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Questions tagged [bessel-functions]

Questions related to Bessel functions.

2
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1answer
43 views

Bessel differential equation from integral

It is a relatively well-known fact that $$\int_{0}^{2\pi}e^{-ikr\cos\theta}d\theta=2\pi J_{0}(kr),$$ where $J_{0}$ is the Bessel function of the first kind and order zero. I'm trying to show that this ...
0
votes
1answer
18 views

Recurrence relation for Bessel function

How does $2J'_n(x) = J_{n-1}(x) -J_{n+1}(x) $ and $\frac{2n}{x}J_n(x) = J_{n-1}(x) + J_{n+1}(x)$ imply $J'_1(r) + \frac{1}{r}J_1(r) = J_0(r) = \frac{1}{r}\frac{d}{dr}(rJ_1(r))$ which implies $rJ_0(kr)...
1
vote
1answer
28 views

Verify $J'_0(x)=-J_1(x)$

I need some help with my math coursework. I am given a Bessel function, $x^2y'' + xy' + (x^2 − n^2)y = 0$, and I am told that it has two independent solutions, $J_n(x)$ and $Y_n(x)$. I am told I need ...
0
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0answers
9 views

perpendicular function in a given interval

Considering $Y_n$ and $J_n$ as nth order Bessel functions of the second and first kind, respectively, I want to find a function $f(x)$ such that below proportion holds: $\int_{x_0}^{\infty} f_{a'}(x)...
4
votes
2answers
53 views

Generalizing the solution to an ODE

Is there a way to solve the following ODE for general integral values of $m$ \begin{align} \frac{\partial A(x)}{ \partial x} = -A(x)^m + \frac1x \label{rec}\tag{1} \end{align} I have some ways to ...
0
votes
2answers
60 views

Bessel function identity

I was trying to find this identity of Bessel function $$e^{-2i\gamma t} J_{\left|n\right|}(2\gamma t) = e^{\large \frac{\pi i}{2}} \sum_{k=|n|}^{\infty} \frac{(-i\gamma t)^k}{k!}\binom{2k}{k-n}$$ on ...
1
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2answers
86 views

Inverse Fourier Transform of a half-integer Bessel function

Is there an analytical solution for the following inverse Fourier transform? $$f(x)=\frac{1}{\sqrt{2\pi}}\,\mathrm{j}^n \int_{-\infty}^\infty\frac{1}{\sqrt{k}\,(k^4-\lambda^4)}\mathrm{J}_{n+\frac{1}{...
0
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0answers
25 views

Modified Bessel function of first kind

While reading a research article, I come accros this expression: $$ {\sum\limits_{m = 0}^\infty {\left( {\frac{{p - \sqrt {{p^2} - {\alpha ^2}} }}{{\alpha \beta }}} \right)} ^m} = \frac{\lambda }{{{\...
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0answers
16 views

Green's function in cylindrical coordinates, a small question

Here is a nice derivation for Green's function of a Laplacian in cylindrical coordinates. For the $r$ coordinate, the equation looks like this: $$\frac{1}{r}\,\frac{d}{dr}\!\left(r\,\frac{dg_m}{dr}\...
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1answer
67 views

How can I compute these hard integrals? [closed]

$$\int_0^1 \sqrt{1-x^2} \cos kx \, \mathrm dx$$ $$\int_0^1 \sqrt{1-x^2}\sin sx \, \mathrm dx $$
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1answer
24 views

Hermite polynomials, prove the solution [closed]

$ \text { The Hermite polynomials, } H_{n}(x) \text { , satisfy the following: } $ \begin{array}{l}{\text { i. }<H_{N}, H_{M}>=\int_{-\infty}^{\infty} e^{-x^{2}} H_{n}(x) H_{m}(x) d x=\sqrt{\...
0
votes
1answer
15 views

Integration of modified bessel function $\int_{x_1-\sigma}^{x_1+\sigma}x\exp(-t_1x^2)\cdot I_0(t_2x)\ dx$

In fact I was originally trying to evaluate the following integral \begin{align} \int_{x_1-\sigma}^{x_1+\sigma}\int_0^{2\pi}x\exp\left(-c\left(x\sin\phi-a\right)^2-c\left(x\cos\phi-b\right)^2\right)\ ...
6
votes
1answer
72 views

Inconsistency of limits

Let $I_n(x)$ and $L_n(x)$ be the modified Bessel and modified Struve functions of order $n$, respectively. Assuming $x$ is real, I am interested in the following limit: $$ \lim_{x\to\infty} \frac{I_0(...
0
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0answers
54 views

Integral involving a power function and $\exp(-a/x^2-b x^2)$.

