Questions tagged [bessel-functions]
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87
questions
2
votes
1answer
42 views
Asymptotic behavior of maximum of bessel function
Let $J_n$ be the Bessel function of the first kind. Let $J_n^{(\max)} = \max_{x>0} J_n(x)$. What is known about the asymptotic behavior of $J_n^{(\max)}$ at large $n$? Specifically, I am looking ...
0
votes
0answers
40 views
Integral of an expression including a fraction having modified Bessel functions of the first kind on both numerator and denominator
I am looking for an analytic result of the following integral
$$\iint_0^\infty {{\rm{d}}x{\rm{d}}y{x^2}{y^2}\exp \left\{ { - {\alpha _1}{x^2} - {\alpha _2}{y^2}} \right\}\frac{{{\rm{I}}_1^2\left[ {\...
0
votes
0answers
50 views
Integral with 4 Bessel functions and an exponential
I would like to solve the following integral
$$
\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,
$$
where $a,b,c,f,r > 0$, and $J_\nu(x)$ is the Bessel ...
3
votes
0answers
50 views
Limit of Hankel function for large complex order, fixed real argument
Consider the Hankel function $H_\nu(z)$ where $\nu=re^{i\theta}$ (real $z>0$, $r>0$, $0\leq\theta<\pi$) as $r\rightarrow\infty$. I am aware that the Bessel function $J_\nu(z)$ has the ...
3
votes
1answer
164 views
Almost periodicity of Bessel functions
We know that a periodic function (e.g. a trigonometric function) has the property
$$
f(x+n\Lambda)=f(x) \qquad n\in\mathbb Z
$$
A Bessel function is not exactly periodic, because the value of the ...
1
vote
0answers
44 views
A Bessel-like integral
I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...
2
votes
1answer
111 views
Integral involving associated Laguerre polynomial and Bessel function
In a quantum mechanics problem I encountered the following integral
\begin{equation*}
\int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,,
\end{equation*}
where $L$ denotes the ...
4
votes
1answer
81 views
Limit for series of Bessel functions evaluated at zeros
The following series arises in an electrostatics problem for a conducting cylinder:
$$
V=\sum_{n=1}^\infty\frac{J_0(k_n\rho)e^{-k_nz}}{k_nJ_1(k_n)^2}
$$
where $J_i$ is the Bessel function of $i^{th}$ ...
0
votes
1answer
256 views
Reciprocal expansion of modified Bessel function
I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are ...
4
votes
1answer
199 views
Gegenbauer's addition theorem for Jacobi polynomials
I have the following identity,
$$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z)
= 2 \, j_n(x) \, j_n(y) \;,$$
where $x, y > 0$, $P_n$ is a Legendre polynomial, and $...
0
votes
1answer
62 views
The ratio of Hankel functions
I have to obtain an asymptotic solution for small real positive $x$ for the ratio of Spherical Hankel functions ($n=0,1,2....)$
${h^{(2)}_n(x)}/{h^{(1)}_n(x)}$
I found that series should be
$-1 + i ...
2
votes
1answer
79 views
Can the integral $ \int_0^R\quad J_{m-n}(a r)J_m(b r) dr$ be explicitly represented in a closed form?
Doe the following definite integral have an explicit representation in terms of a Bessel functions or a generalized hypergeometric function ${}_pF_q$?
$$
\int_0^R\quad J_{m-n}(a r)J_m(b r) dr, \quad \...
0
votes
0answers
120 views
A complex integration formula
I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula:
$f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) \operatorname{...
6
votes
3answers
272 views
A hypergeometric identity related to Bessel functions
The identity in my recent answer can be stated in a particularly neat form:
$${}_2F_0\left({-n, n+1\atop{}};\frac{x}{2}\right) ~\cdot~ {}_2F_0\left({-n, n+1\atop{}};-\frac{x}{2}\right) ~=~ {}_3F_0\...
6
votes
1answer
175 views
Spherical Bessel functions. Sum of squares
In (1) there is a property of spherical Bessel functions, which's derivation I can not find in the literature.
${\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=%
0}^{n}\...
2
votes
2answers
105 views
How to compute means μ1 and μ2 knowing sum of Skellam distributions f(k;μ1,μ2) and sum μ1+μ2, where k is from 2 to n?
The probability mass function for the Skellam distribution for a count difference $ k=n_1-n_2 $ from two Poisson-distributed variables with means $\mu_1$ and $\mu_1$ is given by:
$$ f(k;\mu_1,\mu_2)=...
6
votes
0answers
118 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 ...
0
votes
0answers
52 views
Construction of a special sequence [duplicate]
I m looking for a sequence $(f_j)\in C^\infty(\Bbb{R})$ such that
$$
\int^\infty_0\Big|4r\partial^2_r f_j(r)+4\partial_r f_j(r)+rf_j(r)\Big|^2dr\to 0,
$$ and
$$\int_{\Bbb{R^+}}|f_j(r)|^2 dr=1\quad\...
2
votes
1answer
128 views
How to get asymptotic expansion of the sum of modified Bessel function $\sum_{n=1}^\infty K_0(s\, n)$ as $s\to 0^+$?
I guess for the modified Bessel funcion $K_0(z)$,
$$\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^+,$$
if taking
$$\...
0
votes
1answer
110 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://...
