# Questions tagged [bessel-functions]

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### Asymptotic behavior of maximum of bessel function

Let $J_n$ be the Bessel function of the first kind. Let $J_n^{(\max)} = \max_{x>0} J_n(x)$. What is known about the asymptotic behavior of $J_n^{(\max)}$ at large $n$? Specifically, I am looking ...
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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 ...
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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 ...
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### Series representation of multiplication of two modified Bessel function

Series representation of multiplication of two Bessel function $J_{\mu}(az) J_{\nu}(bz)$ is in terms of sum of hypergeometric functions $_2F_1$, it given in book Treatise on Theory of Bessel Functions ...
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This was originally posted on Math Stack Exchange, but no responses were received. I recently came across the following remarkable identity, due to Hardy: $$\displaystyle \int_{-\infty}^{\infty} \... 1answer 253 views ### Integral of exp(-2cosh(x)) Is there some way to express:$$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}...
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 ...