Questions tagged [bessel-functions]

Questions related to Bessel functions.

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6 views

Leading order solution for integral of $e^{z\sin(\omega t)}$

I am trying to show that $\int e^{z\sin(\omega t)}dt = I_0(z)+O(\frac{1}{\omega})$ using a Bessel function expansion. Using the expansion $e^{z \sin(\omega t)}=I_0(z)-2\sum_{k=1}^\infty(-1)^k\bigg(...
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1answer
22 views

How to calculate the following Laplace transform: $\mathcal{L}[te^{-3t}J_0(2t)]$?

I'm trying to calculate the Laplace transform of this function. $$ \mathcal{L}[te^{-3t}J_0(2t)] $$ where $J_0(t)$ is the zeroth-order Bessel function. Solution Attempt The p-Bessel function is ...
2
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0answers
35 views

Showing Bessel integral function is solution of $x^2y''+xy'+(x^2-n^2)y=0$.

I'm trying to show that $J_n(x)=\frac{1}{\pi}\int_0^{\pi} \cos(n\theta - x\sin \theta) d\theta$ is solution for $x^2y''+xy'+(x^2-n^2)y=0$. I try the following: $$J'_n(x)=\frac{1}{\pi}\int_0^{\pi} \...
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1answer
20 views

How to calculate the following Laplace transform: $ \mathcal{L}[\frac{1-J_0(t)}{t}] $?

I'm trying to calculate the Laplace transform of this function. $$ \mathcal{L}[\frac{1-J_0(t)}{t}] $$ where $J_0(t)$ is the zeroth Bessel function. Solution Attempt The p-Bessel function is ...
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31 views

can you help me solve this problem [closed]

Use $J_n(x)=\sum\limits_{k=0}^{\infty}\frac{(-1)^r}{r!\Gamma (n+r+1) }(x/2)^{n+2r}$ to prove that: $\begin{align} \int_{0}^{\infty} J_{0}(bx) e^{-ax} \, dx &= \sum_{0}^{\infty}(-1)^{r} \frac{b^...
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34 views

Prove using Bessel function [closed]

use \begin{align} J_n(x)=\sum\limits_{r=0}^{\infty}\frac{(-1)^r}{r!\Gamma (n+r+1) }(x/2)^{n+2r} \end{align} to prove that: \begin{align} \frac{d}{dx} [x^{2}J_{n-1}(x)J_{n+1}(x)] &= 2x^{2}J_n(x)...
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45 views

Fourier transform of $\frac{t}{(a^2+t^2)^{3/2}}$.

Im trying to find the Fourier transform of the function: $$\frac{t}{\left(t^2+a^2\right)^{\frac{3}{2}}};$$ As well the Fourier transform of the function: $$\frac{1}{\left(t^2+a^2\right)^{\frac{3}{2}}}....
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1answer
23 views

Double integral of product of Bessel functions in Python

I would like to perform a numerical integral (in Python) of the form $$\displaystyle F_\nu(k) = \int_0^\infty dk' \ k' \int_0^\infty dr\ r \ f(k', r) \ J_\nu(k' r) \ J_\nu(k r),$$ where $J_\nu$ is ...
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1answer
55 views

Asymptotics of $\int xdx ~ f(x) J_\nu(x) J_\nu(\alpha x)$ type integral for $\nu \to \infty$

I am interested in obtaining the asymptotic expansion of integrals of the form $$ I_\nu(\alpha) = \int_0^\infty xdx ~ f(x) J_\nu(x) J_\nu(\alpha x),$$ for $\nu \to \infty$ and some fixed, real, $\...
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73 views

Asymptotics of the Bessel function $J_\nu(z)$ for $\nu \to \infty$ starting from complex contour representation.

Asymptotic behavior of the Bessel function $J_\nu$ as $\nu \to \infty$ is given by $$ J_\nu(z) = \frac 1 {\sqrt {2 \pi}} \left( \frac {e z} {2 \nu} \right)^\nu \left( \nu^{-1/2} - \frac {3 z^2 + 1} {...
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48 views

What is the closed-form of $\int_{0}^{\pi}\text{sin}(n\tau-x\text{sin}\tau)d\tau$

I know the Bessel function can be formatted as follows. $$J_n\left(x\right)=\frac{1}{\pi}\int_{0}^{\pi}\text{cos}\left(n\tau-x\text{sin}\tau\right)d\tau$$ So can we get a Bessel-similar expression for ...
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14 views

how can i integrate a modified BesselI with exponential

I am trying to integrate this by substituting the Bessel i as the summation , but I don't reach to the formula of gamma function to use it. $\int_0^{\infty } \frac{z e^{-\frac{z}{\left(1-c^2\right) \...
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2answers
35 views

