Questions tagged [bessel-functions]

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Integral with 4 Bessel functions and an exponential

I would like to solve the following integral $$\int_0^\infty e^{-a k^2} J_{3/2}(b k) J_{3/2}(c k) J_{3/2}(f k) J_{1/2}(r k) k^{-3} dk,$$ where $a,b,c,f,r > 0$, and $J_\nu(x)$ is the Bessel ...
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Limit of Hankel function for large complex order, fixed real argument

Consider the Hankel function $H_\nu(z)$ where $\nu=re^{i\theta}$ (real $z>0$, $r>0$, $0\leq\theta<\pi$) as $r\rightarrow\infty$. I am aware that the Bessel function $J_\nu(z)$ has the ...
283 views

Almost periodicity of Bessel functions

We know that a periodic function (e.g. a trigonometric function) has the property $$f(x+n\Lambda)=f(x) \qquad n\in\mathbb Z$$ A Bessel function is not exactly periodic, because the value of the ...
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A Bessel-like integral

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...
289 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 ...
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Limit for series of Bessel functions evaluated at zeros

The following series arises in an electrostatics problem for a conducting cylinder: $$V=\sum_{n=1}^\infty\frac{J_0(k_n\rho)e^{-k_nz}}{k_nJ_1(k_n)^2}$$ where $J_i$ is the Bessel function of $i^{th}$ ...
382 views

Reciprocal expansion of modified Bessel function

I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are ...
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In (1) there is a property of spherical Bessel functions, which's derivation I can not find in the literature. ${\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=% 0}^{n}\... 1answer 137 views How to compute means μ1 and μ2 knowing sum of Skellam distributions f(k;μ1,μ2) and sum μ1+μ2, where k is from 2 to n? The probability mass function for the Skellam distribution for a count difference$ k=n_1-n_2 $from two Poisson-distributed variables with means$\mu_1$and$\mu_1$is given by: $$f(k;\mu_1,\mu_2)=... 0answers 139 views Computing the difficult integral \int_0^\infty J_0(x)^4\log(x)dx Computing numerically integrals of oscillating functions from 0 to \infty is a well-known and difficult problem. Here is an example for which I do not know a solution: I know how to compute (to ... 1answer 139 views How to get asymptotic expansion of the sum of modified Bessel function \sum_{n=1}^\infty K_0(s\, n) as s\to 0^+? I guess for the modified Bessel funcion K_0(z),$$\sum_{n=1}^\infty K_0(s\, n) \sim \frac{-2\log 2 - \log \pi + \gamma}{2} + \frac{\log s}{2} + \frac{\pi}{2\, s}, \quad s\to 0^+,$$if taking$$\... 1answer 161 views Alternate forms of the Bessel equation I have a question regarding an alternate form of the Bessel equation and how that alternate form translates to the modified Bessel equation and its solution. The modified form is from: http://... 1answer 238 views Closed form of integration of modified Bessel function composed with trigonometric function times a linear term Assuming one draws two points from a von Mises distribution on a circle, I am looking for the expected distance between two such points. Given the pdf of a centered von Mises distribution $$f_X(t \... 2answers 323 views Asymptotics for the first zero of the Bessel functions Let J_\nu be the standard Bessel function of the first kind and let x_\nu be its smallest zero. Is there a simple reference or result for the asymptotic expansion of x_\nu when \nu goes to +\... 3answers 421 views Approximation of half-integers modified Bessel function of the second kind I am trying to optimise the calculation of the probability distribution poisson-inverse-gaussian its calculation involves a half-integers modified Bessel function of the second kind. Here's a formula ... 3answers 235 views ODE with Bessel decay This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily. I would like to estimate the asymptotic behaviour of the ... 0answers 56 views Bessel decay for nonhomogeneous PDE I'm interested in the following nonhomogeneous PDE$$ (\Delta-k^{2})u=-g $$on the upper-half plane with smooth and integrable Dirichlet boundary condition, where g is a smooth positive function ... 0answers 95 views Is the ratio of two distinct zeros of a Bessel function of the first kind an irrational number? Let J_1 be the Bessel function of the first kind with parameter \alpha=1. Namely, J_1 satisfies the differential equation for y given by x^2y''+xy'+(x^2-1)y=0 and J_1(0)=0. Is it true ... 0answers 58 views Integral reducible to Bessel function I wonder whether one could reduce the following integral to some combination of special (e.g. Bessel) functions:$$\int_{m}^{\infty}d\epsilon \exp\left(-s\epsilon\right)\left( \epsilon^{2}-m^{2}\right)... 2answers 754 views Kuznetsov trace formula, orthogonality of Bessel functions Sorry if this is a vague question. I remember from my younger days that before proving his trace formula, Kuznetsov had a pretty result on orthogonality of Bessel functions. The formulas that I am ... 2answers 427 views An integral involving three Bessel functions I am looking for a closed form for the following integral $$I = \int_0^\infty \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx)$$ which can be thought of as a particular case of the more general integral ... 0answers 188 views Exponential decay of a convolution Let$z=(x,y) \in \mathbb{R}^N \times (0,+\infty)$, and let $$P_m(z)=y^{2s} |z|^{-\frac{N+2s}{2}} K_{\frac{N+2s}{2}}(m|z|),$$ where$N \geq 3$is an integer,$0<s<1$and$K_{\frac{N+2s}{2}}$... 2answers 141 views Literature about the integral of Bessel$\int_0^x I_{0,1}(u) e^{-a u}du$? Thanks to sound remarks here and here, and looking again at these equations, I noticed that my puzzle boils down to these 2 special functions: $$\int_0^x I_i(u) e^{-a u}du, \quad i=0,1$$ where$I_n(u)$... 1answer 212 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$-\...
Consider a function from the Bessel family, for concreteness say $f(x) := J_0(x)$, depicted in blue below (the question can be asked for any order of the first or second kind): I'm interested in the ...