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Questions tagged [bessel-functions]

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2
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1answer
92 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 $+\...
2
votes
2answers
93 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 ...
3
votes
3answers
200 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 ...
2
votes
0answers
39 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 ...
1
vote
0answers
71 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 ...
1
vote
0answers
48 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)...
9
votes
2answers
642 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 ...
5
votes
2answers
229 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 ...
1
vote
0answers
81 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 $K_{\frac{N+2s}{2}}$ ...
0
votes
2answers
123 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 $I_n(u)$...
2
votes
1answer
129 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 $-\...
7
votes
2answers
195 views

The tangent curve to Bessel functions?

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 ...
4
votes
1answer
277 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 ...
2
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0answers
139 views

Series representation of multiplication of two modified Bessel function

Series representation of multiplication of two Bessel function $J_{\mu}(az) J_{\nu}(bz)$ is in terms of sum of hypergeometric functions $_2F_1$, it given in book Treatise on Theory of Bessel Functions ...
4
votes
1answer
216 views

Infinite summation formula of Bessel functions

I would like to find a closed form for the following series involving the Bessel function $J_k(z)$: $$ \sum_{k=0}^{+\infty}\frac{(\mu)_{k}}{k!(\lambda)_{k}}t^k\left(\frac{z}{2}\right)^{k}J_{k+\nu}(z),...
2
votes
2answers
315 views

Integral with Bessel function and hypergeometric function ${}_2F_2$: explicit expression for these polynomials?

This question follows this one, where the general problem has apparently no simpler form than the integral one. I focus now on the limit case: \begin{align} \int_0^T e^{-x}\frac{nI_n(x)}{x}dx=\int_0^T ...
5
votes
1answer
308 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 ...
4
votes
0answers
281 views

On modified Bessel solutions to complex ODE's using Kummer's series

I am trying to reduce the following ODE to Bessel's ODE form and hence solve it: $$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$ I tried to solve it via the standard method, i.e.,...
2
votes
1answer
276 views

Indefinite summation of multiplication of two Bessel functions

Could anyone give an insight on how to prove the following formula? $$\sum_{n=-\infty}^{+\infty}J_{n}(\alpha)J_{N+n}(\alpha)=\delta_{N0} \, ,$$ where $N$ is an integer. I checked many references ...
0
votes
1answer
163 views

The limitation of derivation of modified Bessel function of second kind

The final result I draw is related to the integral of modified Bessel function of the second kind. But I can not solve it, and I need a explicit solution Are you willing to help me? Thank all $I = \...
2
votes
2answers
204 views

Asymptotics of Bessel functions

With $J_n$ standing for the Bessel function of first kind, $n\in \mathbb N$, I define $$ f_n (\rho) =\int_0^π J_n(\rho \sin \theta) \sin \theta \ d\theta.$$ Assuming $1\ll\rho\ll n$, I would like to ...
3
votes
0answers
59 views

Convergence of Bessel (Sturm-Liouville) Expansions at the End Points

I have asked this question before on MSE but received no answer at all. So I assume that it is proper to ask it here. I am not a mathematician so my language may not be too precise, please correct me ...
2
votes
1answer
224 views

Orthogonality with Bessel functions of rational order

I tried to ask this question on MSE (link), but got no comments or answers. So, I hope someone on MO would advise. Given a set of functions $f_{mv}(r,\phi)=J_{v}(k_{mv}r)\cos(v \phi)$ in polar ...
5
votes
0answers
245 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: ...
1
vote
0answers
86 views

Kontorovich Lebedev transform

By the title I mean [reference: ``Spectral methods of Automorphic forms" by Iwaniec (B.41)-(B.43)] for $f\in C^\infty_c(\mathbb{R^+})$, one has $$f(x)=\pi^{-2}\int_{-\infty}^\infty K_{it}(x)F_f(t)t\...
1
vote
0answers
47 views

Generalising a one-dimensional integral identity involving Bessel functions to higher dimensions

This was originally posted on Math Stack Exchange, but no responses were received. I recently came across the following remarkable identity, due to Hardy: $$\displaystyle \int_{-\infty}^{\infty} \...
1
vote
1answer
214 views

Integral of exp(-2cosh(x))

Is there some way to express: $$I(t) = \int_{-\infty}^{t} e^{-2\mathrm{cosh}(x)}~\mathrm{d}x$$ From Bessel functions? By substituting $y = \mathrm{cosh}(x)$ we get $$I(t) = \int_{1}^{\mathrm{cosh}...
2
votes
0answers
404 views

Integrating a product of integrals involving Bessel functions

I have asked similar questions on Math Stack Exchange, but not been able to receive many helpful responses. Therefore, I am posting this problem here, and any input would be extremely valuable. I ...
0
votes
0answers
114 views

Asymptotic of fourier transform of a oscillatory kernel

I am reading Harmonic analysis written by Stein. In page 426, there is a result about the asymptotic of the fourier transform of $\frac{e^{2\pi i |\xi|}}{|\xi|^\alpha}$ at the unit sphere. Suppose $\...
1
vote
1answer
243 views

integral representation of second solution of Bessel differential equation

Let $n$ be an integer and consider the Bessel function of order $n$ $J_n(z)=\frac{1}{2\pi i} \int_{|u|=1} e^{\frac{z}{2}(u-\frac{1}{u})}\frac{du}{u^{n+1}}$ This satisfies the linear differential ...
1
vote
0answers
162 views

Estimating an integral involving Bessel functions

I would like to preface this question by saying that I have asked a series of questions on this topic on Math Stack Exchange, but have almost never received any fruitful responses, with the exception ...
2
votes
1answer
232 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 ...
-1
votes
3answers
104 views

Is there any integer $n$ such that the bessel function J_n(1)=0?

