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Questions tagged [bessel-functions]

Questions related to Bessel functions.

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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{...
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1answer
28 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}) ...
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39 views

PDE Linearization

I'm trying to linearize the following PDE:$$\frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)+\rho_T\left(\frac{\partial z(r,t)}{\partial ...
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34 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{...
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1answer
11 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 ...
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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 ...
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1answer
33 views

Infinite Integral of a Product of Bessel Functions

I am interested in any analytic information about the following integral: $i^{4m+1} \int_0^{\infty} t^{1/4} J_m^4(t) J_{\nu}(\alpha t) dt$ where $i = \sqrt{-1}$ is the imaginary unit $m$ is a ...
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Compute $\int_0^\infty e^{-az} \sum_{n=0}^\infty \frac{z^{n+2}(u+z)^n}{(n+1)!(n+2)!} dz$

I got stuck in this problem: $$\int_0^\infty e^{-az} \sum_{n=0}^\infty \frac{z^{n+2}(u+z)^n}{(n+1)!(n+2)!} dz ~~(1)$$ where $a>0$. My thoughts: By Binomial expansion we have $$ \int_0^\infty e^{-...
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1answer
47 views

How to express cosine of Fourier series as Fourier series again

I have the following Fourier series exapansion: \begin{equation} \phi(t) = a_0 + \Sigma_{n=1}^\infty (a_n\cos pnt + b_n\sin pnt). \end{equation} I want to express $\cos(\phi(t))$ as Fourier series ...
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Laplace transform of products of Marcum Q, Bessel J and power functions

I want to solve this integral likes: $\int_{0}^{\infty} Q_{\mu_1}\left( \alpha, \beta\sqrt{t}\right) t^{\frac{\mu_2-1}{2}}J_{\mu_2-1} \left(c\sqrt{t}\right) \exp(-pt)dt$ where $Q(\cdot)$ is ...
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1answer
44 views

How to find the exact solution of a Sturm Liouville form, 2nd order ODE?

I have a second order ordinary differential equation reducible to Sturm Liovelle form. The equation is given by $\frac{f(x)}{x^2} - (\frac{1}{x}+x)f'(x) + f''(x) =0$ and boundary conditions are : $...
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2answers
60 views

Integral of a bessel function composed with a trigonometric function

Is there any standard way or approximated way to calculate an integral of the form $$ \int_0^T J_n^2[a\cos(2\pi t/T)]dt $$ where $J_n$ is the bessel function of first kind of order $n$?
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1answer
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Bessels Function at non-zero $x$ are entire function of $\alpha$?

I found the following statement from wikipedia about Bessel functions $J_{\alpha}(x)$ but have no idea how to prove it. If $x$ is held fixed at a non-zero value, then the Bessel functions are ...
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1answer
27 views

Integral of the product of Bessel functions of different kinds, but with same argument

I am looking to finding the solution to indefinite integrals \begin{align} \int z J_m(a z) Y_m(a z)\, \text{d}z, \end{align} and \begin{align} \int z I_m(a z) K_m(a z)\, \text{d}z \end{align} I know ...
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40 views

Diverging Integral with Bessel Function

I am looking for the solution to the integral: $$\int_{a}^{\infty} x J_n(\alpha x)\;dx$$ where $a<< \alpha$ and $n$. I get something out of Mathematica for $a=1,2,0.1...$ in terms of the ...
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Derivation and integral of Bessel's function

Let $f(x)$ function define by $$ f(x)=x^m e^{-bx}K_{n+1}(ax)^{'} $$ where $K_v(⋅)$ is the $v$-th order modified Bessel function of the second kind and $L^{'}(x)$ is the derivation of function $L(x)$...
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1answer
50 views

Alternate forms of 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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Analytical solution to a nonlinear second order PDE

I am trying to determine an outer boundary condition for the following PDE at $r=r_m$: $$ \frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,...
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On the expected distance of two von Mises distributed random variables

About this thread I opened a couple of threads here and at MathOverflow regarding the following question, but always put in a different context. Since there has been no satisfactory answer yet, I ...
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1answer
44 views

How to determine series expansion for $x^2 \left[ K_{i\alpha+1}(x) K_{i\alpha-1}(x) - K_{i\alpha}(x)^2 \right] $ for small $x>0$?

Supposing that $\alpha>0$ and $x>0$, define the function: $$ f(x,\alpha) = x^2 \left[ K_{i\alpha+1}(x) K_{i\alpha-1}(x) - K_{i\alpha}(x)^2 \right] $$ where $K_{i \alpha}(x)$ is the modified ...
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Closed form of $\int_0^\infty \sin(x)\sin\left(\frac{1}{x}\right)dx$?

