Questions tagged [bessel-functions]

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Infinite sum of even Bessel functions - Identities

Recently, I came across the following identities among first-kind Bessel functions, namely $$ 2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left[x\,J_1(x)-J_0(x)\right] \label{1}\tag{1} $$...
Alessandro Pini's user avatar
3 votes
2 answers
217 views

Question about the Bessel operator

For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by \begin{equation*} L_\...
Tony419's user avatar
  • 401
2 votes
2 answers
159 views

Integral of nested trigonometric function $\frac{\cos\left(p\cos\left(x\right)\right)}{p\cos\left(x\right) + c}$

while still working on my problem $\int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x$ I came across the following definite integral \begin{equation} \int_{0}^{\pi}\frac{\...
Dennis Marx's user avatar
2 votes
0 answers
141 views

Integral of product of zeroth-order bessel functions times cosine $\int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x$

I am new to Bessel functions and need to solve the following integral \begin{equation} \int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x \end{equation} with $J_{0}$ ...
Dennis Marx's user avatar
2 votes
1 answer
180 views

Sum of Bessel function with integer parameters and fixed argument

Question. Let $J_{\nu}$ be a standart Bessel function of the first order. What is the asymptotic of the sum $\sum_{n\ge 0}|J_n(t)|$ as $t\to\infty$? An upper bounds stronger than $O(t)$ are also ...
Pavel Gubkin's user avatar
3 votes
1 answer
212 views

Closed-form expression for definite integrals involving modified Bessel functions K_1 and K_0

I am attempting to derive a closed-form expression for the following two integrals involving the modified Bessel functions $K_1$ and $K_0$, but I can't find a solution (I don't know if there is one). ...
vbarros's user avatar
  • 43
3 votes
0 answers
48 views

Matrix argument K Bessel functions at half integral orguments

As a working definition I will define: $$ K_\nu^{(n)} (z) = \frac{1}{2^n}\int_{\mathcal{P}} e^{- \operatorname{tr}( y + y^{-1}) z/2} \det y^\nu d \mu(y) $$ where $\mathcal{P}$ represents the space of ...
Max K's user avatar
  • 31
1 vote
1 answer
131 views

$n$th Derivative of $_p F_q(a_1,...,a_p; b_1,...,b_q;x^{-m})$, $p \le q$

Maple seems to suggest the following formula for $n>0$, $p \le q$: \begin{align} \frac{d^n}{d x^n} & {}_p F_q (a_1,\ldots,a_p;b_1,\ldots,b_q;1/x) \\[8pt] = {} & (-1)^n \hspace{1pt} n!\...
japalmer's user avatar
  • 141
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0 answers
36 views

Bessel functions of matrix argument in the scalar case

Herz (1955) provides the following equality: $$ A_\delta(-\lambda) - A_\delta(-\lambda)\lambda^{-\delta} = -\sin(\pi\delta)B_\delta(\lambda)/\pi $$ where $A_\delta$ and $B_\delta$ are the Bessel ...
Stéphane Laurent's user avatar
1 vote
1 answer
204 views

Approximation for a Bessel function integral

I'm trying to calculate hit probabilities on a dart board if the dart thrower has some Gaussian angle distribution function with width $\Delta$ and some systematic angle offsets $\phi_0, \theta_0$. ...
mzw's user avatar
  • 19
0 votes
0 answers
83 views

A bound for the Bessel function of the first kind J_0

I have proved the following bound for the Bessel function of the first kind: $$ J_0(x)=\sum_{m=0}^\infty \frac{(-1)^m\,(x/2)^{2m}}{(m!)^2} $$ which is $$ |J_0(x)|\le \frac1{\sqrt[4]{1+x^2}} $$ but I ...
van der Wolf's user avatar
1 vote
0 answers
39 views

Understanding a Bessel function gluing argument of Simon

I would like to construct a real-valued function $f$ on $(0, \infty)$ with the following properties: $f(r)$ is $C^1$ on $(0,\infty)$ and $C^\infty$ on $(0,1) \cup (1, \infty)$, $-f'' + \tfrac{3}{4}r^...
JZS's user avatar
  • 469
4 votes
1 answer
359 views

Derive the solution of the diffusion equation from the solution of a random walk

Summary The probability distribution (pdf) of a random walk in 1 dimension is represented by a Bessel function. On the other hand, the pdf of a Brownian motion in free space is represented by a ...
Sam's user avatar
  • 281
5 votes
2 answers
355 views

Sum over Bessel functions: what is $\sum_{n=1}^\infty J_n(u)J_n(v)\sin(n\alpha)$?

By Neumann's addition theorem, we know that the following identity holds (including for complex $\alpha$): $$J_0(u)J_0(v)+2\sum_{n=1}^\infty J_n(u)J_n(v) \cos(n\alpha) = J_0 \left( \sqrt{u^2+v^2-2uv \...
Alex Lupsasca's user avatar
0 votes
0 answers
287 views

Is $\sum\limits_{k=0}^{\infty}\frac{1}{(mk)!^{n+1}}$ irrational?

