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Questions tagged [bessel-functions]

Questions related to Bessel functions.

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Finding a general expression for the improper integral $\int_0^\infty K_1( ( k^2+\alpha^2)^{1/2}r)\sin(kz)\,\mathrm{d}k$

$\newcommand{\on}[1]{\operatorname{#1}}$ In solving a classical fluid mechanics problem involving flow in porous media, I encountered a delicate infinite integral ...
Siegfriedenberghofen's user avatar
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Asymptotic estimate from integral representation of modified Bessel function

While trying to prove $$\sum_{n \leq x} d(n) = x \log x + (2\gamma - 1) x + O(x^{1/3} \log x) \tag{1}$$ using Voronoi's formula, I had to use the estimate $$K_0(x) \ll e^{-x} \tag{2}$$ as $x \to \...
Giovanni's user avatar
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Exact evaluation of a sum over product of Bessel functions [closed]

I am interested in the exact evaluation of the sum $\sum_{k=-\infty}^{\infty} \,k\, J_{m-k}(x) J_{n-k}(x) $. Here $k,m,n$ are integers and $J_p(x)$ is the Bessel function of the first kind. A similar ...
AAN's user avatar
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2 answers
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Asymptotics of modified Bessel function of second kind

Let denote $K_\nu$ the modified Bessel function of second kind of argument $\nu\in(0,\infty)$. It is kown that for $x\in\mathbb{R}$, we have: $$K_\nu(x) \sim \sqrt{\frac{\pi}{2x}} e^{-x}$$ as $x\to+\...
NancyBoy's user avatar
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Fourier Transform of product of unit vectors

I have a general integral of the form $$\int_0^{2\pi} d\phi e^{-i F \cos(\phi)} \left(\cos(\phi) \hat{x}+ \sin(\phi) \hat{y}\right)^m$$ That is, in the end, I get bunch of tensors with chains of $\hat{...
Quantization's user avatar
1 vote
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19 views

Definite integral of Modified Bessel function, exponential and trigonometric functions

I am trying to solve the following integral; $$ \int_{0}^{\frac{\pi}{2}} e^{\gamma \cos\theta} I_{1}(\epsilon\sin\theta)d\theta,$$ where $\gamma\in\mathbb{R},\epsilon\in\mathbb{R}^{+},$ and $I_{1}$ is ...
Nelly Clark's user avatar
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Two-dimensional wave equation and Bessel function [closed]

Show that the solution $w(x_1,t)$ to the initial-value problem $\begin{equation}\begin{cases} w_{tt}-c^{2}w_{x_{1}x_{1}}=c^{2}\lambda^{2}w\\ w(x_{1},0)=0, w_{t}(x_{1},0)=g(x_1)=\psi(x_1) \end{cases}\...
logarithm's user avatar
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Hankel transform of $f(r)=\mathrm{erfc}(r)J_{1}(r)/r$

I encountered a Hankel transform dealing with the Ewald method. The Hankel transform of zero order is defined as $$F(k)=\int_{0}^{\infty}f(r)J_{0}(kr)\,r\,dr,$$ where $J_{0}$ is the Bessel function of ...
Wz S's user avatar
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Hankel transform of $f(r)=\frac{\mathrm{erf}(a+r/b)-\mathrm{erf}(a-r/b)}{r}$

I enconter a Hankel transform dealing with the Ewald method. the Hankel transform of zero order is defined as $$F(k)=\int_{0}^{\infty}f(r)J_{0}(kr)rdr,$$ where $J_{0}$ is the Bessel function of the ...
Wz S's user avatar
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Could the product of two Bessel functions of the first kind be expressed in terms of infinite series $J_n(x)J_m(\alpha x)$, where $n,m\in\{0,2\}$?

It is well known that the square of the Bessel function of the first kind of order zero has the Maclaurin series expansion $$ J_{0}(x)^{2} = \frac{1}{\sqrt{\pi}}\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k!}\...
Siegfriedenberghofen's user avatar
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Integral the product of exponential function and Bessel functions [duplicate]

How to solve $\int_{0}^{R} r e^{a r^2}J_{0}(br)dr$. The solution of $R=\infty$ can be found in How do I integrate this exponential + Bessel function term?
wen y's user avatar
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Limiting behavior of integral representation of $(\sqrt{\alpha^2-\partial_x^2}-\alpha)f(x)$

While studying pseudo-differential operators of type $\left(\sqrt{\alpha^{2} - \partial_{x}^{2}}-\alpha\right)\operatorname{f}\left(x\right)$, I came across the following integral representation of ...
Caesar.tcl's user avatar
2 votes
2 answers
107 views

