# Questions tagged [bessel-functions]

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### Mellin transform of the Bessel function $Y_n$ of order $n \geq 2$

The Mellin transform of the function $h$, locally integrable on $(0,\infty)$, is defined by $$M[h,z] = \int_0^\infty t^{z-1} h(t) dt \tag{1}$$ For some functions $h$ the above integral is not ...
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### Airy-type integrals (with different power $\neq 3$)

I am looking for integrals closely related to the Airy function \begin{eqnarray} && A_1= \int _0^\infty x \sin \Phi dx \nonumber \\ && A_2= \int _0^\infty \cos \Phi dx \nonumber \\&...
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### Solving an integral involving a Bessel function, Laguerre function and Gaussian

We want to calculate the expectation value $\langle q^2\rangle$ in polar coordinates which gives us the following integral, for integer values of $p$: \begin{equation}\int_0^\infty dq~q^3 \left(\int_0^...
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### Two questions about an integral involving double product of Bessel functions

Let us define the following integral : $$W_n(r)=r\int_0^{+\infty} J_1(rt)[J_0(t)]^n dt,$$ with $r>0$ a real number and $n\in\mathbb{N}$ and where $J_0(x)$ and $J_1(x)$ are Bessel functions of the ...
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Let $J_\nu$ be a Bessel function of the first kind and let $\{\lambda_{n, \nu}\}_{n\ge 1}$ be a sequence of its zeroes. I claim that $$\inf_{n\ge 1}\bigg|\sqrt{\lambda_{n,\nu}} J_{\nu+1}(\lambda_{n,\... 1 vote 0 answers 99 views ### A series with zeroes of Bessel functions Consider a finite sum$$ S_n(t)=\sum_{m=1}^n \frac{J_\nu(z_{m,\nu} t)}{J_{\nu+1}(z_{m,\nu})}, \nu>0, 0\leq t <1, $$z_{m,\nu} are ordered real positive zeroes of the Bessel function J_\nu(t).... 0 votes 0 answers 67 views ### Can the Bessel functions tend to a plane wave? Can the Bessel functions tend to a plane wave? If I have this function:$$ y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6) $$... 0 votes 2 answers 310 views ### Orthogonality of Bessel function \int_0^bxJ_a(\ell x)J_a(\ell' x)=0 for \ell\neq\ell' How do I show the above relation with Sturm-Liouville theory (assume the usual boundary conditions for the identity)? Here is what I have tried: if we start with$$ \big(xJ_a'(\ell x) \big)'+\left(\... 346 views

### Upper bounds for Bessel functions

Cosider the K-Bessel function $$K_\nu(x):= \frac\pi 2 \frac{I_{-\nu}(x)-I_\nu(x)}{\sin(\nu\pi)}.$$ See also Watson, G. N., A treatise on the theory of Bessel functions., Cambridge: University Press, ...
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### Discrete random walk and SDEs

My advisor has some vague ideas about the relation between discrete random walks and SDEs, and advise me to read a little bit about them. To be more precise, ( if I understand correctly what my ...
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### Integral involving Bessel function

I have to work out the integral: $$\int_0^{\infty} dq \frac{J_0(q \xi)}{q+1}$$ where $J_0(z)$ is the Bessel function of the first type and order zero, $\xi \in \mathbb{R}$, $\xi \ge 0$. So far, ...
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### Variant of modified Bessel functions

Consider the integral \begin{align*} g_f(x)=\int_{\phi=0}^{2\pi} f(\phi) ~e^{x cos(\phi)}~\mathrm{d}\phi, \end{align*} where $f(\phi)$ is a probability density functions defined over $[0,2\pi]$, \...
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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 ...
I am looking for an analytic result of the following integral $$\iint_0^\infty {{\rm{d}}x{\rm{d}}y{x^2}{y^2}\exp \left\{ { - {\alpha _1}{x^2} - {\alpha _2}{y^2}} \right\}\frac{{{\rm{I}}_1^2\left[ {\... 3 votes 1 answer 467 views ### Integral with 4 Bessel functions and an exponential I would like to solve the following integral$$ \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, $$where a,b,c,f,r > 0, and J_\nu(x) is the Bessel ... 3 votes 0 answers 107 views ### Limit of Hankel function for large complex order, fixed real argument Consider the Hankel function H_\nu(z) where \nu=re^{i\theta} (real z>0, r>0, 0\leq\theta<\pi) as r\rightarrow\infty. I am aware that the Bessel function J_\nu(z) has the ... 3 votes 1 answer 482 views ### Almost periodicity of Bessel functions We know that a periodic function (e.g. a trigonometric function) has the property$$ f(x+n\Lambda)=f(x) \qquad n\in\mathbb Z $$A Bessel function is not exactly periodic, because the value of the ... 1 vote 1 answer 278 views ### A Bessel-like integral I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, 0\le\lambda\le1, p\ge0, q\ge0 are ... 2 votes 1 answer 470 views ### Integral involving associated Laguerre polynomial and Bessel function In a quantum mechanics problem I encountered the following integral \begin{equation*} \int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,, \end{equation*} where L denotes the ... 4 votes 1 answer 124 views ### Limit for series of Bessel functions evaluated at zeros The following series arises in an electrostatics problem for a conducting cylinder:$$ V=\sum_{n=1}^\infty\frac{J_0(k_n\rho)e^{-k_nz}}{k_nJ_1(k_n)^2} $$where J_i is the Bessel function of i^{th} ... 0 votes 1 answer 480 views ### Reciprocal expansion of modified Bessel function I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are ... 4 votes 1 answer 450 views ### Gegenbauer's addition theorem for Jacobi polynomials I have the following identity,$$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z) = 2 \, j_n(x) \, j_n(y) \;,$$where x, y > 0, P_n is a Legendre polynomial, and ... 0 votes 1 answer 123 views ### The ratio of Hankel functions I have to obtain an asymptotic solution for small real positive x for the ratio of Spherical Hankel functions (n=0,1,2....) {h^{(2)}_n(x)}/{h^{(1)}_n(x)} I found that series should be -1 + i ... 2 votes 1 answer 96 views ### Can the integral  \int_0^R\quad J_{m-n}(a r)J_m(b r) dr be explicitly represented in a closed form? Doe the following definite integral have an explicit representation in terms of a Bessel functions or a generalized hypergeometric function {}_pF_q?$$ \int_0^R\quad J_{m-n}(a r)J_m(b r) dr, \quad \... 