I have stumbled onto an interesting integral$$\int_0^\infty \sin(x)\sin\left(\frac{1}{x}\right)dx$$ which I noticed graphically that it appears to be $1$, but I have no idea on how to evaluate it. ...
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2answers
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Can Bessel functions be represented as a single function with two variables?

The typical way to represent a Bessel function of first kind is $ J_{\alpha}(z)$, i.e. $ J_{\alpha}: \mathbb{C}\to \mathbb{C}$. Is there any good reason that prevents us to write it as a function of ...
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1answer
218 views

Infinite summation formula for modified Bessel functions of first kind

I was trying to find a closed form for the integral $$4\int_0^{\pi/2} t \, I_0(2\kappa\cos{t}) dt \; ,$$ where $$I_{\alpha}(z) := i^{-\alpha}J_{\alpha}(iz) = \sum_{m=0}^{\infty}\frac{\left(\frac{z}{...
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1answer
61 views

Laplace transform of “shifted” modified Bessel function

Dear all: i'm trying to derive a closed form the following integral, $$ X_n(R)=\int_0^\infty \exp(-p\, r)I_n(\omega (r+R))\, dr, $$ where $I_n$ is the standard modified Bessel function of the first ...
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1answer
38 views

What is the Fourier transform of the 2 dimensional airy function?

What is the Fourier transform for the given two dimensional airy function, $$f(x,y) = \frac{J_1(r)}{r}\,.$$ Where $J_1$ is the Bessel function of the first kind, order one. And $r=\sqrt{x^2+y^2}$. ...
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1answer
74 views

Bernoulli Polynomial BesselJ expansion

I have been reading a paper on classes of polynomials and it gives the following series: $$J_{\nu }(x)=\sum _{n=0}^{\infty } \frac{\left(x^{\nu } B_n\left(x^2\right)\right) \left(\frac{(-1)^{n+2} 2^{-\...
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1answer
92 views

Is there a name for the integral $\int_1^\infty e^{-a(x+\frac{b}{x})}dx$?

It is known that the integral $\int_0^\infty e^{-a(x+\frac{b}{x})}dx$ where $a$ and $b$ are two constants is BesselK-like function. Is there a name for the integral $\int_1^\infty e^{-a(x+\frac{b}{x})...
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166 views

Is there a closed form for the integral $\int_0^\infty \frac{e^{-x^2} I_0 \left(\beta x \right) d x}{\sqrt{ \alpha^2+x^2}}$

I encountered this integral in my work, and it would be really convenient if it had a closed form in terms of any known special functions (which Mathematica could handle): $$J(\alpha,\beta)=\int_0^\...
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34 views

Bessel equation solution in $(-\infty,0)$

Let $x \in \mathbb{R}$. Homogeneous Bessel differential equation $$x^2 \frac{\mathrm{d}^2 f(x)}{\mathrm{d}x^2} + x \frac{\mathrm{d} f(x)}{\mathrm{d}x} + (x^2 - n^2) f(x) = 0 \label{a} \tag{1}$$ has ...
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40 views

Definite integrals involving Bessel functions

I am looking for a closed-form expression for the following integral: $$ I_1 = \int_{0}^{\infty} J_{2m} \left( a x\right) x \dfrac{ e^{-j \sqrt{k^2-x^2} b}}{ \sqrt{k^2-x^2}} \textrm{d} x $$ ...
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1answer
49 views

Inverse Laplace Tranform of a function involving Bessel functions

I need to evaluate (if it exists) the inverse Laplace transform of the following complex function $F(s)$: $$ F(s)=\sqrt{\frac{s}{a}} J_{1}(\sqrt{as}) $$ where $J_{1}(\cdot)$ is the Bessel function of ...
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Approximating Modified Bessel functions of the first kind

I have come across expressions such as $I_\nu(x_1x_2)$ where $x_1,x_2$ are positive reals, $\nu$ is non-negative integer, and $I$ is the modified Bessel function of the first kind. I was wondering if ...
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Convolution between modified Bessel function and sinc function

Would it be possible to calculate the following convolution analytically $$ K_0(\xi r) * \frac{\sin(r)}{r}, $$ where $K_0$ is a modified Bessel function of the second kind and $*$ denotes ...
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Bessel integral invovling algebraic and hyperbolic functions

I am desperate in evaluating the following Hankel transform $$ \int_{0}^{\infty} \frac{J_0(kr)}{k^2+\xi^2} \frac{\cosh(ky)}{\cosh(k)} k\mathrm{d} k, $$ where $J_0(kr)$ is the Bessel function of ...
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1answer
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Domain of validity for the asymptotic expansion of the Bessel function of the first kind

