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

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2
votes
2answers
651 views

Definite integral involving modified bessel function of the first kind

I would like to solve the following integral that is a variation of this one (Integral involving Modified Bessel Function of the First Kind). Namely, I have: $$\frac{1}{\sqrt{2\pi ...
0
votes
0answers
27 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} ...
1
vote
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 ...
0
votes
0answers
14 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} + ...
0
votes
0answers
24 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 ...
7
votes
1answer
193 views

Closed form double integral $ \int_{a}^{c}dr \int_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{r_<^{\ell}}{r_>^{\ell+1}}$

Is there a closed form expression for $$ S_\ell = \int\limits_{a}^{c}dr \int\limits_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{[\min( r , r')]^{\ell}}{[\max(r,r')]^{\ell+1}} ...
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)}\ ...
1
vote
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.
1
vote
0answers
55 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 ...
1
vote
3answers
144 views

Hypergeometric function integral representation

How to prove the following relation? $$ \, _2{F}_1(K,K;K+1;1-m) = \frac{\Gamma (K+1)}{\Gamma (K)} \int_0^{\infty } \frac{1}{(1+x) (m+x)^K} \, dx $$ where $_2{F}_1(.,.;.;.)$ is the hypergeometric ...
0
votes
1answer
53 views

Show some properties of the Digamma Function

Let $\psi(z)$ denote the Digamma function, $\psi(z)=\frac{d}{dz}\ln \Gamma(z)=\frac{\Gamma'(z)}{\Gamma(z)}$. I am meant to show the following properties of $\psi$: $\psi$ is meromorphic in ...
4
votes
1answer
63 views

Integral representation of Bessel function $K_v(y) = \frac{1}{2} \int_{0}^{\infty} t^{v-1} \text{exp}(-\frac{1}{2}y(t+t^{-1}))\text{d}t$.

How does one find the following representation of the bessel function $K_v(y)$: $$K_v(y) = \frac{1}{2} \int_{0}^{\infty} t^{v-1} \exp \left(-\frac{1}{2}y\left(t+t^{-1}\right) \right)\,\mathrm{d}t.$$ I ...
5
votes
1answer
104 views

An elliptic integral?

I ran into an integral a little while ago that looks like an elliptic integral of the first kind, however I am having trouble seeing how it can be put into the standard form. I've tried messing ...
3
votes
5answers
87 views

What are some functions that respect the following criteria? : $f(1/x) = f(x)$ and $\int_{0}^{+\infty} f(x) dx = 1$

I'm looking for some functions that respect these six criteria: $f$ is defined on $[0 ; +\infty[$ $f$ is differentiable everywhere in $[0 ; +\infty[$ $f(0) = 0$ $\lim\limits_{x \to +\infty} f(x) = ...
1
vote
4answers
62 views

integrating this infinite gaussian integral

How does one integrate $\int_{-\infty}^{+\infty}x e^{-\lambda ( x-a )^2 }dx $ where $\lambda$ is a positive constant. My integral tables are not returning anything useable. The best it return is ...
3
votes
0answers
48 views

Expansion of some singular kernel with the help of Bessel and Neumann spherical harmonic functions

With the following notations: $j_n$: spherical Bessel functions, $y_n$: spherical Neumann function, $P_n$: Legendre polynomial, $r$, $\rho$, $\theta$, $\lambda$ arbitrary complex, ...
0
votes
1answer
53 views

Perturbation of the Upper Incomplete Gamma Function

The Upper Incomplete Gamma function, for $t \in \mathbb{R}$, is defined as: \begin{equation} \Gamma(α,β)=\int_{β}^{\infty}t^{α-1}e^{-t}dt \end{equation} For the problem which I am studying it takes ...
8
votes
2answers
539 views

Approximation of $\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$ [duplicate]

I am reading about the Riemann hypothesis, and the article mentioned the Li function: $$\mathrm{Li}(x) = \int\limits_{0}^x \frac{dt}{\ln t}$$ They said that this function can be approximated: ...
2
votes
1answer
24 views

Normalisation of Bessel functions

I've done the integration by parts and obtained $$ \frac{-1}{\alpha^2} \int z^2 J J'$$ but I have no idea how to use Bessel's equation to simplify this as it only appears to get far more ...
0
votes
1answer
54 views

Solving differential equation $y''(x)+Q(x)y(x)=0$ [closed]

How to solve the following differential equation $$y''(x)+Q(x)y(x)=0$$ And how to find exact solution $y(x)$ in terms of special functions?
0
votes
0answers
17 views

Division of half-integer order legendre functions of the second kind with different arguments

I'm in search of a formula for: $\frac{Q_{n-\frac{1}{2}}(\chi_1)}{Q_{n-\frac{1}{2}}(\chi_2)}= ??$ where I am hoping the result to be a function of $\frac{\chi_1}{\chi_2}$. Does anyone know of such ...
1
vote
1answer
167 views

May I know how this integral was evaluated using hypergeometric function?

