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

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0answers
17 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 ...
0
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1answer
30 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 ...
3
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2answers
128 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 ...
2
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1answer
49 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 ...
3
votes
1answer
31 views

A formula for length of representation of a number in a “base” without zeros

If you had 2 items the sequence would go like this: $$1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5, \ldots$$ This is $\lfloor\log_2(n+2)\rfloor$. What if I ...
1
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1answer
37 views

Function with infinite maxima and minima [on hold]

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 ...
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0answers
6 views

Giving two examples of functions with some properties.

This is a question from a list. Obtain two $\mathcal{C}^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ satisfying these properties: $f(x)=0 \Leftrightarrow 0\leq x\leq 1$; $g(x)=x$ if $|x|\leq 1$, ...
0
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1answer
47 views

Book recommendation on special functions

I am currently studying real analysis from rudin and really like the chapter on special functions. But Rudin does not give much knowledge about those topics. Reading the references I found book by ...
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1answer
9 views

Legendre functions - Derivation of the recursion relation

From the following: $$\sum_0^\infty [ n(n-1)a_nx^{n-2} - n(n-1)a_n x^n -2na_nx^n + l(l+1)a_n x^n ] = 0$$ (a) I'm trying to get to: $$\sum_0^\infty [ (n+2)(n+1)a_{n+2} - [n(n+1) + l(l+1)]a_n]x^n = ...
1
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1answer
64 views

Compute $\int_0^{\infty} Q_1(y,b) \frac{y}{\sigma^2} \exp{(-y^2/(2\sigma^2))} \, dy$

We know that the first order Marcum Q-function can be represented as $$Q_1(y, b)=\int_{b}^{\infty} x \exp{(-(x^2+y^2)/2)} I_0(y x) \, dx ,$$ where $I_0(\cdot)$ is the modified Bessel function of the ...
0
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0answers
24 views

integration of product two lower incomplete gamma function and exponential [closed]

i need helping to find the value of this integration : $$ \int_0^{\infty}e^{-\delta x}\gamma(\alpha,\theta x){\gamma(\beta,\theta x)\ }dx $$ where all parameters are positive. Can anyone help me how ...
1
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1answer
24 views

About Beta function $B(\alpha,r\alpha +1)‎\rightarrow‎ 0$

I want to show that $$B(\alpha,r\alpha +1)‎\rightarrow‎ 0$$ when $r‎\rightarrow‎ \infty$ and $0< \alpha <1$. with thanks
2
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1answer
57 views

Integral with incomplete gamma function

I am trying to solve this integral: \begin{equation} \frac{1}{c^{b}}\int_{0}^{\infty} x^{n}\, e^{-a x}\, \gamma(b,c(-d+x)) \ \mathrm{d}x \end{equation} where, $n>0$ is an integer, and $a$, $b$, ...
-1
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0answers
22 views

integral with two lower incomplete gamma

Can I get a step by step answer to this integration ? $$ \int_0^{\infty}e^{-\delta x}\gamma(\alpha,\theta x){\gamma(\beta,\theta x)\ }dx $$
1
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1answer
36 views

integral include lower incomplete gamma square [closed]

I am trying to calculate the following integral: $$ \int_0^{\infty}e^{-\beta x}\{ \gamma(\alpha,\theta x)\}^{2} dx $$ where all parameters are positive. i need helping , thanks!
0
votes
0answers
7 views

Interpretation of diagonal detail in Haar Wavelet Transforms

I am a statistics grad student, and I have just begun exploring the topic of wavelet regression (specifically, Haar wavelets for discrete functions). I understand the generalization from a one ...
1
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1answer
37 views

Integrals involving whittaker functions.

I want to compute the following integrals: $$ \int y^{a} e^{\frac{1}{2}y}M_{k,m}(y)dy $$ where a is an arbitrary constant and $M_{k,m}$ is a whittaker function of the first kind. I already know that ...
3
votes
1answer
48 views

Computing the limit of an integral

Consider the following integral $$ \int_{-\infty}^{\infty}f(t) K(\frac{a-t}{h})dt $$ where (1) $h>0$, $a \in \mathbb{R}$ (2) $f:\mathbb{R}\rightarrow[0,\infty)$ is such that ...
0
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1answer
40 views

integral include lower incomplete gamma

I am trying to calculate the following integral: $$ \int_0^{\infty}e^{-\beta x}\gamma(\alpha,\theta x)dx $$ where all parameters are positive. Any help , Thanks!
0
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0answers
69 views

Limit of this monster function [closed]

Here's a function, and I am interested in finding its limit when $z\rightarrow \infty$. Any idea would be really appreciated. First thing I need to know is if this limit really exists. ...
4
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1answer
86 views

Calculate (or estimate) $S(x)=\sum_{k=1}^\infty \frac{\zeta(kx)}{k!}$.