Is there any integer $n$ such that $t=1$ is a root of the Bessel function of the first kind $J_n (t)$, i.e. $$J_n(1)=\int_{0}^{\pi}\cos (nx-\sin x)=0\,?$$
20
votes
3answers
474 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 ...
3
votes
1answer
244 views

Decay relationship with modified Bessel functions of the second kind

I think that the following inequality holds for all $x > 0$ and all $\nu \ge \frac{1}{2}$: $$ K_\nu(2 x) \le \frac{2^{2 - 2 \nu}}{\Gamma(\nu)} x^\nu K_\nu^2(x) ,$$ where $K$ is a modified Bessel ...
4
votes
1answer
127 views

Monotonicity of integral of Bessel functions

Is it known, and if yes how does one show, that the function $$ \psi(n):=n\int_0^{+\infty} e^{-x}I_0\left(\frac{x}{n}\right)^{n-1}I_1\left(\frac{x}{n}\right)\mathrm{d}x$$ is decreasing for all $n\ge ...
3
votes
0answers
138 views

Green's Function for a Kernel with Symmetric Fourier Transform $\nabla^2-x^2$

I am trying to find the inverse of the following kernel in 3 dimensions $$ \nabla^2-x^2, $$ where, $$ x^2=\vec{x}.\vec{x} $$ It seems quit simple and one would think there should already be solutions ...
3
votes
0answers
166 views

Factoring Bessel functions into an amplitude and a phase

Take some $\nu>0$. Let $J_\nu(x)$ be the Bessel function of the first kind. Let's restrict its domain to $\mathbb R^+$. Is it possible to find a pair of functions $A_\nu(x), \phi_\nu(x):\mathbb R^+\...
1
vote
1answer
344 views

Integration of Bessel Function of the first kind

I have encountered a problem during my thesis study. I need to find $F(x)$ in terms of $G(y)$. My statement as follows: $$\int_{0}^\infty F(x)[Bx^3J_0(xy)+x^4J_1(xy)]dx=G(y)$$ where $B$ is a constant, ...
1
vote
1answer
97 views

Definite intergal with two K-Bessel functions and x

I would like to calculate the definite integral with K-Bessel funcitons and a and b complex (n and k integers): $$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(kx) \; dx$$ I could not find it in ...
1
vote
0answers
216 views

Integration involving modified bessel function, exponential and power

I need to find the following integration. $$ \int_0^a e^{-(N-1)x}(\sqrt{4x}K_{1}(\sqrt{4x}))^N $$ where $$ a>0, \quad N \geq 1 $$ Any help will be much appreciated. BR Frank
0
votes
0answers
317 views

Integral involving modified bessel function of second kind, exponential and power

I need to compute the following integral. $$ \int_0^a e^{-bx}\sqrt{4(a-x)}K_1(\sqrt{4(a-x))}dx\,. $$ where $$ a>0$$ and $b$ can be greater than zero or less than zero but it is not a complex ...
3
votes
1answer
556 views

Definite integral with modified Bessel functions, trigonometric function and a power

I require the following integral involving the modified Bessel functions of the first and second kinds of order one $$I(a, b, c) = \int_0^{\infty} \frac{\sin(ax)}{x} I_1(bx) K_1(cx) \mathrm{d}x, \...
4
votes
0answers
224 views

Bounds for Bessel functions

Let $0 < \delta < 1$, and let $I_\delta$ be the set of all complex numbers $\mu$ such that $-1/2 + \delta < \Re \mu < 1/2 -\delta$. Is there a polynomial $P_\delta$ such that for all $\mu \...
4
votes
0answers
110 views

Asymptotic expansion of Mellin transform of products of modified Bessel function K

Let $n\ge 1$ be an integer, let $$F(x,y)=\int_0^\infty u^{n(x+y)} (K_{x-y}(u))^n du$$ for $x,y\ge 0$. When $n=1$, this is just Mellin transform of the Bessel K function. When $n=2$, $F(x,y)$ has ...
3
votes
1answer
261 views

Summation of an integral involving Laguerre polynomial and Bessel function

In an engineering setting, I reduced my problem to calculating the following sum: $$\sum_{n=0}^\infty \frac{n!}{(k+n)!}\left[\int_0^a \left(\frac{x}{u}\right)^kL_n^{(k)}\left(\frac{x^2}{u^2}\right)\...
3
votes
2answers
336 views

The Identity of the Modified Bessel Function of the first kind

Recently, I read a letter, containing the following identity: $$ \sum _{q=-\infty }^{\infty } \frac{(-1)^q I_q\left(\left| \alpha \right| ^2\right) I_q\left(\left| \alpha \right| ^2\right)}{2 q+1}=\...
2
votes
0answers
147 views

Approximating a divergent integral with modified Bessel functions of the first and second kinds

I am a physicist who needs to evaluate the following (divergent at the origin) integral involving the modified Bessel functions of the first and second kinds $$I = \int_0^{\infty} \frac{\cos(ax)}{x} ...
7
votes
1answer
501 views

Proof of a Fourier pair with Bessel functions?

How can we prove that the Fourier transform of the function $$ f(x) = \begin{cases} (a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\ 0 & \text{otherwise} \end{cases} $$ ...
2
votes
0answers
269 views

What am I missing in this highly oscillatory integral? [closed]

I want to numerically integrate this equation (in python): $\int_{0}^{\infty}{\rm d}k f(k) J_v(r k)J_v(s k) $, where f(k) is a non-smooth function, and $J_v$ are the Bessel function of the fist kind....