Questions tagged [bessel-functions]

Question 1

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 ...

Question 2

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 ...

Question 3

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}...

Question 4

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) \;+\; ...

Question 5

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 ...

Question 6

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^{...

Question 7

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}\...

Question 8

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 ...

Question 9

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 ...

Question 10

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(\...

Question 11

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^...

Question 12

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 ...

Question 13

$$a_{m} = \frac{2}{R^{2}J^{2}_{n + 1}(\alpha_{mn})}\int_{0}^{R}xf(x)J_{n}(\alpha_{mn}x/R)dx \tag{1}$$ As an exercise I've been trying to write GNU Octave code that will take an arbitrary function that ...

Question 14

Within the context of studying fluids turbulence in wakes, I stumbled upon an integral in the form $$S(\alpha,\beta)=\int\limits_0^\infty x \exp{(-x^2)} I_0(\alpha x) \cosh(\beta x)~dx,$$ where $I_0$ ...

Question 15

I would like to evaluate analytically the improper integral $$ f(z,r,\rho,\nu) = \int_0^\infty \sin(k z)\, K_{i\nu}(k \rho)\, K_{i\nu}(k r)\, dk \, , $$ where $K_{i\nu}$ is the modified Bessel ...

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Questions tagged [bessel-functions]

Analytic sum of an alternating series involving $n/(n+\sqrt{a^2+n^2})^2$

Bessel differential equation?

Question in the proof of Hankel's integral representation of the Bessel function of the first kind

Analytic resummation of a series involving modified Bessel functions $K_\nu $

Where do the constants $\cos(\nu \pi)$ and $\sin(\nu \pi)$ appear in the definiton of the Neumann function from Bessel functions?

A truly arcane definite integral. Closed form needed for an application

On reducing this Meijer G-function

Bessel satisfies linear pde but can't find any references for it

Lack of closed form of Rayleigh functions for $n > 10$

Closed form expression for the Inverse Fourier transform of products of (Spherical) Bessel functions

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.

Express Legendre functions in terms of Bessel functions

Confusion in calculating coefficients in a Fourier-Bessel series expansion of an arbitrary function.

Integral of linear, Gaussian, modified bessel function of first kind, and hyperbolic cosine functions

How to evaluate the improper integral $\int_0^\infty \sin(k z)\, K_{i\nu}(k \rho)\, K_{i\nu}(k r)\, dk $ analytically?

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