# Questions tagged [bessel-functions]

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $i^{th}$ ...
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### 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 ...
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### 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 ...
1answer
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### 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 ...