Questions tagged [bessel-functions]
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152 questions
4
votes
1
answer
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views
Infinite summation formula of Bessel functions
I would like to find a closed form for the following series involving the Bessel function $J_k(z)$:
$$
\sum_{k=0}^{+\infty}\frac{(\mu)_{k}}{k!(\lambda)_{k}}t^k\left(\frac{z}{2}\right)^{k}J_{k+\nu}(z),...
- 147
2
votes
2
answers
855
views
Integral with Bessel function and hypergeometric function ${}_2F_2$: explicit expression for these polynomials?
This question follows this one, where the general problem has apparently no simpler form than the integral one. I focus now on the limit case:
\begin{align}
\int_0^T e^{-x}\frac{nI_n(x)}{x}dx=\int_0^T ...
- 634
5
votes
1
answer
522
views
Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$
EDIT: Some additional details and corrections, I would appreciate any information about the highlighted expression.
I try to solve $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$ where $I_n(x)$ is the ...
- 634
4
votes
0
answers
391
views
On modified Bessel solutions to complex ODE's using Kummer's series
I am trying to reduce the following ODE to Bessel's ODE form and hence solve it:
$$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$
I tried to solve it via the standard method, i.e.,...
- 449
2
votes
1
answer
667
views
Indefinite summation of multiplication of two Bessel functions
Could anyone give an insight on how to prove the following formula?
$$\sum_{n=-\infty}^{+\infty}J_{n}(\alpha)J_{N+n}(\alpha)=\delta_{N0} \, ,$$
where $N$ is an integer. I checked many references ...
- 23
0
votes
1
answer
379
views
The limitation of derivation of modified Bessel function of second kind
The final result I draw is related to the integral of modified Bessel function of the second kind. But I can not solve it, and I need a explicit solution Are you willing to help me? Thank all
$I = \...
2
votes
2
answers
662
views
Asymptotics of Bessel functions
With $J_n$ standing for the Bessel function of first kind, $n\in \mathbb N$, I define
$$
f_n
(\rho)
=\int_0^π J_n(\rho \sin \theta) \sin \theta \ d\theta.$$
Assuming
$1\ll\rho\ll n$, I would like to ...
- 16.2k
3
votes
0
answers
108
views
Convergence of Bessel (Sturm-Liouville) Expansions at the End Points
I have asked this question before on MSE but received no answer at all. So I assume that it is proper to ask it here. I am not a mathematician so my language may not be too precise, please correct me ...
- 271
3
votes
1
answer
598
views
Orthogonality with Bessel functions of rational order
I tried to ask this question on MSE (link), but got no comments or answers. So, I hope someone on MO would advise.
Given a set of functions $f_{mv}(r,\phi)=J_{v}(k_{mv}r)\cos(v \phi)$ in polar ...
- 183
5
votes
0
answers
569
views
Why are Bessel function and Kloosterman sum similar?
It is a convention to say Kloosterman sums and Bessel functions are similar.
There are papers talking about Bessel functions on $p$-adic group (associated with a representation) such as Baruch's: ...
- 3,804
1
vote
0
answers
141
views
Kontorovich Lebedev transform
By the title I mean [reference: ``Spectral methods of Automorphic forms" by Iwaniec (B.41)-(B.43)] for $f\in C^\infty_c(\mathbb{R^+})$, one has
$$f(x)=\pi^{-2}\int_{-\infty}^\infty K_{it}(x)F_f(t)t\...
- 1,692
1
vote
0
answers
71
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:
$$\displaystyle \int_{-\infty}^{\infty} \...
- 161
2
votes
2
answers
389
views
Integral of exp(-2cosh(x))
Is there some way to express:
$$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}...
- 1,902
2
votes
0
answers
571
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 ...
- 161
1
vote
1
answer
743
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 ...
- 11
1
vote
0
answers
312
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 ...
- 161
3
votes
1
answer
386
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 ...
- 1,271
-1
votes
3
answers
318
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.
$$J_n(1)=\int_{0}^{\pi}\cos (nx-\sin x)=0\,?$$
- 101
21
votes
3
answers
758
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 ...
- 1,271
3
votes
1
answer
721
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 ...
- 135
4
votes
1
answer
170
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 ...
- 41
3
votes
0
answers
225
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 ...
- 31
5
votes
2
answers
485
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^+\...
- 6,829
1
vote
1
answer
954
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, ...
- 11
1
vote
1
answer
187
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 ...
- 1,199
1
vote
0
answers
302
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
- 119
0
votes
0
answers
454
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 ...
- 119
3
votes
1
answer
1k
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
$$I(a, b, c) = \int_0^{\infty} \frac{\sin(ax)}{x} I_1(bx) K_1(cx) \mathrm{d}x, \...
- 285
4
votes
0
answers
337
views
Bounds for Bessel functions
Let $0 < \delta < 1$, and let $I_\delta$ be the set of all complex numbers $\mu$ such that $-1/2 + \delta < \Re \mu < 1/2 -\delta$. Is there a polynomial $P_\delta$ such that for all $\mu \...
- 1,362
4
votes
0
answers
221
views
Asymptotic expansion of Mellin transform of products of modified Bessel function K
Let $n\ge 1$ be an integer, let
$$F(x,y)=\int_0^\infty u^{n(x+y)} (K_{x-y}(u))^n du$$
for $x,y\ge 0$.
