# Questions tagged [bessel-functions]

The bessel-functions tag has no usage guidance.

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### 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 $+\...

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votes

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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 ...

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votes

**3**answers

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 ...

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votes

**0**answers

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 ...

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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 ...

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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)...

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**2**answers

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 ...

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votes

**2**answers

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
...

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vote

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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 $K_{\frac{N+2s}{2}}$ ...

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votes

**2**answers

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 $I_n(u)$...

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**1**answer

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 $-\...

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**2**answers

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 ...

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votes

**1**answer

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 ...

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**0**answers

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 ...

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votes

**1**answer

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),...

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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 ...

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**1**answer

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 ...

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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.,...

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votes

**1**answer

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 ...

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**1**answer

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 = \...

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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 ...

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votes

**0**answers

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 ...

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votes

**1**answer

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 ...

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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: ...

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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\...

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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} \...

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**1**answer

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}...

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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 ...

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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 $\...

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**1**answer

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 ...

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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 ...

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**1**answer

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 ...

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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.
$$J_n(1)=\int_{0}^{\pi}\cos (nx-\sin x)=0\,?$$

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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 ...

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**1**answer

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 ...

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votes

**1**answer

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 ...

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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 ...

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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^+\...

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vote

**1**answer

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, ...

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vote

**1**answer

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 ...

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**0**answers

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

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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 ...

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votes

**1**answer

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, \...

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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 \...

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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

**1**answer

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)\...

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**2**answers

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}=\...

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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} ...

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votes

**1**answer

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

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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....