I can not solve the following integral using the hypergeometric function: $$\int_a^b (\sin x)^{(1/n)}dx$$ Wolframalpha showed the following result. but I do not understand how Wolframalpha came ...
2
votes
0answers
46 views

Asymptotic behavior of zeros of a function

Let $f(x,m)=(2m-1)\Gamma(m)\,x^{-m}$ where $x>0$ and $\Gamma(z)$ denotes the Gamma function. Let $g(x,m)=f(x,m)+f(x,-m)$. I'm interested in the solution $m=m(x)>0$ of the equation $g(x,m)=0$ ...
0
votes
1answer
29 views

Shift of dirac delta function involving a sphere

Alright, I'm clueless on how to kickstart this question. The idea of the dirac delta function by itself is understandable, at least at my current level. But once the question starts throwing in ...
0
votes
2answers
41 views

Prove $\frac{-4 \sqrt x + 2 e^x \sqrt x + \sqrt \pi \operatorname{erfi}\sqrt x}{2 \sqrt x}\leq \frac{e^{3x}-3x-1}{3x}$.

From Wikipedia, the imaginary error function, denoted erfi, is defined as $$\operatorname{erfi}(x) = \frac{2}{\sqrt\pi} \int_0^xe^{t^2}\,\mathrm dt.$$ Prove that $$\frac{-4 \sqrt x + 2 e^x \sqrt x + ...
1
vote
1answer
26 views

Recursive function including Bessel functions

I was wondering if anybody knows how to solve (numerically) the following recursive equation (found in http://dx.doi.org/10.1109/3.250392): $$E^{o}_{k}=\sum^{\infty}_{q=-\infty}J_{q-k}(2m)E^{o}_q,$$ ...
0
votes
1answer
26 views

Evaluating a difficult 3-dimension dirac delta

Currently doing problem 1.48 of "Introduction to electrodynamics by David Griffith" I've read the examples, the theory and understood but come the exercise the author has a terrible habit of dishing ...
11
votes
1answer
169 views

Reducing multi-variable functions to a composition of 1- or 2-variable functions

There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta ...
1
vote
0answers
23 views

Refences to Sturm–Liouville theory with a singular weight function.

For $\alpha,\beta$, nonpositive integers at least one of which is non-zero, define $\omega\colon (-1,1)\to \mathbb{R}$ as $\omega(x) = (1-x)^\alpha(1+x)^\beta$. Then $\omega$ blows at at $x=-1$ or ...
1
vote
1answer
39 views

Can you justify the existence of a $x_{*}$ solving $\mbox{li}(x_{*})=\mbox{erf}(x_{*})?$

Can you justify the existence of a $x_{*}$ solving $$\DeclareMathOperator{\li}{li}\DeclareMathOperator{\erf}{erf}\li(x_{*})=\erf(x_{*})?$$ Here $\li(x)$ is a special function, the so called ...
3
votes
5answers
115 views

When actually $f(g(x))=g(f(x))$ holds?

We can see that if $f(x)=g(x)=x$ then $f(g(x))=g(f(x))$. I would like to see other examples of functions $f(x)$ and $g(x)$ such that $f(g(x))=g(f(x))$. P.S. By definition we also must have ...
0
votes
1answer
30 views

Simple form of LegendreQ function

for any n is positive integer LegendreP function can be expressed as $\displaystyle P_n(x)=\frac{1}{2^n n!}\frac{d^n}{dx^n}\left[(x^2-1)^n\right]$. Let $\displaystyle ...
2
votes
1answer
36 views

How to prove this relation?

Is the relation $$\lim_{x\rightarrow 1}\frac{Q_n^m(x)}{P_n^m(x)}=\frac{\pi}{2}\cot m\pi$$ correct? Here P and Q are the associated Legendre polynomials of the first and second kind respectively. Does ...
1
vote
0answers
22 views

Prove that the special Hermite functions are eigenfunctions of $R_{x,y}$?