Bessel-like integral, with exponential and trigonometric functions involved

Does anyone know if the following integral can be written in a closed form expression and how? Thank you all; $$I = \int_{\phi=0}^{2\pi} \sin\phi e^{\alpha\cos\phi+\beta\sin\phi} d\phi, \alpha, \beta \...
Nelly Clark's user avatar
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A root finding problem

While solving a partial differential equation, I obtained the following function: $$ F(z)=(k^2 - z^2)J_1^2(z)-z^2J_0^2(z) \tag 1$$ where $k \in (1, \sqrt2)$ and $z$ is a complex variable. I need to ...
FriendlyNeighborhoodEngineer's user avatar
2 votes
0 answers
37 views

Sum of Gaussian weighted modified Bessel functions of the first kind.

through the course of some calculations arising from computing a partition function in the Hubbard Model under certain conditions. I get this expression showing up $$ \mathcal{I}(x,\gamma) = \sum_{n=-\...
rmathguy's user avatar
4 votes
0 answers
179 views

Definite integral involving K Bessel function and a square root

I have recently been trying to evaluate some integrals involving the modified Bessel function $K_0(x)$. The specific integrals are $$L(x,u) = \int_0^{1} K_0\left( 2x \sqrt{r(1-r)} \right) \exp(2ixur) ...
lewismcombes's user avatar
6 votes
1 answer
302 views

Evaluting $\int_0^\infty K_\nu(ax) K_\nu(bx) \sin(cx) dx$ missing from Erdelyi

There is a known integral, which converges for $\text{Re}(a+b)>0$, $c>0$ and $|\mathrm{Re}(\nu)| < \frac{1}{2}$ such that $$ \int_0^\infty K_\nu(ax) K_\nu(bx) \cos(cx) dx = \frac{\pi^2}{4\...
QuantumEyedea's user avatar
0 votes
2 answers
58 views

2 dimensional Fourier transform of $\frac{x}{(x^2+y^2+c^2)^{3/2}}$

I'm trying to calculate the 2D Fourier transform of this function: $$\frac{x}{(x^2+y^2+c^2)^{3/2}}$$, where $x$ and $y$ are independent variables and $c$ is a positive constant, I know the answer ...
Wz S's user avatar
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Is there a closed form for the following integral?

I want to find a closed form of the following integral: $$ I \equiv \int_{0}^{R}\frac{b\operatorname{J}_{1}\left(ax\right) \operatorname{J}_{0}\left(bx\right) + a\operatorname{J}_{1}\left(bx\right)\...
CfourPiO's user avatar
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2 answers
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Integral of Bessel Function [closed]

I am interested in the following integral: $$\int_0^L dx\;\frac{J_0(x)}{x}$$ where $J_0(x)$ is a Bessel function of the first kind. Does the integral converge, and if so is there a simple way to write ...
Sam's user avatar
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1 answer
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Bessel function of first and second kind as solution of an integral

Needing to solve this definite integral $$ \int_{0}^{\infty} \frac{1}{\sqrt{u}}\exp\left[{-\frac{(t-u)^2}{2}} \right] du$$ where $t$ is real, someone make me notice that, although it seems that there ...
umby's user avatar
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0 answers
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how to calculate $\sum\limits_{n=0}^{\infty}\sum\limits_{m=-n}^{n}j_n(r)^2Y_n^m(\theta,\psi)^2$

I am wondering how to calculate $\sum\limits_{n=0}^{\infty}\sum\limits_{m=-n}^{n}j_n(r)^2Y_n^m(\theta,\psi)^2$, where $j_n(r)$ is the Spherical Bessel function, and $Y_n^m(\theta,\psi)$ is the ...
madao's user avatar
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Approximating asymptotically the Laplace inverse of $\frac{\exp\left(-\sqrt{s^2 + 1}\right)}{\sqrt{s^2 + 1}}$ for larger t

I was trying to find the behavior of the inverse laplace transform of the function $$F(s)=\frac{\exp\left(-\sqrt{s^2 + 1}\right)}{\sqrt{s^2 + 1}}$$ for larger $t$. So basically here is my approach: I ...
MB17's user avatar
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2 votes
1 answer
96 views

Convert the Laplace transform of the Bessel function to a Fourier transform

I want to calculate the Fourier transform of the function $f(t)$, defined as $f(t)=0$ if $t<0$ and $f(t)=J_{n}(t)$ if $t\ge0$, in which $J_{n}(t)$ is the Bessel function of the first kind. That is, ...
Lucas Bitencourt's user avatar
1 vote
1 answer
34 views

Jacobi-Anger expansion for non-integer power

One way of writing the Jacobi-Anger expansion is $$ \frac{1}{2\pi} \int_0^{2\pi} e^{i (n \theta - a \sin \theta)} d \theta = J_n(a) $$ for real $a$ and integer $n$. Is there a corresponding formula ...
Bio's user avatar
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3 votes
2 answers
130 views

