gamma and digamma function

In my answer of the previous OP, I'm able to prove that \begin{align} I(a)&=\int_0^\infty e^{-(a-2)x}\cdot\frac{1-e^{-x}(1+x)}{x(1-e^{x})(e^{x}+e^{-x})}dx\tag1\\[10pt] &=\int_0^1\frac{y^{a-1}}...
0
votes
1answer
41 views

Unconventional Differentiation Rules

We all know the stock-standard and conventional differentiation rules, such as the Sum and Difference Rule, Product Rule, Chain Rule etc. But are there other more advanced rules that are not treated ...
58
votes
1answer
2k views

Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + \...
0
votes
0answers
8 views

proving the output of cantor pairing function is un correlated if one bit of input is changed

Pairing function $\pi :N \times N\rightarrow N$ is defined as: $\pi(a_{1},a_{2})=\frac{1}{2}(a_{1}+a_{2})(a_{1}+a_{2}+1)+a_{2}$. Exp 1. If $a_{1},a_{2}$ are input then output is $\pi_{1}$ Exp 2. If ...
2
votes
4answers
51 views

How to express $2x^3-x^2-3x+2$ as a linear combination of Legendre polynomials

I have used the formula \begin{align}p_0(x)&=1\\ p_1(x)&=x\\ p_2(x)&=\frac12(3x^2−1)\\ p_3(x)&=\frac12(5x^3−3x) \end{align} $$2x^3-x^2-3x+2=Ap_3(x)+Bp_2(x)+Cp_1(x)+Dp_0(x)$$ EDIT- \...