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

learn more… | top users | synonyms

1
vote
1answer
14 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
11 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
97 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 ...
1
vote
0answers
44 views

Solve the nth zero of a function. [on hold]

Say I have a mystery continuous function, could be anything. f(x) Assuming we don't know the distribution of the zeros of the function, Is there a known way to solve the nth zero (hits the x-axis)? ...
0
votes
0answers
28 views

Series expansion of elliptic integral involving n th order polynomial in the denominator

My goal is to find an expansion in powers of 1/ρ of the integral: \begin{equation}I_n(\rho)=\int_\rho^{+\infty}\frac{dt}{(E_n(t))^2\sqrt{t^2-h_2^2}\sqrt{t^2-h_3^2}},\quad \rho \ge h_2\end{equation} ...
1
vote
1answer
25 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 ...
0
votes
0answers
28 views

The sum $\sum_{i=1}^n \frac{x^i}{\Gamma(\frac{i}{\sqrt{n}})}$? [on hold]

I'm interested in evaluating the following sum : $\displaystyle{\sum_{i=1}^n \frac{x^i}{\Gamma(\frac{i}{\sqrt{n}})}}$ where $x>0$. The existence of a closed form would be great but is perhaps too ...
2
votes
2answers
32 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 ...
-2
votes
0answers
11 views

Mittag Leffler Stability [closed]

How to prove "Mittag-Leffler stability implies asymptotical stability" ? What should be done? Thanks.
2
votes
0answers
54 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.
2
votes
0answers
26 views

2D Cauchy Distribution Peak [closed]

Is the general form of a 2D Cauchy Peak, if A is the amplitude: $$\frac{A}{1+\frac{(x-x_0)^2}{\gamma_x^2}+\frac{(y-y_0)^2}{\gamma_y^2}}$$ $?$
4
votes
0answers
26 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 ...
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)}{...
2
votes
0answers
29 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, ...
0
votes
3answers
37 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
48 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
31 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
52 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
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}\...
3
votes
1answer
264 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 ...
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 ...
0
votes
0answers
24 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
63 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} ...
1
vote
0answers
72 views

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

This question is a sequel to this previous question. As before, some background information is needed first as follows from my textbook: The standard form of Bessel's differential equation is $$x^...
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
3
votes
0answers
60 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 ...
2
votes
1answer
61 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
26 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
66 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 ...
0
votes
0answers
6 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- \...
0
votes
1answer
30 views

The function $\zeta(\frac{1}{2}+it) \left[ \sqrt{2}\left( \cos(t\log 2)+i\sin(t\log 2) \right) -2 \right]$ has a numerical root

Using the complex exponentiation (this is the MathWolrd's Page) one can deduce for $t>0$ $$2^{\frac{1}{2}+it}=\sqrt{2}e^0(\cos(t\log 2)+i\sin(t\log 2)),$$ since $a=2,b=0,c=\frac{1}{2}, d=t$ and $\...
2
votes
1answer
36 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
57 views

Integral of incomplete gamma function and limit of hypergeometric function

Let $a > 0$ and consider the integral $$\int_x^\infty \frac{\Gamma(a,t)}{t}\,dt$$ where $\Gamma(a,t)$ is the upper incomplete gamma function $$\Gamma(a,t) = \int_t^\infty x^{a-1} e^{-x} \, dx.$$ ...
13
votes
1answer
136 views

Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

The Dedekind eta function is denoted by $\eta(\tau)$, and is defined on the upper half-plane ($\Im \tau >0$). Put $\tau = i x$ where $x$ is a positive real number. The function has the following ...
0
votes
1answer
31 views

The Bessel function and finding expression

The Bessel Function $J_v$ of the first kind of order $v$ can be defined by the series expresion $$J_v(x)=\sum_{n=0} ^{\infty} \frac{(-1)^n}{n!\Gamma{(1+v+n)}}\left(\frac{x}{2}\right)^{2n+v}$$ (i) if ...
2
votes
1answer
53 views

