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Questions tagged [bessel-functions]

Questions related to Bessel functions.

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Summation of a series which include Bessel function

Is there any way I can compact this summation without using computer. \begin{equation} S = \sum_{n=1}^{\infty}-2(-1)^{n}\frac{J_n(ka)}{H_n^{(2)}(ka)}\cos{[n(\phi-\phi_{0})]} , \end{equation} where $...
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6 views

Modified Bessel function if the half integer order [on hold]

enter image description here can anyone help me with a first step or two to this question?
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1answer
20 views

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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0answers
14 views

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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1answer
31 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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16 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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1answer
33 views

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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21 views

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
42 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
50 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
79 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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217 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
59 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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transform the equation into Bessel equation

$$\frac{d}{dx} \left(x^a \frac{dy}{dx}\right)+bx^h y=0$$ Show that equation can be transformed into a Bessel equation in terms of $t$ and $u$ by transforming both independent and dependent variables ...
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1answer
41 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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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
115 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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47 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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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 ...
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1answer
48 views

Integral representation for the solution of a difference equation

The following is an example from my textbook book on asymptotic studies. How is this true? I've never worked with the Bessel functions directly and the author isn't clear on which definition of $J_0$ ...
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16 views

Using Initial Conditions to find Solutions to Differential Equation Involving Bessel Functions

I began with the two-dimensional wave equation on a half-disc with fixed edges. After some work, I found the solution, $u(x,r,t)$. So now I know that: $$u(x,r,t) = \sum_{n=1}^{\infty}\sum_{m=1}^{\...
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How to integrate the following combination of modified Bessel function of second kind

How to integrate the following where all the constant are positive real numbers(https://i.stack.imgur.com/E4UPy.jpg) $$ \int_0^\infty \frac{K_\mu (a \sqrt{x^2+p^2})}{K_{\mu} (b \sqrt{x^2+p^2})} \cos (...
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1answer
23 views

Expressing the solution of a differential equation in terms of the Bessel equation of order 2

I found this question on a problem set for my Differential equations course: Show that the differential equation $y''+x^{-3/2}y=0$ has a non-trivial solution in terms of Bessel function $J_2(4x^{1/...
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Solution of, an ODE related to the modified bessel equation. [duplicate]

I would like to know the general solution of $$x^2 y^{''}+x y^{'} -k^2 x^2 y=0\tag{1}$$ for $y(x)$. (1) is related to the equation $$\quad x^2 y^{''}+x y^{'} - x^2 y=0$$ , which is the $n=0$ form of ...
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0answers
17 views

Neumann expansion of seemingly simple series

I know there is this Neumann expansion (Watson - A Treatise on the Theory of Bessel Functions p.525) $$ z^\nu = \Gamma(\nu+1) \sum_{n=0}^\infty \frac{z^n}{n!} \, J_{\nu+n}(2z) $$ where $J$ is the ...
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1answer
28 views

Bessel differential equation, Bessel functions and variable sign change

The homogeneous Bessel differential equation $$x^2 f''(x) + xf'(x) + (x^2 - \nu^2) f(x) = 0$$ does not change if $x$ is substituted with $-x$. So, it could be expected that it is the same as regards ...
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1answer
49 views

Hints on evaluating $\int_{0}^{k} x J_{n}^2(x) dx$

I am wondering how to do the integral $$\int_{0}^{k} u J_n^2({u}) du, \ n \in \mathbb{Z}^+,$$ and express my answer in terms of other first kind Bessel functions. I have searched here for useful ...
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53 views

Series expansion of ratio of modified bessel functions

Let $I_{0}(z)$ and $I_{1}(z)$ be modified bessel functions of zero/first order (and first kind). How can I show that for the large value of $z$ it holds $$\frac{I_{1}(z)}{I_{0}(z)}\sim 1 - \frac{1}{...
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Finding relation between the solutions of two PDE's

Suppose $\delta$ is a vector in $\mathbb{R}^d$ with all the components being positive. Assume that $x\in\mathbb{R}^d_+$ and $Q(x)$ denotes the diagonal matrix formed by the vector $x$. It is known ...
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1answer
35 views

Asymptotic property of Bessel function

I am wondering, are these properties of modified Bessel functions correct? $\frac{I_{1}(z)}{I_{o}(z)}\approx 1$ and $\frac{I_{0}(2z)}{I_{o}^{2}(z)}\approx z^{1/2}\pi^{1/2},$ for large $z.$ If so,...
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1answer
55 views

Integrals involving Bessel function

I am getting stuck in two integrals involving Bessel functions and hoping someone to help me out. We know that the Bessel function, $$J_v(x)=x^v\sum_{r=0}^{\infty}\frac{(-1)^rx^{2r}}{2^{2r+v}r!\Gamma(...
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1answer
37 views

integral involves positive definite function and Bessel function

Could the following integral be $0$? \begin{eqnarray*} \lim_{x \to 0^+} \int_0^\pi \vartheta^{\alpha+\frac{1}{2}} \frac{J_{\alpha-\frac{1}{2}} ( x \vartheta )}{x^{\alpha-\frac{1}{2}}} g(\...
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36 views

Mode shapes and frequencies of Sturm-Liouville problem [closed]

I have been asked the following problem in my homework. Given a differential equation $$ xy^{''}+y^{'}+k^{2} y(x)=0$$ with condition $y(1)=0$ and $y(x)$ is finite when $x \to 0$ find mode shapes ...
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1answer
57 views

