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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2 votes
1 answer
148 views

$|\hat\mu(\xi)| \lesssim |\xi|^{-1/2}$ where $\mu$ is $f\mapsto \int_{\mathbb R} \psi(r) \int_{S^{1}} f(rx,r)\, d\sigma(x)\, dr $

I have questions about the proof of Theorem $2.1$ here. The proof is on Pg. $10$. I am trying to work out the $d = 2$ case in particular. $$\mathcal C^d = \{(x_1, \ldots, x_{d+1}): |(x_1, \ldots, x_d)|...
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1 vote
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How to relate this integration with the integral expansion of special functions?

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, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
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1 vote
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33 views

Class of spectral zeta functions whose analytic extension takes a particular form

In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...
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Bounds for ratio of Bessel functions

I know several good papers where bounds of Bessel function ration considered. For instance, the following Bessel function ratio, $$h(z) = \frac{I_{\nu}(z)}{I_{\nu+1}(z)}$$ has bounds $$h(z)<\frac{z}...
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2 votes
1 answer
121 views

From a sum of cosines to an integral of Bessel function

In a physics paper (pubs.acs.org/doi/10.1021/j100210a011), I see the following transformation: $$\sum_q \frac{2[1-\cos(\textbf{q} \cdot \textbf{r})]}{q^2} =\frac{1}{\pi} \int_0^{+\infty}[1-J_0(qr)]\...
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2 votes
1 answer
245 views

Can $\int \Big{(} \frac{1}{e^{x}-1} - \frac{1}{e^{x}} \Big{)} \Big{(} I_{0}(2 \sqrt{x}) - 1 \Big{)} dx $ be evaluated?

Currently, I'm working on a problem pertaining to certain integrals involving the modified Bessel function of the first kind. On p. 59 of this book by Rosenheinrich, it is stated that $$\int e^{-x} I_{...
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60 views

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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2 votes
1 answer
101 views

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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3 votes
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114 views

On analogues of Weber's formula

Let $J_0(x)$ be the $0$-th Bessel function of the first kind. Weber's formula states that $$ \int_0^{+\infty}e^{-x}J_0(2\sqrt{\alpha x})J_0(2\sqrt{\beta x})dx=e^{-\alpha-\beta}I_0(2\sqrt{\alpha\beta})....
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97 views

Is there any known relationship between logarithm of modified Bessel function of second type and its ratio?

Is there a known relationship between $\log(K_s(x))$ and $\frac{K_{s-1}(x)}{K_s(x)}$ for $s < 0$ and $x > 0$?
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1 answer
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Log-concavity of the modified Bessel function of a second kind

I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...
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2 votes
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126 views

Integral rewritten in terms of a modified Bessel function

I am reading this paper by Kunz and Shapiro: they state that the integral (Eqs. 3.17-3.19) $$\int_{-\infty}^\infty\frac{dy}{2\pi}e^{ib(y-i\delta)}\left[\exp\left(-\frac{ia}{y-i\delta}\right)-1\right]\...
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1 answer
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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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7 votes
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171 views

Reference for proof of an integral from the "Tables of Integral Transforms" involving a Gaussian and a Laguerre polynomial

I am looking for a proof of one of the integrals presented in Harry Bateman's Tables of Integral Transforms. The specific integral in question is presented on page 42 in chapter 8.9 as equation (3): $$...
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10 votes
1 answer
398 views

An integral identity involving cotangents and Bessel functions

Numerical experiments suggest that the following integral identity holds for Bessel functions of the first kind, $$J_2(t) = 12 \int_0^{1/2}\mathrm{d}x\,\cot \pi x \int_0^x \mathrm{d}y\, \cot \pi y \, ...
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  • 3,070
2 votes
2 answers
59 views

Asymptotic behavior of a Bessel function on a sequence on zeros with a shifted parameter of type

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,\...
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  • 347
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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 $J_\nu(t)$....
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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: $$ y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6) $$ ...
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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(\...
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6 votes
2 answers
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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2 votes
2 answers
163 views

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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1 vote
0 answers
288 views

What is the Jacobi-Anger expansion of $\sin^{[k]} (\theta) $?