What is the motivation behind the Bessel function of second kind

I am studying Bessel function and found the good reference by G.N. Watson At some point in page 58 he introduces the following expression due to Hankel: \begin{eqnarray} \lim_{\nu \to n} \frac{J_{\...
3
votes
0answers
67 views

Solving a functional equation $2 f(2x)=f(x)(1+\cos(x))+f(x+\pi)(1-\cos(x))$

I am trying to solve the following functional equation, which appears in some of my physics calculations : $f(x)=\frac{1}{2}\left(f(\frac{x}{2})(1+\cos(\frac{x}{2}))+f(\frac{x}{2}+\pi)(1-\cos(\frac{x}...
3
votes
1answer
140 views

What are some common ways to express in mathematical notation the indefinite integral of the $\Gamma$ function? [closed]

I checked on WolframAlpha (I am not good at calculating integrals) and it said there are no elementary functions to express it and gave me a horribly complex (I think... I didn't examine it very ...
0
votes
0answers
29 views

Incomplete Gamma Asymptotics

my question is simple : if $a_n$ and $z_n$ are both real positive sequences tending to $+\infty$, what is the asymptotic ($n \to +\infty$) behaviour of $\Gamma(a_n,z_n)$ when 1) $a_n \neq z_n$ and $\...
2
votes
1answer
56 views

How to prove that $e^{-\gamma}=\prod_{n=1}^\infty\left(1+\frac1n\right)e^{-1/n}$

Suppose we defined the Gamma function $$\frac1{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^\infty\left(1+\frac zn\right)e^{-z/n}$$ where $\gamma$ is just a constant. I want to prove that $\Gamma(1)=1$, so I ...
0
votes
2answers
43 views

Approximation of $\ln(x+1)$ with $\Psi$ function

I found the following approximation for the function $$f=\ln(x+1)$$ $$f\simeq\Psi\left(x+\dfrac{3}{2}\right)-2+\gamma+\ln(2)$$ where $\Psi(x)$ is the 'Digamma' function: $\Psi(x)=\dfrac{\dfrac{d}{dx}\...
1
vote
0answers
28 views

On series of the kind $\sum_{n=1}^\infty\frac{1}{n^2}\cdot\frac{1}{(1+nx)^s}$ and Frullani's theorem

I would like to know if my computations, in this post are not required justifications for the computations unless if there is a mistake in my claims, were rights and solve a question as if are known a ...
7
votes
2answers
165 views

Integral ${\large\int}_0^\infty\big(2J_0(2x)^2+2Y_0(2x)^2-J_0(x)^2-Y_0(x)^2\big)\,dx$

I'm interested in the following definite integral: $$\int_0^\infty\big(2J_0(2x)^2-J_0(x)^2+2Y_0(2x)^2-Y_0(x)^2\big)\,dx,\tag1$$ where $J_\nu$ and $Y_\nu$ are the Bessel functions of the first and the ...
4
votes
2answers
89 views

Solving $y^y = x$ for large $x$

I was playing around with recurrence relations and noticed that $\sqrt x$ has the fun property that $$\frac{x}{f(x)} = f(x)$$ ($\sqrt{x}$ and its negation are the only functions $f(x)$ that satisfy ...
6
votes
4answers
272 views

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

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^{\...
2
votes
1answer
38 views

Upper bound for ratio of modified Bessel functions of second kind

I was wondering if someone has an idea if for $0 < x < y$, and $0< \nu \leq \frac{1}{2}$, one can obtain an upper bound for the ratio $$ \frac{K_{\nu}(x)}{K_{\nu}(y)} $$ Thanks.
3
votes
0answers
40 views

Are these Infinite Series Representations of Special Functions?

I am not sure how to google the answer for this question. Anyway, in trying to compute the velocity of a charged particle in an electromagnetic field, I came across these two infinite series (...