Limit of a Bessel function is a Gaussian

Let $x$ and $\nu$ be real. Is the following true? $$ e^{-\frac{x^2}{4}}=\lim_{\nu\rightarrow\infty}\Gamma(\nu+1)\left(\frac{2}{\sqrt\nu x}\right)^\nu J_\nu\left(\sqrt{\nu} x\right), $$ It appears to ...
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67 views

can we integrate the following combination of modified Bessel function of second kind.

$$\int_0^\infty\frac{\gamma K_\nu(\gamma(a+x))}{K_\nu(\gamma a)}\,d\gamma$$
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2answers
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The error for approximation of bessel function

The sixth degree polynomial $1-{x^2\over 4}+{x^4\over 64}-{x^6\over 2304}$ Is sometimes used to approximate the Bessel function $J_0(x)$ of the first kind of order zero for $0 \leq x\leq 1$....
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The sum of infinite sum of modified Bessel function

I would like to ask a question about the following problem: $T(r,z)=\sum_{n=1}^{\infty}[C_{1n}I_0(\lambda_nr)+C_{2n}K_0(\lambda_nr)]cos(\lambda_nz)$ in which, $C_{1n}$ and $C_{2n}$ are coefficients, ...
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0answers
16 views

Compute Infinite sum of Zero Order modified Bessel Function with COS

I am looking for the close integral series expression of following Modified Bessel Function. $T(r,z)=\sum_{i=0}^n[C_1I_0(\lambda_nr)+C_2K_0(\lambda_nr)]cos(\lambda_nz)$ I didn't find a closed ...
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124 views

Bessel integral asymptotics

I'm looking at the integral $$I(\alpha,f)=\int_{-\pi}^\pi K_0 \left ( \alpha \sqrt{1+f(x)^2-2f(x)\cos(x)} \right ) dx.$$ Here: $\alpha \gg 1$. $f$ is a smooth, $2\pi$-periodic function. $f(0)=1$. ...
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1answer
52 views

Relations between Bessel functions

Consider the following two couples of functions: Bessel function of the first kind $J_{\nu}$ and Bessel function of the second kind $N_{\nu}$, also known as $Y_{\nu}$; modified Bessel function of the ...
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1answer
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Question on Laplace Transform of a Bessel Integral of the First Kind

I am trying to understand an integral I saw in a book. The book is "Earth Resistances" by G.F. Tagg. Unfortunately it's difficult to get a hold of this book, but I am confused by equation 3.9 is in ...
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49 views

Is there a closed form expression for the first zero of the first Bessel function?

$j_{1,1}$ denotes the first zero of the first Bessel function of the first kind. (That's a lot of firsts!) It's approximately equal to $3.83$. My question is, is there any closed form expression ...
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1answer
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second order differential equation of special function

Bessel function $J_n(x)$ and $Y_n(x)$ obeys the following differential equation: $x^2 y''(x)+x y'(x)+(x^2-n^2)y=0,$ where superscript ' denotes differentiation with respect to $x$. In general, ...
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Definite integration of spherical Bessel function of radical argument

I have to solve some integrals of the form: $$\int_0^{x_0} dx \, j_n( R ) \cdot \frac{p(x)}{R^n}$$ where $R=\sqrt{x^2 + 2 a c x + c^2}$ , $j_n$ is the spherical Bessel function of order n, p(x) is a ...
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Under which conditions the following infinite integral involving a zeroth order Bessel function is convergent / defined?

Everybody hello, The goal is to determine the non-trivial conditions between the real parameters $a$, $b$, and $c$, for which infinite integral below is convergent / defined: $$ \int_0^\infty q J_0(...
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2answers
34 views

Integrating a Bessel Function $K_0(ax)$ without the constant 'a'.

Thanks for reading ! I am having a problem with the numerical integration of the Bessel function $$y = K_0(ax)$$ Since my constant is too large ($a = 6800$) I am getting the large arguments ...
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1answer
43 views

Modified Bessel function of order 0 and 1 question

@skbmoore Prove that $$\sum_{n=0}^{\infty}\frac{\Gamma{(n+3/2)}\,r^n}{(n!)^2}=\frac{\sqrt{\pi }}{2} e^{r/2} \left((r+1) I_0\left(\frac{r}{2}\right)+r I_1\left(\frac{r}{2}\right)\right)$$ ...
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1answer
71 views

Solving $I=\int_{0}^{1} x^2 J_0(k_nx) \ dx.$

I am trying to solve, $$I=\int_{0}^{1} x^2 J_0(k_nx) \ dx.$$ My attempt: I choose make the substitution $u=k_nx$, which has led to, $$I=\frac{1}{k_n^3}\int_{0}^{k_n} u^2 J_0(u) \ du.$$ Looking at an ...
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1answer
26 views

Hankel Transform with Bessel function of the first kind

I am trying to understand a substitution in a paper I am reading. The paper is "The Application of Linear Filter Theory to the Direct Interpretation of Geoelectrical Resistivity Sounding Measurements" ...
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1answer
65 views

Wronskian of Bessel Functions at $x =1$

If $u(x)$ and $v(x)$ are any two solutions of Bessel's equation of order $\alpha$, then the Wronskian $W(x;u,v)=\dfrac{c}{x}$ (See for example here). I am trying to calculate $c$ for when $u(x)$ and $...