Asymptotic behavior of a ratio involving Bessel K functions

Consider the ratio $$f(x) = \frac{\textrm{BesselK}^{(2,0)}(0,x)}{\textrm{BesselK}(0,x)}$$ Empirically $$f(x) \sim \frac{1}{x} - \frac{1}{2x^2} + \frac{1}{1.846(5) x^3} + \mathcal{O}\left(\frac{1}{x^4}...
Arthur B.'s user avatar
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0 answers
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Challenging Problem: Showing integral is positive

Does anyone have any advice or help on how to tackle the following problem ?: $$ \mbox{Show that the function}\ \operatorname{f}\left(r\right) = \int_{0}^{\infty}\...
Sabiske's user avatar
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2 votes
2 answers
126 views

Modified Bessel Function and Dirac Delta function

Is there a representation of the Dirac delta function of the form $$ \int_0^\infty \frac{d z}{z} K_{i \nu} (z) K_{i \nu'} (z) = f(\nu) \delta ( \nu - \nu' ) , \qquad \nu,\nu'>0. $$ for some ...
Prahar's user avatar
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3 votes
1 answer
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Compute the integral: $\int_{0}^{\infty}e^{-x}J_{0}(x)dx$ [duplicate]

At first, I was computing the following integral: $$\int_{0}^{\infty}e^{-x}J_0(\sqrt{x})dx$$ which can be easily solved using the taylor series of $J_0(x)$: $$J_0(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{...
Silver's user avatar
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1 answer
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Bessel-look-alike differential equation

This looks like a Bessel differential equation with an extra $x^4,x^6$ term but I am not able to figure out to proceed in solving it. Any idea how to proceed to solve this equation? Any help/reference ...
SiPh's user avatar
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5 votes
1 answer
158 views

Closed form of $ \int_0^\infty dx r J_1(r x) \left [ J_0(x) \right ]^Q $

I'm looking for a closed form for the following definite integral $$ I(r,Q) := r \int_0^\infty dx J_1(r x) \left [ J_0(x) \right ]^Q $$ where $r$ is a positive real, $Q$ is a positive integer and $J_a$...
lcv's user avatar
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1 vote
1 answer
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(Infinite) sum of products of 'offset' Bessel functions (an application to calculating the RMS power of a harmonic "FM" oscillator)

TLDR version: I'm trying to calculate $\sum_{n=1}^{\infty} J_{n-1}(a) J_{-n-1}(a)$ but can't seem to get it right. For motivation & attempt, read the rest. Consider the harmonic "FM" ...
got trolled too much this week's user avatar
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0 answers
53 views

calculate$\sum_{n=-\infty}^{\infty}J_n(\omega)^2e^{jn\psi}$

I am wondering how to calculate the following expression: $$\sum_{n=-\infty}^{\infty}J_n(\omega)^2e^{jn\psi}$$ I have tried to use the Jacobi-Anger Expansion, also the equation below: $$\sum_{n=-\...
madao's user avatar
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0 votes
2 answers
72 views

Prove that for the Bessel function it holds: $J_{3/2}(x) = \sqrt{\frac{2}{\pi}} \left( \frac{\sin(x)}{\sqrt{x^3}} - \frac{\cos(x)}{\sqrt{x}} \right).$

Prove that for the Bessel function it holds: $ J_{3/2}(x) = \sqrt{\frac{2}{\pi}} \left( \frac{\sin(x)}{\sqrt{x^3}} - \frac{\cos(x)}{\sqrt{x}} \right). $ Attempt: To prove the given identity for the ...
user avatar
1 vote
2 answers
81 views

Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?

As the title says, I would like to know if there is a closed form for the integral: \begin{align*} \int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\...
learner123's user avatar
2 votes
0 answers
25 views

Reference request for Bessel function of the second kind with matrix argument

As the title says, I would like to know if anyone could provide a reference which provides the definition and properties of the Bessel function of the second kind with matrix argument. If possible, I ...
learner123's user avatar
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0 answers
22 views

Mellin-Barnes representation of the ratio of first kind Bessel functions

I have a question regarding the Mellin-Barnes representation of Bessel functions of the first kind. I know that the product of these functions admits the following integral representation $J_{\mu}(x)...
Alessandro Pini's user avatar
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0 answers
39 views

Weighted Bessel potential space is a Banach space?

