Questions tagged [bessel-functions]

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21 votes
3 answers
729 views

Proof of an identity involving $\int \exp(-|x-s|)dx$ over an even sphere

I want to prove the following identity calculating the integral of an exponential over an even dimensional sphere in terms of functions $\chi_i(R)$ and $\tilde\psi_i(s)$ (described below) which are ...
Simon Willerton's user avatar
13 votes
2 answers
2k views

$\mathrm{Bessel}^3$ Integral

I'm trying to calculate the following integral: $\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$ ($\mathrm{BesselJ}[n,x]$ is ...
Mikhael's user avatar
  • 133
5 votes
1 answer
507 views

Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$

EDIT: Some additional details and corrections, I would appreciate any information about the highlighted expression. I try to solve $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$ where $I_n(x)$ is the ...
Alexandre's user avatar
  • 368
33 votes
1 answer
1k views

$\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?

QUESTION Numerical calculation with gp (first to the default 38-digit precision, then tripled) supports the conjecture that $$ \int_0^\infty x \, [J_0(x)]^5 \, dx = \frac{\Gamma(1/15) \, \Gamma(2/15) ...
Noam D. Elkies's user avatar
7 votes
3 answers
436 views

A hypergeometric identity related to Bessel functions

The identity in my recent answer can be stated in a particularly neat form: $${}_2F_0\left({-n, n+1\atop{}};\frac{x}{2}\right) ~\cdot~ {}_2F_0\left({-n, n+1\atop{}};-\frac{x}{2}\right) ~=~ {}_3F_0\...
Max Alekseyev's user avatar
5 votes
0 answers
517 views

Why are Bessel function and Kloosterman sum similar?

It is a convention to say Kloosterman sums and Bessel functions are similar. There are papers talking about Bessel functions on $p$-adic group (associated with a representation) such as Baruch's: ...
7-adic's user avatar
  • 3,764
5 votes
1 answer
950 views

Integral involving Laguerre, Gaussian and modified Bessel function

I am trying to prove that the integral \begin{align} \int_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n} L_n^{l-n}\left(\frac{r^2}{C}\right) I_{l-n}\left(\rho r \right) r dr \end{align} has ...
Katie's user avatar
  • 53
3 votes
2 answers
217 views

Question about the Bessel operator

For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by \begin{equation*} L_\...
Tony419's user avatar
  • 401
3 votes
1 answer
384 views

What can we know about "the half" of the generating series of Bessel function

I am interested in the series $$\sum_{n\geq 1}I_n(x)\lambda^n$$ which is not the full generating series of the modified Bessel function of the first kind because it starts from $n=1$ and not at $-\...
Alexandre's user avatar
  • 368
3 votes
1 answer
355 views

A Bessel function integral identity involving $\int_0^\pi \frac{K_{j-1/2}(w)}{w^{j-1/2}}\sin^{2p-1}(\theta)\, d\theta$

Suppose that $w=\sqrt{R^2 + s^2 -2Rs\cos\theta}$ with $R\ge s>0$, that $p$ is a positive integer and that $j$ is an integer with $0\le j\le p$. Let $I$ and $K$ denote the modified Bessel functions ...
Simon Willerton's user avatar
2 votes
0 answers
142 views

Integral of product of zeroth-order bessel functions times cosine $\int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x$

I am new to Bessel functions and need to solve the following integral \begin{equation} \int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x \end{equation} with $J_{0}$ ...
Dennis Marx's user avatar
2 votes
1 answer
698 views

Integral involving associated Laguerre polynomial and Bessel function

In a quantum mechanics problem I encountered the following integral \begin{equation*} \int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,, \end{equation*} where $L$ denotes the ...
Tom26's user avatar
  • 23