This is a generalization of Double integral containing $e^{(b+ic)/z^2}$ . Let us take $a >0$, $b>0$. The following results are well known: \begin{eqnarray} \int\limits_0^\infty e^{-\frac{a}{x^2} ...
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0answers
10 views

Fourier bessel's expansion V/S Fourier series?

I'm currently working on a project involving Fourier Bessel's Expansion. When I try to solve the Coefficients using substitutions of bessel's functions as in the attached picture. I observe that the ...
0
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0answers
15 views

Proof of the Normality of Bessel

This is a my proof of the normality of bessel functions . But I want to check it and I prefer another proof if exist. Help me please. The Bessel function $J_{\nu}(x)$ satisfies the following ...
0
votes
2answers
26 views

Identity for the Bessel function

I'm sure this is easy, but I can't find it yet. What is $J_n(-x)$? Where $J_n$ is the usual Bessel function of integer order. In particular, I'm looking to the sum $\sum_{n=-\infty}^\infty J_n(-x) = ...
4
votes
1answer
71 views

What is the Puiseux series of the Bessel function $J_n(n)$?

By numerical experimentation I find the first three terms of the Puiseux series of the Bessel function of the first kind $$ J_n(n) = \frac{\Gamma(\frac13)}{2^{2/3}\cdot 3^{1/6} \cdot \pi}n^{-1/3} - \...
1
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2answers
66 views

Solve the following differential equation: $xy''-y'+k^2y=0$

It is not an Euler equation. I tried power series, but I can't develop them around $x=0$. I also tried dividing Bessel's equation by $x$, but it doesn't seem to be possible to arrive at my equation. ...
0
votes
3answers
54 views

Solve differential equation: $y''-\dfrac{1}{x}y'+\dfrac{\alpha^2}{x^2}y=0$

The Bessel differential equation can be written like this $$y''+\dfrac{1}{x}y'+(1-\dfrac{\alpha^2}{x^2})y=0$$ and one of the linearly independent solutions is the bessel function of the first kind, $...
2
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0answers
23 views

Coefficients of the Fourier-Bessel Series if $f(r,\theta)$

This is directly related to my last post General Solution for $u(r,\theta, t)$. I have found that the general solution for $u(r,\theta,t)$ is, $$\sum_{k=1}^{\infty}\sum_{m=1}^{\infty} C_ke^{-(\mu^n_m)...
4
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0answers
41 views

General Solution for $u(r,\theta, t)$

I am trying to find a general solution for the PDE $$\frac{\partial u}{\partial t}=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2 u}{\...
1
vote
2answers
61 views

Please compute $\int_0^{\infty}e^{-(x+ai)^2}dx$, where, $i^2=-1$.

I'm trying to do the following Fourier (Hankel?) transform for a cylindrincally symmetric function: $$ \int_0^{\infty} \!\int_0^{\infty} \! \left( Are^{-(r^2+z^2)/\delta^2}\right)J_1(k_r r) e^{-ik_z ...
0
votes
0answers
23 views

Bessel function limit

I was doing a physics problem and encountered with following limit of Bessel function : $\lim_{R\to\infty} R^2J_n(\lambda R)$ and $\lambda = \sqrt{\omega^2 - \frac{1}{R^2}}$ I got this limit from ...
3
votes
0answers
43 views

Solving an ODE to find the general solution as an infinite series for $u(r,t)$

I am trying to solve the following problem. I have the two ODEs \begin{align} T'+\alpha\lambda T&=0 \\ \frac{d}{dr}\left(r\frac{dR}{dr}\right)+\lambda rR&=0, \end{align} with boundary ...
1
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1answer
24 views

Uniform convergence of Bessel series - question concerning proof in Watson 3.13

Good afternoon, I am currently studying on Bessel functions using Watson's Treatise on the Theory of Bessel functions. In chapter 3, the series defining the Bessel function of first kind for ...
0
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0answers
30 views

Spherical expansion of an exponential function?