I'm studying weighted $L^p$ spaces. In O. Christensen, Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering (Birkhäuser, Boston, Mass, 2010), weighted $L^p$ space is ...
eraldcoil's user avatar
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0 votes
0 answers
32 views

Use General Properties of Bessel Functions to Solve Bessel Equation

The Bessel Functions $C_\nu (z) = J_\nu(z), Y_\nu(z), H_\nu ^{(1)}(z), H_\nu ^{(2)}(z)$ fulfil the properties (see e.g. A. Apelblat, "Bessel and Related Functions - Volume 1 Theoretical Aspects&...
Tütü's user avatar
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2 votes
0 answers
60 views

Out-of-phase exponential integral of cosine and double-angle cosine

I am trying to solve the following integral: $${\frac {1}{2\pi }}\int _{0}^{2\pi }e^{z\cos(2\theta )+y\cos (\theta+\theta_0) }d\theta$$ This is similar to this fairly standard integral that can be ...
Beno94's user avatar
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0 votes
0 answers
22 views

Can we prove the monotonicity of a function which involves modified Bessel function of the second kind?

I find in Matlab that function $$f(x)=\frac{c}{x}+\frac{K^\prime_\lambda(\sqrt{x})}{K_\lambda(\sqrt{x})}$$ for $c>0$, when $\lim_{x\rightarrow0}=+\infty$, this function is strictly decreasing at ...
Paul's user avatar
  • 1
5 votes
1 answer
205 views

How to prove the result of the following integral? [duplicate]

How to prove that $$ \int _0^{\infty }\frac{K\left(\frac{1}{2}-\frac{1}{2 \sqrt{x+1}}\right)}{\sqrt[4]{x+1}}e^{-x}{\rm d}x = \frac{1}{2} \sqrt{e \pi } K_{1/4}\left(\frac{1}{2}\right) $$ where $K(x)$ ...
Jie Zhu's user avatar
  • 239
3 votes
1 answer
71 views

How to evaluate this definite integral in terms of Bessel functions.

In the context of Green's functions for the Free Klein-Gordon field, the following integral occurs: $$\int_m^{\infty}{\rho e^{-\rho r}\over\sqrt{\rho^2-m^2}}\; d\rho.$$ Here $m$, is a positive ...
Albertus Magnus's user avatar
0 votes
1 answer
20 views

Smoothness of Matérn kernel

I want to find the number of smoothness of the Matérn Kernel, which means the maximum $s$ such that the function is $s_*=\lfloor s -1\rfloor$ times differentiable and the last derivative is $(s-s_*)-$...
Davide Maran's user avatar
  • 1,199
0 votes
1 answer
115 views

Identify the special function with this sum and integral form

There is a multivariate generalization of the Bessel function that has both a sum and integral form. Both are functions of a vector $\mathbf{0}\leq\mathbf{x}\in\mathbb{R}^n$, with parameters defined ...
Victor V Albert's user avatar
0 votes
1 answer
73 views

closed form of $\sum_{n=1}^\infty \frac{J_{2n-1}(z)}{2n-1}$

Does the following infinite series have a closed form: \begin{equation} \sum_{n=1}^\infty \frac{J_{2n-1}(z)}{2n-1}? \end{equation} Here, $J_n$ is the Bessel function. (If the denominator does not ...
user1239110's user avatar
2 votes
1 answer
97 views

these pde's and the Dirichlet divsor problem

I noticed that $$t^2 \frac{\partial^3}{\partial t^3}\Delta_t(s)+s^2 \frac{\partial}{\partial s} \Delta_t(s)=0 $$ is satisfied by $$\Delta_t(s)= - \sqrt{\frac{t}{s}}Y_1{(4\pi\sqrt{ts})}+ \sqrt{\frac{t}{...
zeta space's user avatar
1 vote
0 answers
120 views

How to evaluate the integral $\int_0^\pi e^{i(a\sin x + b\cos x)} dx$

We know that (from e.g. here) $$ \int_0^{2\pi}{\rm e}^{{\rm i}\left[a\sin\left(x\right) + b\cos\left(x\right)\right]}{\rm d}x = 2\pi\operatorname{J}_{0}\left(\sqrt{a^{2} + b^{2}}\right) $$ where $\...
userflux9674's user avatar
2 votes
0 answers
138 views

Divisors sum and Bessel Function related sums

Discovered the following relation: $$\sum _{k=1}^{\infty } \sigma (k) \left(K_2\left(4 \pi \sqrt{k+y} \sqrt{y}\right)-K_0\left(4 \pi \sqrt{k+y} \sqrt{y}\right)\right)=\frac{\pi K_1(4 \pi y)-3 K_0(...
Gevorg Hmayakyan's user avatar
1 vote
1 answer
103 views

Inverse Mellin transform of $\Delta(t):=\sum_{s=1}^\infty d(s)\sqrt{\frac{t}{s}}M_1(4\pi \sqrt{ts})$

Define $$ \Delta(t):=\sum_{s=1}^\infty d(s)\sqrt{\frac{t}{s}}M_1(4\pi \sqrt{ts}) $$ where $M_1(z)=-Y_1{(z)}-\frac{2}{\pi}K_1(z)$ and $d(s)$ is the divisor function. What is the inverse Mellin ...
zeta space's user avatar

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