Questions tagged [bessel-functions]
Questions related to Bessel functions.
1,914 questions
0
votes
0
answers
58
views
Swapping sum and integral with an infinite series of modified Bessel functions
I am studying the following integral
\begin{align}
\int_0^{\infty} I_1(\sqrt{x}) \sum_{k=1}^{\infty} \Big( k K_1(k\sqrt{x}) - k^2 \sqrt{x}\, K_0(k\sqrt{x})
\end{align}
where $I_1$ and $K_\nu$ are ...
- 305
1
vote
0
answers
79
views
Definite integral of the two Bessel functions / (x-a)
In my work, an integral of the following type arose:
$$\int_0^\infty dx J_l(a x) J_l(b x) \frac{1}{x - c + i 0}.$$
Here $J$ is the Bessel function of the first kind. Assume that $a$, $b$, and $c$ are ...
2
votes
1
answer
49
views
Asymptotic Expansion of Bessel Function using Sommerfeld Contour
On pages 291-294 of Bender & Orszag (Advanced Mathematical Methods for
Scientists and Engineers-Asymptotic Methods and Perturbation Theory) they
develop the full asymptotic expansion of $J_0(x)$ ...
- 388
3
votes
3
answers
268
views
Analytic sum of an alternating series$\sum\limits_{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}$
I recently came across the following series with a positive real number $a$:
\begin{align}
S(a) = \sum _{n=1}^{\infty}(-1)^{n} \frac{n}{\left(n+\sqrt{a+n^2}\right)^2}
\end{align}
Does anyone know if ...
- 305
1
vote
2
answers
112
views
Bessel differential equation?
he Bessel functions of the first kind $J_n(x)$ are defined as the solutions to the Bessel differential equation:
$$ x^2y''(x)+ xy'(x)+(x^2-n^2)y(x)=0.$$
The special case of $n=0$ gives $J_0(x)$ as the ...
- 1,577
1
vote
1
answer
73
views
Question in the proof of Hankel's integral representation of the Bessel function of the first kind
I am trying to understand the proof of Hankel's integral representation of $J_\alpha(x)$:
$$ J_\alpha(x) = \frac{(x/2)^\alpha}{2\pi i} \int_{c-i\infty}^{c+i\infty} t^{-\alpha -1} \exp\left(t-\frac{x^2}...
- 23
2
votes
1
answer
56
views
Analytic resummation of a series involving modified Bessel functions $K_\nu $
I am interested in whether the following infinite series can be resummed or expressed in a more compact analytic form:
\begin{align}
S(t) = \sum_{n = 0}^{\infty} \left[
K_0\!\big((1 + 2n)t\big) \;+\; ...
- 305
3
votes
1
answer
143
views
Where do the constants $\cos(\nu \pi)$ and $\sin(\nu \pi)$ appear in the definiton of the Neumann function from Bessel functions?
I'm studying the relation between the Bessel function of the first kind $J_\nu(x)$ and the Neumann function (or Bessel function of the second kind) $Y_\nu(x)$. I know that $Y_\nu(x)$ can be expressed ...
- 85
6
votes
1
answer
228
views
A truly arcane definite integral. Closed form needed for an application
I am trying to find a closed form for this definite integral for a certain application:
$$ J=\int_0^1 \sqrt{\log x} \sqrt{\frac{1+x}{1-x}}\log\bigg(\frac{e^{\frac{1}{\log x}}+e^{-\frac{1}{\log x}}}{e^{...
- 1,209
5
votes
1
answer
108
views
On reducing this Meijer G-function
One can prove rather straightforwardly, by Mellin transforms, that
$$I=\int\limits_{0}^{\infty}\frac{J_{0}^{2}(t)J_{1}(t)}{t}\mathrm{d}t=\frac{1}{2\sqrt{\pi}}G^{1,2}_{3,3}\left(\left.\begin{matrix}\...
- 881
6
votes
0
answers
112
views
Bessel satisfies linear pde but can't find any references for it
The form
$$\Phi_s(p)= \int_0^\infty e^{-px} e^{-s/x} \, dx = 2\sqrt{\frac sp} K_1(2\sqrt{sp})$$
is a standard representation for the $K_\nu(\cdot)$ Bessel function ($\nu=1$). It appears in analytic ...
- 1,209
3
votes
1
answer
244
views
Lack of closed form of Rayleigh functions for $n > 10$
The Rayleigh function is defined as follows for integers $n$: $\displaystyle \sigma_n(\nu) = \sum_{k=1}^{\infty} j_{\nu,k}^{-2n}\ $, where the $j_{\nu,k}$ are the zeros of the Bessel function of the ...
0
votes
1
answer
102
views
Closed form expression for the Inverse Fourier transform of products of (Spherical) Bessel functions
I am interested in the following (inverse) Fourier transform of a function involving a product of spherical Bessel functions:
$$\mathcal{I} \equiv \frac{1}{2\pi}\int d\omega e^{-i\omega(t-t_0)}~I(\...
- 49
1
vote
0
answers
45
views
Compute $ \int_{-\pi/2}^{\pi/2} y_0(z) h_0^{(1)} \left (k \sqrt{a^2 + (b - z)^2 } \right)z \; dz $ involving Spherical Bessel Functions.
I want to solve the following integral:
$$
\int_{-\pi/2}^{\pi/2} y_0(z) h_0^{(1)} \left (k \sqrt{a^2 + (b - z)^2 } \right)z \; dz
$$
Where $a$, $b$ and $k$ are real parameters. Also, $y_0(z)$ and $h_0^...
- 823
0
votes
0
answers
84
views
Express Legendre functions in terms of Bessel functions
The solution to the Legendre differential equation
$$ (1-x^2) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 2x\frac{\mathrm{d}y}{\mathrm{d}x} + n(n+1) y = 0 $$
is a linear combinations of the Legendre ...
- 469