When $n=1$, this is just Mellin transform of the Bessel K function. When $n=2$, $F(x,y)$ has ...
- 453
3
votes
1
answer
602
views
Summation of an integral involving Laguerre polynomial and Bessel function
In an engineering setting, I reduced my problem to calculating the following sum:
$$\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)\...
- 917
3
votes
2
answers
637
views
The Identity of the Modified Bessel Function of the first kind
Recently, I read a letter, containing the following identity:
$$
\sum _{q=-\infty }^{\infty } \frac{(-1)^q I_q\left(\left| \alpha \right| ^2\right) I_q\left(\left| \alpha \right| ^2\right)}{2 q+1}=\...
- 43
2
votes
0
answers
194
views
Approximating a divergent integral with modified Bessel functions of the first and second kinds
I am a physicist who needs to evaluate the following (divergent at the origin) integral involving the modified Bessel functions of the first and second kinds
$$I = \int_0^{\infty} \frac{\cos(ax)}{x} ...
- 285
7
votes
1
answer
909
views
Proof of a Fourier pair with Bessel functions?
How can we prove that the Fourier transform of the function
$$
f(x)
=
\begin{cases}
(a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\
0 & \text{otherwise}
\end{cases}
$$
...
- 71
2
votes
0
answers
517
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....
- 121
7
votes
1
answer
1k
views
Fourier expansion of Eisenstein Series
I have been reading a bit about the Fourier expansion of Eisenstein series (weight 1/2). I came across the fact that the coefficients contain Modified Bessel functions.
Further reading I found ...
0
votes
0
answers
92
views
Energy Oscillations in a One Dimensional Crystal
Good day!
Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)?
article, that I have
Especially ...
0
votes
0
answers
49
views
Non interacting complex unit
How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...
- 1
1
vote
0
answers
78
views
Related to derivative of Modified Bessel I function wrt the order
I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that $Re(\dfrac{I'...
33
votes
1
answer
1k
views
$\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?
QUESTION
Numerical calculation with gp (first to the default 38-digit
precision, then tripled) supports the conjecture that
$$
\int_0^\infty x \, [J_0(x)]^5 \, dx =
\frac{\Gamma(1/15) \, \Gamma(2/15) ...
- 79.9k
3
votes
1
answer
260
views
Is it possible to get an equation with two exponentials and a bessel function in closed form?
Is it possible to get the equation below into closed form? I have tried using integration tables but I haven't found anything that matches. Are there any other methods to achieve a closed form ...
2
votes
0
answers
2k
views
convolution integral involving modified Bessel functions of the first kind
I'm stuck with this convolution integral ($z \geq 0$)...
\begin{equation}
f_{Z}(z)=\int^{\infty}_{-\infty}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ???
\end{equation}
which represents the pdf of the sum $Z = ...
- 163
2
votes
1
answer
824
views
Limit involving modified Bessel Function of the second kind
I'm looking for the following limit
$$\lim_{x\rightarrow 0} \frac{\sqrt{\frac{\text{BesselK}^{(2,0)}(0,x)}{\text{BesselK}(0,x)}}}{\log (x)}$$
I believe the limit is finite, and is near -0.578. ...
- 1,902
4
votes
1
answer
146
views
Estimate on sum of $J_n^4$
If $J_n(x)$ is the Bessel function of order $n$, we know that for all $x$, $$\sum_{n=-\infty}^{\infty} J_n^2(x)=J_0^2(x)+2\sum_{n=1}^{\infty} J_n^2(x)=1.$$
What is known about
$$
\sum_{n=-\infty}^{\...
- 2,494
2
votes
1
answer
627
views
Maximal minimum of Bessel functions
This comes from a scattering problem. Consider the usual non singular Bessel functions of the first kind, $J_n(x)$. It is known that their zeros are countable, and all zeros are distinct. My question ...
- 2,494
0
votes
0
answers
146
views
Riemann surface of $K_0(z)$
The question concerns the modified Bessel function of the second kind of order zero ($K_0(z)$).
How does its Riemann surface looks like? How can one evaluate its values on the "other" sheet(s)?
- 175
2
votes
1
answer
1k
views
Expression for infinite sum of two Bessel functions and a power
I'm searching for an expression of the following sum (all indices are integers and all variables are positive real numbers):
$$\sum_{\lambda=-\infty}^{+\infty} A^{|l-\lambda|} B^{|\lambda|} J_{\lambda+...
- 201
4
votes
2
answers
2k
views
Integrals of two Bessel functions of the first kind and a modified bessel function of the second kind
I'm searching for a suitable (hopefully simple enough) solution to the following form of integral:
$$\int_0^\infty \mathrm{d}x~x^n J_\nu(a x) J_\nu(b x) K_\mu(c x) $$
Where $n$, $\nu$, and $\mu$ are ...
- 201
0
votes
2
answers
1k
views
Indefinite integration of multiplication of two Bessel function
I am trying to calculate this integral. I know it has an analytic expression when $a = 0$. But, is there any analytic expression for this case?
$$\int_{a}^{\infty}J_2(bx)J_1(cx)\,dx$$
Thanks in ...
- 101
2
votes
1
answer
853
views
The approximation of first-ordered modified Bessel function of the second kind
After analysing the outage probability of a single relay selection system, I got to the following form:
$P = 1 + \sum\limits_{k = 1}^K {\left( \begin{array}{l}
K\\
k
\end{array} \right){{\left( { - 1} ...
- 173