A very short question here... I'd like to use an asymptotic approximation of a Bessel functions, which I've found given in several places as $J_\nu(z)=\sqrt{\frac{2}{\pi z}}cos(z-\frac{1}{2}\nu \pi -\...
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Neumann theorem for Bessel function of the first kind (arbitrary order)

There is a Neumann theorem, proving that: $J_0 \big(\sqrt{Z^2+z^2 - 2Zz \cos \phi} \big)=\sum_{k=0}^{\infty} \epsilon_k J_k(Z) J_k (z) \cos k \phi$ where $\epsilon_0=1$, and $\epsilon_k = 2$: $k\geq ...
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38 views

Confusion regarding Kelvin functions

I am trying to implement the following equation from this paper and having some troubles in the interpretation of $bei'$ and $ber'$. I understand from the definition of Kelvin functions that for ...
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21 views

Integral with Bessel function and sine.

How to evaluate the integral with Bessel function: $\int_0^{2\pi}J_1(x\sin\theta)\sin^2\theta d\theta$ Thank you in advance.
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Sign of the derivative of the Hankel Function with purely imaginary arguments

Consider the Hankel function of the first kind $H^{(1)}_\nu (z)$. If I restrict $z$ to be a purely imaginary number of the form $ix$ where $x \in \mathbb{R}$ and $x>0$ and let $\nu =0$. We can see ...
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Integral of Bessel function products times Gaussian times algebraic function

I am searching for a closed form solution to the following integral: $ \int_0^\infty J_m(\rho x ) J_m(a x) \frac{x} {x^2 + c^2} e^{-d x^2} \, dx. $ The tables from ...
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1answer
44 views

Damped vibrations of a membrane stretched over a circular frame

I am given this following PDE with the initial and boundary conditions with $0 < r < 1$, $t > 0$, and $v_0$ being a constant: $u,_t,_t + 2bu,_t = u,_r,_r + \frac{1}{r} u,_r$ $u(t,r=0) = 0, \...
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1answer
51 views

Can we refine this asymptotic for Laguerre polynomials?

I just found an interesting and useful limit for Laguerre polynomials: $$\lim_{n \to \infty} L_n \left( \frac{2r}{n+1/2} \right)=J_0(2 \sqrt{2r})$$ I'm using specifically this form of the argument ...
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1answer
81 views

Why a solution for $c^2 \Delta u = u_{tt}$ must have eigenfunctions as its series terms expansion?

I'm reading this which explains how to arrive at a solution for $u$ as a series expansion involving $J_0, J_1,\cdots$ which are Bessel Functions. It concludes at page 5 saying that $u$ is the ...
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2answers
256 views

Double integral with Hankel transform

Let's say we have a double integral in the following form: $$I=\int_0^\infty \int_0^\infty f(x) g(y) J_0(xy) x y dx dy $$ Using the definition of the Hankel transform, we can write: $$I=\int_0^\...
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1answer
75 views

Find the general solution in terms of Bessel functions: $t^2x'' + x' + x = 0, \quad t < 0, \text{ Hint: } s = 2\sqrt{t}$

I was asked the following question: Find the general solution in terms of Bessel functions: $$t^2x'' + x' + x = 0, \quad t < 0, \text{ Hint: } s = 2\sqrt{t}$$ My approuch I think that what I ...
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1answer
71 views

2D Fourier of Bessel

I need help with the fourier transform of Bessel function of first kind on 2 dimensions. $$G(w_1,w_2) = F[J_0(a\sqrt{x^2+y^2})]$$ where $J_0$ is the bessel function of first kind of order 0, and $a$ ...
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0answers
49 views

Bounds for the error of this approximation to the Bessel function

I found a nice explicit approximation to the Bessel function today, using the integral: $$J_0(x)=\frac{2}{\pi} \int_0^1 \frac{\cos x u}{\sqrt{1-u^2}}du$$ With Chebyshev-Gauss quadrature we can see ...
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1answer
120 views

Fourier transform of “hyperbolically distorted” Gaussian / Bessel-type integrals

Dear Math enthusiasts, I'm trying to see if I can find an analytical expression for the Fourier transform of a Gaussian pulse $p(\tau) = {\rm e}^{-B\tau^2}$ that is distorted by a hyperbolic ...
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0answers
53 views

Integral of two Bessel functions product times Gaussian

Does anyone have a clue about how to solve this integral? Will it have a closed form? $\int_0^\infty e^{-x^2}J_n(ax)J_n(bx)dx$ I've been searching materials and papers for a while, and did find ...
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0answers
86 views

Integrating triple product of Bessel functions over a finite domain

Just in case there's any way to simplify this integral or at least transform it to something which is easy to integrate numerically: $$I(a,b,c)=\int_0^1 r J_0(ar)J_0(br)J_0(cr)dr$$ I'm interested in ...