Cross-post from MSE. The Jacobi-Anger expansion gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \...
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1 vote
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Problem of correctly defining Hankel transforms

I have found the definition of the $v$-th order Hankel transform in the book of A.D. Poularikas "Transforms and applications" in Chapter $9$: $$H_{v}(f(s))=\int_{0}^{+ \infty} rf(r) J_{v}(sr)...
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2 votes
1 answer
220 views

Definite integral of modified Bessel function of second kind

How do I integrate a modified Bessel function of the second kind as shown below? A good approximation of the definite integral is also ok, I do not need an exact solution. $\int_\frac{1}{\lambda}^{\...
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Inverse mellin transform

Let $K_1(t)$ be the K-Bessel function, then we have $\int_{0}^{\infty}K_v(y)y^s\frac{dy}{y}=2^{s-2}\Gamma(\frac{s+v}{2})\Gamma(\frac{s-v}{2})$ See page 106 of Bump's book Automorphic forms and ...
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3 votes
0 answers
201 views

L functions of Symmetric power of elliptic curves

Let $E$ be an elliptic curve over the raional field with conductor $N$, which corresponds to the eigenform $f(z)=\sum a_nq^n$. Let $L(Sym^2E,s)$ be the L function of the symmetric power of $E$.I am ...
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3 votes
0 answers
116 views

L functions of elliptic curves over quadratic fields

Let $E$ be an ellitpic curve over a quadratic field $K/\mathbb{Q}$. Then the L function of $E$ is defined as $L(E_K,s)=\prod_{\mathfrak{p}\nmid \Delta}(1-a_{\mathfrak{p}}N(\mathfrak{p})^{-s}+N(\...
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2 votes
1 answer
166 views

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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0 votes
1 answer
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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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2 votes
1 answer
241 views

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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0 answers
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Integral of an expression including a fraction having modified Bessel functions of the first kind on both numerator and denominator

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[ {\...
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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 ...
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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 ...
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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 ...
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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 ...
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2 votes
1 answer
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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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4 votes
1 answer
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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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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 ...
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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 $...
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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 ...
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2 votes
1 answer
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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 \...
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0 votes
0 answers
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A complex integration formula

I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula: $f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) \operatorname{...
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7 votes
3 answers
340 views

A hypergeometric identity related to Bessel functions

The identity in my recent answer can be stated in a particularly neat form: $${}_2F_0\left({-n, n+1\atop{}};\frac{x}{2}\right) ~\cdot~ {}_2F_0\left({-n, n+1\atop{}};-\frac{x}{2}\right) ~=~ {}_3F_0\...
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6 votes
1 answer
295 views

Spherical Bessel functions. Sum of squares

In (1) there is a property of spherical Bessel functions, which's derivation I can not find in the literature. ${\mathsf{j}_{n}^{2}}\left(z\right)+{\mathsf{y}_{n}^{2}}\left(z\right)=\sum_{k=% 0}^{n}\...
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2 votes
1 answer
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How to compute means μ1 and μ2 knowing sum of Skellam distributions f(k;μ1,μ2) and sum μ1+μ2, where k is from 2 to n?

The probability mass function for the Skellam distribution for a count difference $ k=n_1-n_2 $ from two Poisson-distributed variables with means $\mu_1$ and $\mu_1$ is given by: $$ f(k;\mu_1,\mu_2)=...
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6 votes
0 answers
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Computing the difficult integral $\int_0^\infty J_0(x)^4\log(x)dx$

Computing numerically integrals of oscillating functions from $0$ to $\infty$ is a well-known and difficult problem. Here is an example for which I do not know a solution: I know how to compute (to ...
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2 votes
1 answer
180 views

How to get asymptotic expansion of the sum of modified Bessel function $\sum_{n=1}^\infty K_0(s\, n)$ as $s\to 0^+$?

I guess for the modified Bessel funcion $K_0(z)$, $$\sum_{n=1}^\infty K_0(s\, n) \sim \frac{-2\log 2 - \log \pi + \gamma}{2} + \frac{\log s}{2} + \frac{\pi}{2\, s}, \quad s\to 0^+,$$ if taking $$\...
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  • 21
0 votes
1 answer
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Alternate forms of the Bessel equation

I have a question regarding an alternate form of the Bessel equation and how that alternate form translates to the modified Bessel equation and its solution. The modified form is from: http://...
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  • 3
0 votes
1 answer
296 views

Closed form of integration of modified Bessel function composed with trigonometric function times a linear term

Assuming one draws two points from a von Mises distribution on a circle, I am looking for the expected distance between two such points. Given the pdf of a centered von Mises distribution $$f_X(t \...
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