Let $x\in\mathbb R$, $x>1$ and $$S(x)=\sum_{k=1}^\infty \frac{\zeta(kx)}{k!}$$ where $\zeta(x)$ is the Riemann zeta function. Calculate (or estimate) $S(x)$.
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1answer
27 views

How to prove this limit of Airy Function.

I have no idea how to prove this limit $$\lim_{x\rightarrow \infty }\exp\left ( \frac{2x^{3/2}}{3} \right )\sqrt[4]{x}\mathrm{Ai}\left ( x \right )=\frac{1}{2\sqrt{\pi }}$$ where ...
3
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0answers
112 views

Another beta integral due to Cauchy.

I have the following identity which I want to prove: $$C(x,y):= \int_{-\infty}^{\infty} \frac{dt}{(1+it)^x(1-it)^y} = \frac{\pi \cdot 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ where ...
0
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0answers
13 views

Behavior of $J/I$ w.r.t $m_1$, $I=\int_{m_1k}^{\infty} t^{N-k}e^{-t} dt$ and $J=(m_1k)^{N-k} e^{-m_1k}-\int_{m_1k}^{\infty} \log(t) t^{N-k}e^{-t} dt$

Let us define $I=\int_{m_1k}^{\infty} t^{N-k}e^{-t} dt$ and $J=(m_1k)^{N-k} e^{-m_1k}-\int_{m_1k}^{\infty} \log(t) t^{N-k}e^{-t} dt$. We assume that $m_1 \ge 0$, $k \ge 0$ and $k \le N$. Using the ...
1
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0answers
28 views

Integration involving Error function

I am interested in the following integral $$\int_0^\infty x^n\text{Erf}[ax]j_m(bx)dx,$$ where Erf is the error function $j_n$ is the spherical Bessel function of first kind. Does the analytical ...
1
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1answer
54 views

How to know if I can't solve an equation with “standard” methods?

I'm particularly fascinated by transcendental equations whose posses closed form solutions and when I pose some of them to my friends or teachers I heard a lot of "You can't solve this in closed form" ...
0
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0answers
25 views

What the inverse function of $_2F_1(a,b;c;z)$

What the inverse function of the function $f(z)$ given by $$ f(z) = \, _2F_1(a,b;c;z), \quad \mid z \mid <1, $$ where is the Gauss hypergeometric function given by $$ ...
3
votes
0answers
50 views

Same values for Gamma Function

I was thinking about the Gamma function, which for an integer positive argument is nothing but the factorial function. Using the integral representation, namely $$\Gamma[x] = \int_0^{+\infty}\ ...
1
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3answers
50 views

How could I solve $\int_{-\infty}^{+\infty} x^2e^{-x^2}dx$ apply special function gamma

I try solve the integral $$\int_{-\infty}^{+\infty} t^2e^{-t^2}dt$$ I do not know but I think that I should apply $gamma\ function$, which is $$ \Gamma (x)=\int_{0}^{\infty} t^{x-1}e^{-t}dt$$ Like ...
5
votes
2answers
147 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 ...
0
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2answers
86 views

Evaluating an integral involving $(a-x)^ne^{-1/x}/x^2$

Let $a>0$ be a small parameter and consider a fixed integer $n\geq 0$. Is it true that $$ \int_0^a \frac{(a-x)^n e^{-1/x}}{x^2}\ dx=n! a^{2n}e^{-1/a}(1+O(a)) $$ as $a\to 0$? I have verified this ...
2
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2answers
68 views

Integration of Associated Legendre Polynomial

I am interested in the following integral $$I=\int_{-1}^1P_\ell^2(x)P_n(x)\mathrm{d}x,$$ where $P_n(x)$ is Legendre Polynomial of $n$th order, and $P_\ell^2$ is Associated Legendre Polynomial. Any one ...
2
votes
1answer
42 views

Elliptical Integral that diverges at one point

I have to solve the following integral $$I=\int_{\lambda_1}^yd\lambda\frac{1}{1-\lambda}\sqrt{\frac{(\lambda-\lambda_1)(\lambda-\lambda_2)(\lambda-\lambda_4)}{\lambda-\lambda_3}}$$ where ...
0
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0answers
49 views

Prove that this expression involving $_2 F_1$ and Gamma functions is identically zero

While attempting an answer to the question Evaluation of $\displaystyle \int_{1}^{3}\left[\sqrt{1+(x-1)^3}+(x^2-1)^{\frac{1}{3}}\right]dx$, after a few manipulations I came across the following ...
2
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1answer
61 views

Calculating $I=\int_{-1}^1{\dfrac{1}{\sqrt{1-x}}P_n(x)} \, dx$ where $P_n$ is a Legendre Polynomial.