We know that a normal planewave can be Rayleigh expanded by spherical harmonics as$$e^{i\vec k·\vec r}=4π\sum_{l=0}^∞\sum_{m=-l}^li^lj_l(kr)Y_{lm}(\hat{\vec k})Y_{lm}^*(\hat{\vec r}).$$ Does any body ...
1
vote
1answer
66 views

Integral involving Bessel function and exponential

In the course of a complicated calculation, I encountered the following simple-looking integral $$ \Phi(\beta)=\int_\beta^\infty \frac{dz}{z}e^{-\beta z}I_1(z)\ , $$ where $I_1(z)$ is a Bessel ...
0
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0answers
16 views

Finding an integral to solve wave propogation in circular membrane

I am trying to derive the analytical solution of $u(r,t)$ for a drum struck at it's centre with rigid termination at the boundary. I used this answer to get to the final general solution, which can be ...
0
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1answer
28 views

Solve the Bessel differential equation

Show that $J_{n}(x) / x^{n}$ is a solution of $$\frac{d^{2} y}{d x^{2}}+\left(\frac{1+2 n}{x}\right) \frac{d y}{d x}+y=0$$ and that $\sqrt{(x)} J_{n}(k x)$ is a solution of $$\frac{d^{2} y}{d x^{2}}+\...
0
votes
1answer
20 views

checking the Solution of Bessel differential equation

I want to check the first part of the solution and help in the second part. Obtain the solution $$y_{1}(x)=J_{0}(x)=1-\frac{x^{2}}{2^{2}}+\frac{x^{4}}{2^{2}\cdot 4^{2}}-\ldots+\frac{(-1)^{n} x^{2 ...
1
vote
1answer
15 views

Tranforming an ODE for $R(r)$ to a Bessel Equation of Order $m$

I have the ODE $$\frac{d}{dr}\left(r\frac{dR}{dr}\right)+\left(k^2 r-\frac{m^2}{r}\right)R=0.$$ I am trying to transform this ODE into the following Bessel equation of order $m$, $$p\frac{d}{dp}\...
0
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0answers
64 views

What is spherical Bessel function $j_\ell (x)$ limit when $\ell \to \infty$?

Very often I encounter formula of the following form $$ j_\ell (x) \sim \sqrt{\frac{\pi}{2\ell+1}}\delta\left(x - \ell-\frac{1}{2} \right), ~~{\rm when}~~ \ell \to \infty,$$ where $j_\ell(x)$ is a ...
4
votes
1answer
78 views

Evaluating the integral $\displaystyle \int_0^{\infty}e^{-\alpha\cosh(u-\beta)}\,e^{-n u}\,du$

I'm trying to calculate the following integral $$\int_0^{\infty}e^{-\alpha\cosh(u-\beta)}\,e^{-n u}du$$ with $\alpha\geq 0$ , $\beta\in \mathbb{R}$ and $n=0,1,2,...$ It seems that may be related ...
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0answers
30 views

Inequality for linear combination of trigonometric functions with coefficients involving integrals of Bessel functions.

I have a linear combination of trigonometric functions where the coefficients involve integrals of Bessel functions. Specifically, I am trying to find the set of constants $A,B,C$ such that \begin{...
2
votes
0answers
27 views

A question over Series expansion of function

My question may appear naif or unclear but let me try the same. Is there any possibility that a sufficiently regular function defined from a domain in $\mathbb{R}^2$ bounded to $\mathbb{R}$, has/...
7
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2answers
376 views

A peculiar integral identity

Here I was, innocently trying to solve this daunting-looking integral $$\int_0^\pi e^{v \cos \theta \cos t} \cosh(v \sin \theta \sin t) dt $$ when the inner beauty behind this beast slowly started ...
1
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1answer
53 views

Solutions for the ODE: $x^2y''+xy'+(x^2-v^2)y=0$

According to this site, if $v$ is non-integer the solution of $$x^2y''+xy'+(x^2-v^2)y=0\tag1$$ is $$y(x)=C_1J_v(x)+C_2J_{-v}(x).\tag2$$ Where $J_v(x)$ is the Bessel function of the first kind. In ...
0
votes
0answers
29 views

Bessel function and delta of dirac

I'm trying to make sense of the next integral, I don't know how to proceed when I take into account the value of $m$ specially when it is equal to 0, and$ n =0 $ as well. $$J_{m}(\rho)=\sum^{\infty}_{...
0
votes
1answer
28 views

What is the relationship between the zeroth-order Bessel function of the first kind and nested cosines?