I was using Wolfram Alpha for things, and I came across $I_{0}(2)$. For fun I tried asking Wolfram Alpha if the number was irrational, but said it's unknown. I believe this is an error, as its ...
Frax's user avatar
  • 101
2 votes
0 answers
184 views

Power series of the modified Bessel function of the second kind

I am looking for a power series representation of $$ \frac{1}{K_{\nu}(x)}, $$ where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer. I know that ...
esner1994's user avatar
0 votes
0 answers
69 views

Fourier transform of an exponential function with radical argument divided by a radical

I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...
Ft insat's user avatar
5 votes
0 answers
180 views

Proximity of zeroes of Bessel functions

I have been running into a question for which I found no reference in the litterature. I do not have a strong background in number theory ; for me this is motivated by a question in PDEs (how close ...
Micka's user avatar
  • 51
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0 answers
102 views

Solution to the integral of Bessel $\int^1_0 x \sin(a x) J_1 (b x) dx$

I've been trying to work out the solution of this integral. I have seen in the Gradshteyn (6.669(9)) a similar integral: \begin{equation} \int_0^1 x^\nu \sin(a x)J_\nu(a x)dx= \frac{1}{2\nu+1}\left[\...
Brian Isaac's user avatar
15 votes
1 answer
598 views

Fourier's proof of reality of all roots of Bessel function $J_0(x)$

In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real. I want to ask if there is a modern version of this proof exist in literature? If someone ...
TPC's user avatar
  • 690
1 vote
1 answer
152 views

Integral involving Bessel and Laguerre function

Is there a formulas for the following integral $$\int^\infty_0 e^{-ar^2}L^1_k(b r^2)J_1(cr)r^d dr $$ where $L^1_k$ is the Laguerre polynomials of type 1 and $J_1$ is the Bessel function with $a,...
Ryo Ken's user avatar
  • 11
0 votes
0 answers
25 views

Bessel equation with another type of weight

A research problem led me to consider the following modification of Bessel's equation: $$ - \xi''(r) - \frac{N - 1}{r} \xi'(r) - \left(f(r) - \frac{\theta}{r^2} \right)\xi(r) = 0 \quad \text{in} \quad ...
Danilo Gregorin Afonso's user avatar
2 votes
2 answers
257 views

Definite integral of Bessel function of the first kind times $x^{3/2}$

I am looking for preferably a closed form (or series solution if not possible) for the following integral: $$\int_0^a x^{3/2} J_\nu (bx) dx$$ where $\nu$ is an integer. This 1D integral appears when ...
Alex's user avatar
  • 73
3 votes
0 answers
191 views

How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials

From numerical experiments in Mathematica, I have found the following expression for the integral: $$ \int_{-1}^{1}h_{n}^{(1)}\left(\sqrt{a^{2}+b^{2}+2ab\tau}\right)P_{n}^{m}\left(\frac{a\tau+b}{\sqrt{...
Chris's user avatar
  • 31
4 votes
0 answers
170 views

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 ...
user1029664's user avatar
2 votes
1 answer
191 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)|...
stoic-santiago's user avatar
1 vote
0 answers
35 views

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 $n,a,b$ ...
Rekha K.'s user avatar
1 vote
0 answers
58 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 ...
geocalc33's user avatar
2 votes
1 answer
154 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)]\...
Glxblt76's user avatar
  • 123
2 votes
1 answer
273 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_{...
Max Muller's user avatar
  • 4,485
1 vote
0 answers
66 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 \\&...
Maxim Lyutikov's user avatar
2 votes
1 answer
150 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^...
Corne Koks's user avatar
3 votes
0 answers
154 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})....
Alexander Kalmynin's user avatar
0 votes
1 answer
184 views

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 ...
user808843's user avatar
2 votes
0 answers
144 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]\...
baderi's user avatar
  • 21
2 votes
1 answer
298 views

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 ...
fbrx's user avatar
  • 21
7 votes
1 answer
255 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): $$...
schade96's user avatar
10 votes
1 answer
421 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 \, ...
Timothy Budd's user avatar
  • 3,545
2 votes
2 answers
133 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,\...
Tony419's user avatar
  • 401
1 vote
0 answers
118 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 $J_\nu(t)$....
SitnikSergei's user avatar
0 votes
0 answers
115 views

Can the Bessel functions tend to a plane wave?

Can the Bessel functions tend to a plane wave? If I have this function: $$ y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6) $$ ...
Gin's user avatar
  • 23
0 votes
2 answers
703 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(\...
user avatar
6 votes
2 answers
693 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, ...
mape's user avatar
  • 63
2 votes
2 answers
274 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 ...
Paresseux Nguyen's user avatar
1 vote
0 answers
558 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 \...
Max Muller's user avatar
  • 4,485
1 vote
0 answers
71 views

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)...
Adam Hammam's user avatar
2 votes
1 answer
350 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}^{\...
Ale's user avatar
  • 21
4 votes
0 answers
184 views

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 ...
Dianbin Bao's user avatar
3 votes
0 answers
291 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 ...
Dianbin Bao's user avatar
3 votes
0 answers
157 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(\...
Dianbin Bao's user avatar