Proof of the Bessel $ J_n $ using the generating function

Given that $\sum\limits_{n=-\infty}^{+\infty}t^nJ_n(x)=e^{x(t-t^{-1})/2}$, prove that $J_n(x)=\sum\limits_{k=0}^{+\infty}\frac{(-1)^k}{k!(n+k)! }(x/2)^{n+2k}$ by expanding $$e^{x(t-t^{-1})/2}=\sum_{...
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1answer
99 views

Sup norm of Fourier transform of $ \frac{\sin |x|}{|x|^\lambda} \mathbb 1_{\{2^k\le |x| <2^{k+1}\}}, \ 0<\lambda<n $

It seems to me that in a paper of Charles Fefferman (open access), it is claimed in the introduction that (3rd page of the PDF file, 'page 11', $\lambda\in(0,n)$) $$\sup_{\xi\in\mathbb R^n}\left| \...
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5 views

Bessel bridge and process - transition density and SDE

I am somehow confused about what is going on here on page 186-187: https://books.google.de/books?id=7vZ0DwAAQBAJ&lpg=PA111&ots=Q-CelGtKTX&dq=transition%20dimensional%20bessel%20bridge&...
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13 views

spherical Bessel - Legendre relation, incident wave

I was reading the partial wave expansion for incident and scattered wave. I cannot understand two things: 1. Why in this process it indicates that the relation indicated in the following picture is ...
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1answer
26 views

Evaluating Sum of Modified Bessel Functions

In deriving some approximate solutions, I've come across the following sum I need to evaluate $$S=\sum_{n=0}^\infty(-1)^nI_n(\alpha)e^{-n^2\beta},$$ where $I_n$ is the modified Bessel function of ...
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18 views

Relation between the Bessel function of the first kind and fractional derivatives

Introduction The Bessel function of the first kind is defined as follows: $${\displaystyle J_{\alpha }(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!\Gamma (n+\alpha +1)}}{\left({\frac {x}{2}}\right)}^...
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36 views

Expression of $\int_{\mathbb S^{n-1}}\, e^{-i\left<v,\omega\right>}\,d\sigma(\omega) = ?$

I would like to know what the following integral equals $$\int_{\mathbb S^{n-1}}\, e^{-i\left<v,\omega\right>}\,d\sigma(\omega) = ?,$$ where $S^{n-1}$ is the two-dimentional sphere of $\mathbb R^...
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1answer
21 views

$K_{\frac{3}{2}}(z)=?$ where $K_{\nu}$ is the modified Bessel function

The modified bessel function of the second kind is the function $K_n(z)$ which is one of the solutions to the modified Bessel differential equation. Using Wolframalpha, we get $$K_{\frac{3}{2}}(z)= \...
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54 views

Integral of a Hankel function of a sine

Does anyone know how to show that $$ \int_{0}^{2\pi}H_{0}^{(1)}\left(2sin\left(\theta/2\right)x\right)e^{in\theta}d\theta=2\pi J_{n}(x)H_{n}^{(1)}(x) $$ where $n$ is an integer and $x>0$? It arose ...
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1answer
31 views

Mellin transform and Bessel function

In this article, the following identities are stated without proof: $$\int_0^\infty x^{s - 1}K_0(4\pi x^{1/2}) dx = \frac{1}{2}(2\pi)^{-s}\Gamma(s)^2$$$$\int_0^\infty x^{s - 1}Y_0(4\pi x^{1/2}) dx = -\...
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1answer
26 views

Hankel integral modulated by cosine

This interesting integral arises in the calculation of wave reverberation in air ducts, and I believe it evaluates to: $$ \int_{-\infty}^\infty \frac{e^{ik\sqrt{x^2+w^2}}}{\sqrt{x^2+w^2}}\cos(\alpha x)...
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16 views

Modified Bessels Function of the Second Kind into Meijer G Function

I want to know how is the function $\int_{0}^{d} e^{-bx}\left(\sqrt{\frac{a-bx}{c}}\right)K_1\left(\sqrt{\frac{a-bx}{c}}\right) dx$ converted into Meijer G form ? I know the Meijer G form of ...
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27 views

Orthogonality of Bessel functions with different indices

We know that for Bessel functions with same index the orthogonality condition holds: $$ \int_0^{+\infty}J_\mu(k x)J_\mu(k' x) k' x dx=\delta(k-k') $$ Is it possible to obtain an analytic result for($\...
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1answer
139 views