I'm reading a paper where this equality is claimed: $$\cos(a + b \cos(t) ) = \cos(a) J_0(b),$$ where $a$, $b$ are constants and $J_0$ is the zeroth-order Bessel function of the first kind. How can ...
0
votes
0answers
65 views

Expressing $\int_{0}^{2\pi} e^{ikr(\sin\alpha\cos\beta-\sin\theta\cos(\phi-\beta))}\; d\beta$ as a Bessel function

How to express the trigonometric function $$\sin\alpha\cos\beta-\sin\theta\cos(\phi-\beta)$$ inside the integral to become Bessel function? $$I=\int_{0}^{2\pi} e^{ikr(\sin\alpha\cos\beta-\sin\theta\...
1
vote
3answers
140 views

Improper integral involving logarithm, exponent, and modified Bessel function

Question How can this integral, plotted in Fig. 1, be expressed more simply, for evaluation, perhaps in terms of common special functions? $$H = \int_0^\infty \log(a)\,e^{-a^2}I_0(ba)\,da,\tag{1}$$ ...
0
votes
1answer
23 views

Frequency modulation synthesis side bands

I am trying to understand frequency modulation (applied to sound spectrum synthesis, not radio transmission), and all explanations of side bands I have found make a huge leap. For reference, frequency ...
0
votes
0answers
32 views

Closed form for an integral involving a generalized incomplete Gamma function?

I am trying to find a closed form for this integral: $$\int_{0}^{\infty}\int_{0}^{\infty}e^{-d_{p,s}^v\,x-d_{s,p}^v\, y+d_{p,p}^v\,\frac{\xi_1\,\sigma^2\,xy}{P_p\,\xi_2\,xy+\sigma^2}}\mathrm dy\...
0
votes
0answers
51 views

Resolution complex integral with Matlab

I'm trying to solve an equation proposed by a paper. I can't integrate up to infinite cause the code gives me a warning. I am able to integrate until a value as 2000 or 3000, but the solution differs ...
1
vote
0answers
23 views

Orthogonality Relationship for Spherical Bessel Functions

I begin with Wikipedia's identity \begin{equation*} \int_{0}^{\infty} J_{\alpha}(z) J_{\beta}(z) \frac{dz}{z} = \frac{2}{\pi}\frac{sin(\frac{\pi}{2}(\alpha - \beta))}{\alpha^2 - \beta^2} \end{...
1
vote
1answer
31 views

Interpretation of $\tilde{f}(\mathbf{0})$

Given a function in real space $f(\mathbf{r})$, what is the interpretation of the of the value $\tilde{f}{(\mathbf{0})}$? As an example, take the Fourier transform of \begin{equation} V(\mathbf{r}) ...
2
votes
0answers
53 views

Laplace transform of heat conduction PDE in cylindrical coordinates.

I'm trying apply the Laplace transformation to solve the non-dimensional heat conduction PDE for a hollow cylinder with convection boundary conditions and a non-homogenous initial condition. $$\frac{...
0
votes
1answer
37 views

Asymptotic behaviour of Bessel function of the second kind with a negative order

Is there any result on the asymptotic behaviour of Bessel function of the second kind with a negative order? What I have found is the behaviour when the order $Re(\nu)>0$. For example, it is shown ...
0
votes
0answers
21 views

For $\delta(x) = \lim_{e \to 0} \mu(x,e)$ definition, express $\mu(x,e)$ in terms of Bessel functions

Based on the definition of Dirac Delta as: $\delta(x) = \lim_{e\to 0} \mu(x,e)$ Is it possible to obtain an expression of $\mu$ as a series of Bessel's functions $J_n$ or in which satisfy the ...