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 ...

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 ...

On pages 291-294 of Bender & Orszag (Advanced Mathematical Methods forScientists and Engineers-Asymptotic Methods and Perturbation Theory) theydevelop the full asymptotic expansion of $J_0(x)$ ...

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 ...

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 ...

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}...

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) \;+\; ...

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 ...

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^{...

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}\...

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 ...

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 ...

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(\...

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^...

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 ...