Proving $\int_{0}^{\frac{\pi}{2}}\text{erf}(\sqrt{a}\cos(x))\text{erf}(\sqrt{a}\sin(x))\sin(2x)dx=\frac{e^{-a}-1+a}{a}$

Assume $a>0$, how can we show that: $$\int_{0}^{\frac{\pi}{2}}\text{erf}(\sqrt{a}\cos(x))\text{erf}(\sqrt{a}\sin(x))\sin(2x)dx=\frac{e^{-a}-1+a}{a}$$ $$\int_{0}^{\frac{\pi}{2}}\text{erf}\ ^2(\sqrt{...
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0answers
35 views

Bessel function giving incorrect result

I am trying to reproduce a paper and there is a term in it, $x_{nm}$, which is given as : $$ x_{nm} = \int_{0}^{\frac{1}{\sqrt{\pi}}} r dr \int_{0}^{2\pi} d\theta \ \psi_{nk_{1}l}^{*} \ r\ cos(\...
3
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1answer
48 views

Show $K_a(x)=\int^{\infty}_0\exp(-x\cosh(t))\cosh(at)dt$ converges

Let $K_a$ be the modified Bessel function of the second kind of order $a \geq 0$: $$K_a(x)=\int^{\infty}_0\exp(-x\cosh(t))\cosh(at)dt$$ $x\in(0,\infty)$ Fix $a>0$ and use the comparison test to ...
3
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1answer
99 views

Expressing Fourier Series Representation of $e^{j \beta \cos 2 \pi f_m t}$ in terms of Bessel's Function.

I want to represent the signal $e^{j \beta \cos 2 \pi f_m t}$ in terms of its Fourier Series and then represent the result in terms of Bessel's Function. I have computed $c_n$ and would like someone ...
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0answers
39 views

Proving continued fraction for $\frac{I_{\frac{p}{q}}\left(\frac{2}{q}\right)}{I_{1+\frac{p}{q}}\left(\frac{2}{q}\right)}$

I have a question about the following well-known continued fraction:$$p+q+\cfrac1{p+2q+\cfrac1{p+3q+\cfrac1{p+4q+\ddots}}}=\frac{I_{\frac pq}\left(\frac 2q\right)}{I_{1+\frac pq}\left(\frac 2q\right)}\...
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0answers
39 views

Difficult numerical Integral with Besselfunctions: transformation of variables?

For a physics problem that I'm trying to study I would like to exand an eigenproblem in the eigenfunctions of the laplacian over a unit disk with neuman boundry conditions. To do this I need to ...
1
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1answer
48 views

Definite Integral of modified bessel function of second kind

How do I integrate $$\int_{0}^{d} \sqrt{\frac{a-bx}{c}}K_1\left(\sqrt{\frac{a-bx}{c}}\right) dx$$ where $K_1$ represents modified Bessel function of second kind and $a,b,c,d$ are constants? Please ...
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17 views

Solving a Bessel-like equation

I'm trying to find the eigenvalues for a Schrödinger's equation in a potential that looks like $$ V(\ell) = \left(\alpha+\frac{1}{2}\right)^2 + \left(\left(\alpha e^{-\ell}-(\alpha+1)\right)^2 - (\...
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27 views

Integral involving Legendre polynomial and exponential function

I am wondering if there is a nice representation of the following integral $$\int_{-1}^1 P_n(x)\,\mathrm{e}^{\,\mathrm{j}\,\lambda\,|\bar{x}-x|}\,\mathrm{d}x$$ with $\bar{x}\in\mathbb{R}$, $\lambda\...
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0answers
38 views

An integral of a Hankel function

Does anyone know how to find an exact expression for the integral $$ \int_{0}^{2\pi}e^{in\theta}H_{0}^{(1)}\left(\alpha\sqrt{1-\beta cos(\theta)}\right)d\theta $$ where $\alpha>0$, $0<\beta<1$...
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1answer
15 views

Neumann functions of integer order - are they undefined by definition?

I have the following definition for Neumann functions (in terms of Bessel functions) $$ Y_n(z) = \frac{J_n(z) \cos(n\pi) - J_{-n}(z)}{\sin(n \pi)}. $$ Now my problem is concerned only with $n \in \...
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1answer
28 views

Solution of Modified Bessel D.E. with Boundary Value Condition

Find the particular solution from this Modified Bessel D.E \begin{align} x^2\frac{d^2u}{dx^2} + x\frac{du}{dx} - \alpha^2x^2u(x) = 0, \hspace{10pt}x\in[a,b] \end{align} with boundary condition $u(a)=1,...
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1answer
31 views

How to prove $-J_0(\pi \mu) = J_2(\pi \mu)$ where $\mu$ is a solution of $J_1(\pi x) = 0$?