How to prove that the special Hermite functions are eigenfunctions of the rotation operators $$R_{x,y} = x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x}.$$ Where the special Hermite ...
0
votes
1answer
40 views

Convergence of series

Is this series convergent? $$S_{N}=\sum_{n=0}^{N-1}\frac{c_{n}^{N}}{c_{N}^{N}}$$ where $c_{n}^{N}$ is coefficient of $x^{n}$ in chebyshev polynomial $T_{N}(x)$, i.e. ...
3
votes
3answers
169 views

Copulas, implication

Let $C$ be a copula function. Prove that $C(t,1-t)=0$ for all $t\in[0,1]$ implies that $C(u,v)=\max(u+v-1,0)$. I think the implication other way around is easy to see, however I can't see why the ...
6
votes
2answers
221 views

Closed form for a zeta series :$\sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{(k+2)2^{k+2}}$

It is not that diffcult to derive \begin{align} \sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{k2^k}=&-\frac{\gamma}{2}+\ln\left(\frac{2}{\sqrt{\pi}}\right)\tag{1}\\ ...
27
votes
1answer
325 views

What is a closed form of $\int_0^1\ln(-\ln x)\ \text{li}\ x\ dx$

Let $\operatorname{li} x$ denote the logarithmic integral: $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Is it possible to find a closed form of the following integral? $$\int_0^1\ln(-\ln x) ...
0
votes
1answer
21 views

Numerical integration of $E_1(x)$

I want to solve the following integral for $\gamma_0$: $$\int_{\gamma_0}^\infty \frac{1}{t}e^{-at} dt = c$$ for the specific values $a = 0.01$ and $c = 12.1$. As I understand, this is a variant of ...
0
votes
0answers
15 views

inverting a complicated function.

Is it ever possible to rewrite a function, such as $$ x - A\sqrt{y(x)} + B\tanh\left(\sqrt{y(x)}\right) +C =0 $$ in terms of $y(x)$. By invert, I mean, optimistically, express using something like ...
2
votes
1answer
32 views

How can I show that this Jacobi polynomial can be expressed as the sum of these two Legendre polynomials?

Let $n\in \mathbb{N}^+$ be a positive integer. Let $L_n\colon \mathbb{R}\to \mathbb{R}$ be the $n$'th order Legendre polynomial. Let $J_n^{(\alpha,\beta)}\colon \mathbb{R} \to \mathbb{R}$ be the ...
0
votes
1answer
46 views

The derivatives of Riemann xi function

What are the first few values of derivatives of Riemann xi function at zero? Is there any general formula for calculating the nth derivative of the riemann zeta function at zero? What happens to the ...
2
votes
2answers
19 views

Identifying a function

I am reading a piece of a physic paper where a function is mentioned without being given a name or reference - I guess it is a canonical one and that I should be familiar with. The expression goes ...
2
votes
1answer
90 views

Closed form of an integral $\int_0^{\pi/2} \ln^n (\sin x) \, dx$

Let $n \in \mathbb{N}$. May we have a closed form for the integral: $$\mathcal{J}=\int_0^{\pi/2} \ln^n (\sin x) \, {\rm d}x$$ One obvious approach would be to go through beta functions and ...
11
votes
1answer
253 views

Closed-form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$

From the amazing result by Raabe we know that $$LG_1=\int_0^1 \ln \Gamma(x)\,dx = \frac{1}{2}\ln(2\pi) = -\zeta'(0).$$ We also know that $$LG_2 = \int_0^1 \left(\ln \Gamma(x)\right)^2\,dx = ...
0
votes
0answers
33 views

Show that $J_n(x)$ satisfies Bessel equation $ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $

Here is the definition of the Bessel function I am starting with a definition as an integral. $$ J_n(x) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{i n t - x \sin t} \, dt $$ Essentially we have computed ...
27
votes
1answer
888 views

elliptic functions on the 17 wallpaper groups

In doubly periodic functions as tessellations (other than parallelograms), we learned about the Dixonian elliptic functions. There are 17 wallpaper groups -- are there elliptic function analogues for ...
8
votes
2answers
228 views

Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$ This result seems to me digitally correct? Can we prove ...
1
vote
1answer
56 views

Function with infinite maxima and minima [closed]

Can you please give an example of a function with an infinite number of maxima and minima occurring in any finite time interval? Edit: This question came to me as I was reading on the dirichlet ...