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

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0
votes
1answer
45 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
72 views

Integrating a product of complicated exponential function and error function

I have a problem with the following integral $$\int_0^\infty\dfrac{{\rm e}^{-t-(x^2-a^2)/t}}{t}Erf\left(\frac{a}{\sqrt{t}}\right)\,{\rm d}t, $$ where $0\le a<x$ and Erf stands for the error ...
4
votes
1answer
264 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 p}^{-1}\left(\...
1
vote
0answers
38 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 ...
7
votes
1answer
304 views

How to solve the general sextic equation with Kampé de Fériet functions?

It is frequently stated, for example on Wolfram Mathworld, that the general sextic equation $$x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0 = 0$$ can be solved in terms of Kampé de ...
5
votes
0answers
48 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 ...
6
votes
2answers
123 views

Integral of two logs and a power: $\int_0^1 u^c \log(1-au)\log(1-bu)\,\mathrm du$

Does the following integral have a closed form in terms of known functions? $$ f(a,b,c) = \int_0^1 u^c \log(1-au)\log(1-bu)\,\mathrm du.$$ The parameters are possibly complex, and satisfy $$\Re(c)>-...
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
51 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 ...
6
votes
1answer
158 views

The number $\sum\limits_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$ is transcendental

Prove that the number: $$\sum_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$$ is transcendental. I don't have a direct proof but a round one. The series can be expressed in terms of $\vartheta_3$ ...
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 $0$ ...
51
votes
6answers
5k views

Is there a function whose antiderivative can be found but whose derivative cannot?

Does a function, $f(x)$, exist such that $\int f(x) dx $ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if $f(x)=e^{x^2}$, then the derivative is easily ...
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 looking ...
1
vote
0answers
36 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 ...
0
votes
1answer
48 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 ...
2
votes
0answers
65 views

Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity $\eta(-\frac{...
7
votes
1answer
165 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 )\mathrm{d}x=\frac{\pi^3}{96}\ln{2}-\frac{3\pi\zeta{(3)}}{128}-\frac{\pi^2G}{16}+\frac{\beta{(4)}...
2
votes
1answer
43 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 $$H_n(z)=\frac{2^n}{\sqrt{\pi}}\int_{-\infty}^...
0
votes
1answer
57 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). $K_{\nu}(z)=\frac{z^{\nu}}{2^{\nu+1}}\int_{0}^{\infty}t^{-\nu-1}e^{-t-z^{...
0
votes
0answers
32 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 ...
3
votes
0answers
40 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?
7
votes
4answers
272 views

Solving $\ln{x}=\tan{x}$ with infinitely many solutions

Lets take $f(x)=\ln{x}$ and $g(x)=\tan{x}$ When $f(x)=g(x)$ that is $\ln{x}=\tan{x}$, we see that the graph is like: Hence we see that there are infinitely many solutions to $x$ but the two ...
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 ...
10
votes
0answers
293 views

Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$

Prove that: $$ I=\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx=\frac{7}{2}\zeta(3){\log^22}-\frac{\pi^2}{6}{\log^32}-\frac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\frac{\pi^4}{48}\ln2 $$ Using ...
2
votes
1answer
34 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
26 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 ...
3
votes
1answer
101 views

Approximating hypergeometric function F(1,1+a,2+a,z) for z->1

in my studies a normalization constant for a pmf includes the hypergeometric function ${}_2F_1(1,1+a,2+a,z)$ The parameters are in the range $0.99<z<1$ and $0<a<5$. I have tried some ...
2
votes
2answers
662 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 w^2}}\int_{-\infty}...
0
votes
0answers
29 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} \sum_{k=0}^{\infty}(2k-a)g_{0,k}...
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 1 + 1/2 + 1/3 + 1/4 + ...,...
0
votes
0answers
16 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} + [c+n-(a+b+1)]\frac{...
0
votes
0answers
26 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
198 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)}\ \sqrt{\pi}=\frac{\Gamma(2n)}{2\Gamma(n+1/2)}\sqrt{...
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
3answers
152 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
57 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 $\mathbb{...
4
votes
1answer
65 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
116 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 $[0 ; +\infty[$ $f(0) = 0$ $\lim\limits_{x \to +\infty} f(x) = 0$...
1
vote
4answers
64 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, $R=\sqrt{r^2+\rho^...
0
votes
1answer
57 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
559 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
25 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 complicated....
0
votes
1answer
62 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
20 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 ...