I am doing phased array antenna research and am trying to understand a final step in the Taylor circular aperture distribution solution. The solution Taylor presents has removable singularities at: $...
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1answer
16 views

Approximate an equation with Bessel functions when one variable tends to 0

Consider: $$- \frac{\mu_1 \epsilon_1 J_{n - 1}(p) J_{n + 1}(p)}{p^2 J_n^2 (p)} + \frac{(\mu_1 \epsilon_2 + \mu_2 \epsilon_1) J'_n (p) K'_n (q)}{pq J_n(p) K_n(q)} + \frac{\mu_2 \epsilon_2 K_{n - 1}(q) ...
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0answers
19 views

What's the spectrum of $\cos(\sum_{n} A_n\cos(n*\Delta\omega t+n\phi)$?

We all know Jacobi–Anger expansion as follows: $$\cos(z\cdot \cos(\theta))=J_0(z)+2\sum_{n=1}^{\infty}(-1)^n J_{2n}(z) \cos(2n\theta)$$ $$\sin(z\cdot \cos(\theta))=-2\sum_{n=1}^{\infty}(-1)^n J_{2n-1}(...
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1answer
20 views

Simplifying factorials in the Bessel polynomials

Consider the generating formula for the Bessel polynomials (increasing coefficients): $$a_k=\frac{(N+k)!}{2^kk!(N-k)!}$$ I was trying to generate as high an order as possible with ...
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1answer
27 views

Generalization of periodicity

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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1answer
86 views

Show Bessel integral function is solution of $x^2y''+xy'+(x^2-n^2)y=0$.

Show Bessel integral function $$J_n(t)=\frac{1}{2\pi} \int_{0}^{2\pi} e^{i t \sin \theta} e^{-in \theta} d \theta$$ is a solution of $$x^2y''+xy'+(x^2-n^2)y=0.$$ The suggest is that $n^2J_n(x)=-\...
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0answers
32 views

Limiting forms for modified Bessel functions of the 2nd kind when $x \to 0$

Case 1 Consider the behaviour of the modified Bessel function of the second kind for $x \to 0$, when $x \in \mathbb{R}$ and $n = 0$. Abramowitz and Stegun (Handbook of Mathematical Functions, par. 9....
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34 views

A Bessel-like integration

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 ...
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1answer
41 views

Solving dual integral equation

I am trying to solve for $g(y)$ in the following set of dual integral equations: $$\int_0^\infty y \ g(y) \ J_0(xy) \ dy = 0\ \text{ for }\ 0<x<1$$ $$\int_0^\infty g(y)\ J_0(yx) \ dy = x^0 \ \...
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1answer
29 views

Prove the equation is a solution to the second order ODE (Bessel's Equation)

The given equation is $$x^2y^{''}+xy^{'}+(x^2-\frac{1}{4})y=0$$ and the solution we are meant to verify is $$y_1(x)=x^{-1/2}\cos(x)$$ Taking the first and second derivative of this solution yields ...
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1answer
36 views

Explicit value of a Bessel integral

Given the integral : $\int_{0}^{\infty}\frac{xK_{\mu}(x(a+b))}{K_{\mu}(xa)}J_0(cx)dx$ where $K_{\mu}(.)$ is the modified Bessel function of the second kind of order $\mu$ and $J_0(.)$ is the Bessel ...
2
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0answers
57 views

How can one solve this differential equation? (Optical Fibers with a tanh(x) index)

The differential equation for a optical fiber with a refractive index $n(r)$ is given as $$\nabla^{2}_{\perp}A(r,\theta)+(k^{2}n(r)^2-\beta^2)A(r,\theta)=0.$$ which is separable in cylindrical ...
5
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1answer
100 views

Sum of Bessel functions to the fourth power, $\sum_{k\in\mathbb{Z}} J_k(x)^4$

Let $J_k$ denote the $k$-th order Bessel function of the first kind. I know that $$\sum_{k\in\mathbb{Z}} J_{\mu-k}(x) J_{\nu-k}(y) = J_{\mu-\nu}(x-y) \quad \forall x,y\in\mathbb{R},\mu,\nu\in\mathbb{Z}...

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