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
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
votes
0answers
51 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 ...
14
votes
1answer
192 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
vote
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
49 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
vote
0answers
34 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
votes
0answers
47 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
42 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
vote
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 ...
1
vote
0answers
30 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 ...
3
votes
1answer
257 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 ...
7
votes
1answer
141 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 ...
1
vote
0answers
35 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
39 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 ...
0
votes
0answers
24 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
46 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). ...
3
votes
0answers
38 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
vote
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
votes
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
29 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
vote
0answers
24 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
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} ...
0
votes
1answer
38 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
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} + ...
1
vote
2answers
41 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
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 ...
1
vote
1answer
32 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
71 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
43 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 ...
0
votes
1answer
48 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
59 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
95 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
86 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
61 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 ...
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
52 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
14 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 ...
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
27 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 + ...
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 ...
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): $$E^{o}_{k}=\sum^{\infty}_{q=-\infty}J_{q-k}(2m)E^{o}_q,$$ ...
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 ...
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 ...
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 ...
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 ...
2
votes
1answer
35 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 ...