0
votes
1answer
179 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
$$f_X(t \...
3
votes
2answers
224 views
Asymptotics for the first zero of the Bessel functions
Let $J_\nu$ be the standard Bessel function of the first kind and let $x_\nu$ be its smallest zero. Is there a simple reference or result for the asymptotic expansion of $x_\nu$ when $\nu$ goes to $+\...
3
votes
3answers
235 views
Approximation of half-integers modified Bessel function of the second kind
I am trying to optimise the calculation of the probability distribution poisson-inverse-gaussian
its calculation involves a half-integers modified Bessel function of the second kind. Here's a formula ...
3
votes
3answers
232 views
ODE with Bessel decay
This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily.
I would like to estimate the asymptotic behaviour of the ...
2
votes
0answers
55 views
Bessel decay for nonhomogeneous PDE
I'm interested in the following nonhomogeneous PDE
$$ (\Delta-k^{2})u=-g $$
on the upper-half plane with smooth and integrable Dirichlet boundary condition, where $g$ is a smooth positive function ...
1
vote
0answers
86 views
Is the ratio of two distinct zeros of a Bessel function of the first kind an irrational number?
Let $J_1$ be the Bessel function of the first kind with parameter $\alpha=1$. Namely, $J_1$ satisfies the differential equation for $y$ given by $x^2y''+xy'+(x^2-1)y=0$ and $J_1(0)=0$.
Is it true ...
1
vote
0answers
56 views
Integral reducible to Bessel function
I wonder whether one could reduce the following integral to some combination of special (e.g. Bessel) functions: $$\int_{m}^{\infty}d\epsilon \exp\left(-s\epsilon\right)\left( \epsilon^{2}-m^{2}\right)...
9
votes
2answers
708 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
318 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
147 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
136 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
181 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
266 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 ...
5
votes
1answer
514 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
votes
0answers
207 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
627 views
Infinite summation formula of Bessel functions
I would like to find a closed form for the following series involving the Bessel function $J_k(z)$:
$$
\sum_{k=0}^{+\infty}\frac{(\mu)_{k}}{k!(\lambda)_{k}}t^k\left(\frac{z}{2}\right)^{k}J_{k+\nu}(z),...
2
votes
2answers
548 views
Integral with Bessel function and hypergeometric function ${}_2F_2$: explicit expression for these polynomials?
This question follows this one, where the general problem has apparently no simpler form than the integral one. I focus now on the limit case:
\begin{align}
\int_0^T e^{-x}\frac{nI_n(x)}{x}dx=\int_0^T ...
5
votes
1answer
366 views
Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$
EDIT: Some additional details and corrections, I would appreciate any information about the highlighted expression.
I try to solve $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$ where $I_n(x)$ is the ...
4
votes
0answers
335 views
On modified Bessel solutions to complex ODE's using Kummer's series
I am trying to reduce the following ODE to Bessel's ODE form and hence solve it:
$$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$
I tried to solve it via the standard method, i.e.,...
2
votes
1answer
451 views
Indefinite summation of multiplication of two Bessel functions
Could anyone give an insight on how to prove the following formula?
$$\sum_{n=-\infty}^{+\infty}J_{n}(\alpha)J_{N+n}(\alpha)=\delta_{N0} \, ,$$
where $N$ is an integer. I checked many references ...
0
votes
1answer
267 views
The limitation of derivation of modified Bessel function of second kind
The final result I draw is related to the integral of modified Bessel function of the second kind. But I can not solve it, and I need a explicit solution Are you willing to help me? Thank all
$I = \...
2
votes
2answers
374 views
Asymptotics of Bessel functions
With $J_n$ standing for the Bessel function of first kind, $n\in \mathbb N$, I define
$$
f_n
(\rho)
=\int_0^π J_n(\rho \sin \theta) \sin \theta \ d\theta.$$
Assuming
$1\ll\rho\ll n$, I would like to ...
3
votes
0answers
76 views
Convergence of Bessel (Sturm-Liouville) Expansions at the End Points
I have asked this question before on MSE but received no answer at all. So I assume that it is proper to ask it here. I am not a mathematician so my language may not be too precise, please correct me ...
2
votes
1answer
299 views
Orthogonality with Bessel functions of rational order
I tried to ask this question on MSE (link), but got no comments or answers. So, I hope someone on MO would advise.
Given a set of functions $f_{mv}(r,\phi)=J_{v}(k_{mv}r)\cos(v \phi)$ in polar ...
5
votes
0answers
331 views
Why are Bessel function and Kloosterman sum similar?
It is a convention to say Kloosterman sums and Bessel functions are similar.
There are papers talking about Bessel functions on $p$-adic group (associated with a representation) such as Baruch's: ...
1
vote
0answers
103 views
Kontorovich Lebedev transform
By the title I mean [reference: ``Spectral methods of Automorphic forms" by Iwaniec (B.41)-(B.43)] for $f\in C^\infty_c(\mathbb{R^+})$, one has
$$f(x)=\pi^{-2}\int_{-\infty}^\infty K_{it}(x)F_f(t)t\...
1
vote
0answers
54 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:
$$\displaystyle \int_{-\infty}^{\infty} \...
1
vote
1answer
253 views
Integral of exp(-2cosh(x))
Is there some way to express:
$$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
476 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 ...
1
vote
1answer
388 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 ...