Calculating $$I=\int_{-1}^1{\dfrac{1}{\sqrt{1-x}}P_n(x)} \, dx$$ Where $P_n$ is a Legendre Polynomial. My progress: For any integral of the form: $$\int_{-1}^1{f(x)P_n(x)} \, dx$$ Usinng Rodrigues ...
1
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1answer
56 views

Meijer G-function limit for $z\rightarrow\infty$

I am trying to understand if the integral $$f(R)=\int_a^R\frac{K_1(r)dr}{r}$$ has a finite limit for $R\rightarrow\infty$. With Wolfram Mathematica I got the following primitive: $$\frac{1}{4} ...
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0answers
60 views

How To Prove The following equation?

The equation arised in the paper:Exact and asympototic representations of the sound field in a stratified ocean.That is the equation(3.12) for solving the problem $$\Delta ...
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0answers
97 views

How to verfy if the approximations of the complex error function have no poles?

I found an article published few days ago in arXiv:1601.01261 that shows a very simple Matlab code for computation of the complex error function (aka the Faddeeva function) defined as \begin{equation} ...
1
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1answer
29 views

gegenbauer polynomial

Usually, Gegenbuaer polynomial is denoted by $C^{(\lambda )}_{n}(x)$ with $\lambda >-1/2$. My question: is it possible to generalize Gegenbuaer polynomial for $Re(\lambda)>-1/2, \lambda \in ...
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0answers
22 views

Estimate lower incomplete gamma function $\left|\gamma(n,z)-\gamma(n,-z)\right|$.

From Wikipedia, the upper incomplete gamma function is defined as: $$\Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t ,\,\!$$ whereas the lower incomplete gamma function is defined as: ...
3
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0answers
44 views

Airy transform of gaussian on half-line: $\int_{0}^\infty dx\, e^{-x^2}\text{Ai}(y-x)$

Background. The Airy transform of $f$ is defined as $$\int_{-\infty}^\infty dx\, f(x)\,\text{Ai}(y-x)\;.$$ $\text{Ai}$ denotes Airy function, $$\text{Ai}(x)=\frac{1}{\pi}\int_{-\infty}^\infty ...
0
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1answer
79 views

What is the following expression equal to?

What is the following expression equal to? $$z^{\alpha } \left(\, _2F_1\left(1,-\alpha ;1-\alpha ;\frac{1}{z}\right)+\, _2F_1(1,\alpha ;\alpha +1;z)-1\right)$$ The derivative of it with respect to z ...
1
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0answers
44 views

What is the general theory of solving polynomial equations “beyond radicals”?

For example, using Bring radicals or elliptic functions to solve quintic equations. Wikipedia says that similar methods can be used for higher degree polynomials, but I'm struggling on finding ...
2
votes
1answer
114 views

Extract imaginary part of $\text{Li}_3\left(\frac{2}{3}-i \frac{2\sqrt{2}}{3}\right)$ in closed form

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. $\mathrm{Re}[\text{Li}_2(i)]=-\frac{\pi^2}{48}$ Is there a closed form (free of polylogs and ...
0
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0answers
14 views

Wigner 3j for Interchanged $m$

I am given with two wigner j coefficients $$\begin{pmatrix} \ell_1&\ell_2&\ell_3\\ m& 0& -m\end{pmatrix},$$ and $$\begin{pmatrix} \ell_1&\ell_2&\ell_3\\ -m& 0& ...
0
votes
0answers
18 views

Reference for Fractional calculus and Differential Operators

I`ve been struggling with Fractional calculus and differential operators while studying special functions, and got to the conclusion that I need some references for them. So I ask for as many ...
2
votes
0answers
106 views

Limit with the Appell F1 function

While attempting to solve this problem I ran into a nasty limit. Mathematica claims that the indefinite integral ...
13
votes
3answers
206 views

Conjecture $\sum_{m=1}^\infty\frac{y_{n+1,m}y_{n,k}}{[y_{n+1,m}-y_{n,k}]^3}\overset{?}=\frac{n+1}{8}$, where $y_{n,k}=(\text{BesselJZero[n,k]})^2$

While solving a quantum mechanics problem using perturbation theory I encountered the following sum $$ S_{0,1}=\sum_{m=1}^\infty\frac{y_{1,m}y_{0,1}}{[y_{1,m}-y_{0,1}]^3}, $$ where ...
0
votes
0answers
19 views

How to compute this integrale $\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2} } dx$

I would like to calculate the following integral $$I(a,y)=\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2}} dx, \quad a>0, y\in \mathbb R^3 .$$ Here's what I did: In ...
1
vote
0answers
28 views

Integration on complex spheres and Gamma function

I'm studying special functions, especially Jacobi functions, related to the rank one groups ($U(1, n; \mathbb{F})$ where $\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{H}$, the skew-field of quaternions), ...