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Questions tagged [bessel-functions]

Questions related to Bessel functions.

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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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2answers
39 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
28 views

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
32 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
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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
68 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
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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
39 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 $...
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$\phi(x)$ for negative integers

During the the proof of the following formula I faced with $\phi(x)$ for negative integers. That is in order to finish proof of the mentioned formula (which is not proved in the book), after a long ...
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1answer
138 views

Fourier-Bessel Series for $f(x)=1-x$

I am trying to find the Fourier-Bessel series for the function $$f(x)=1-x\ \ \text{for} \ \ 0<x<1.$$ The Fourier-Bessel series has the form $$f(x)=\sum_{n=1}^{\infty} A_nJ_v(k_nx), \ \ 0<x&...
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2answers
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$\left.\left(\frac{\mathrm d}{\mathrm d x}\right)^n J_0(x)\right|_{x=0}={}$?

I am interested in determining a closed expression for the n-th derivative of the Bessel function of the first kind $J_0(x)$, centered in $x=0$: \begin{equation} \left.\left(\frac{\mathrm d}{\mathrm ...
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1answer
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Ricatti differential equation $y'=-13(x-y^{2})$

$$y'=-13(x-y^{2})$$ I'm a beginner. I know, that this is a kind of Riccati equation, but is it possible to solve it with only simple methods? Thank you p.s. I know nothing about Bessel functions. ...
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find the eigen function and eigen value of differential operator

I have an operator, defined in the cylindrical coordinate system with cylindrical symmetry, given by: $\frac{\partial^2}{\partial r^2}+ \frac{\partial}{r\partial r} $ I would like to find the ...
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A closed form for $\int_R^{\infty} x^{2\nu+1}J_{\nu}(x)\exp\{-\frac{x^2}{4\sigma^2}\}dx$

Can the integral $$I(R,\sigma)\equiv \int_R^{\infty} x^{2\nu+1}J_{\nu}(x)\exp\{-\frac{x^2}{4\sigma^2}\}dx$$ be written in a closed form in terms of elementary functions? Here $\nu$ is a positive ...
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Question regarding Hankel functions

I'm studying Hankel functions and I specifically need to calculate the $L^2$ norm of the Hankel function of the first kind (or approximate it), but I'm having real trouble with doing so. I've looked ...
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1answer
68 views

Prove $\lim\limits_{r \to \infty} r-\frac{\sqrt{r}}{e^{r}}\sum_{n=0}^{\infty}\frac{\Gamma{(n+3/2)}r^n}{(n!)^2}=1/4$

This is a follow-up question for the one described here. Evaluate $$\lim\limits_{r \to \infty} r-\frac{\sqrt{r}}{e^{r}}\sum_{n=0}^{\infty}\frac{\Gamma{(n+3/2)}r^n}{(n!)^2}=1/4$$ There is a ...
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1answer
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Singular point in Bessel differential equation

In this site, Bessel differential equation is presented as: $$x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 \label{a} \tag{1}$$ and "equivalently, dividing through by $x^2$", $$\frac{d^...
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Integrating the product of a sine with two Bessel functions

I have a difficult integral again, which I would really prefer to be able to express analytically: $$Y_{nlL}=\int_0^1 \sin \pi n r J_0( \gamma_l r )J_0(\gamma_L r) dr$$ Here $n=1,2,3,4,\dots$ and $\...
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1answer
128 views

Find the Fourier-Bessel Series for $f(x)$ With Respect to the Orthogonal Set: How Was $w(x)$ Found?

I have the following problem: If $f(x) = x$, $0 < x < 2$, find the Fourier-Bessel series for $f(x)$ with respect to the orthogonal set $\{ J_1 (k_n x) \}$, where $k_n$ is the $n$th positive ...
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1answer
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Indicial Equation

So I have here a series solution for SHO and it's given by $$ \sum_{\lambda=0}^{\infty}a_{\lambda}(k+\lambda)(k+\lambda-1)x^{k+\lambda}+\sum_{\lambda=0}^{\infty}a_{\lambda}(k+\lambda)x^{k+\lambda}+\...
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2answers
141 views

How to prove the integral $\int_{0}^{\infty} x^{-1}\sin (x+x^{-1})\,dx=\pi J_0(2)$

$$I=\int_{0}^{\infty} \frac{\sin \left(x+\frac{1}{x}\right)}{x}dx=\pi J_0(2)$$ I'v found:$$I=2\int_{0}^{\infty}\frac{\sin \left(x^2+\frac{1}{x^2}\right)}{x}dx=3\int_{0}^{\infty}\frac{\sin \left(x^3+\...
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decay estimate for a elliptic operator

I have a function $f : \mathbb{R}² \rightarrow \mathbb{C}$ such that $|- \Delta f+ f | < 1/(1+r)^{1+\sigma}$, where $r$ is the distance to the origin and $0<\sigma <1$, and I want to show ...
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1answer
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Questions about modified Bessel function of second kind

$$\int_{0}^{1} e^{\frac{1}{\log(x)}} \, dx = 2 \,K_{1}(2)$$, where $K_1$ is the modified Bessel function of the second kind. I was wondering, $(1)$ Can you express $$\int_{0}^{1} e^{\frac{1}{\log(x)...
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2answers
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Series of inverse zeros of bessel functions

I am interested in the numerical values for the series $\displaystyle\sum_{n=1}^\infty\frac{1}{j_{0,n}^4}$ and $\displaystyle\sum_{l=1}^\infty\sum_{m=1}^\infty\frac{1}{j_{l,m}^4}$. where $j_{k,m}$ ...
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2answers
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Bessel Integration in matlab

I was trying to do integral in matlab with below equation. Could anyone help me please to do with Riemann sum approach in matlab? Thanks in advance!! $$\int_0^\infty{{e}}^{-{k^2}/{4}} J_0(k)k{d} k$$
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How to get this solution in terms of Modified Bessel function?

I have tried to solve a PDE by using Laplace-Carson transformation and obtained the following solution in Laplace-Carson domain. Is it possible to put the solution in the following Modified-Bessel ...
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1answer
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Finding limits of derivative spherical bessel function

The derivative of the spherical bessel function is defined as $f_{n}^{\prime}(z)= - f_{n+1}(z) +(n/z)f_{n}(z).$ The problem occurcs if I try to plot it at z = 0. I want to approximate it using l'...
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1answer
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What is the integral of $e^{x\cos\theta + y\cos\theta}$

Knowing that $$\int_{0}^{2\pi} e^{x\cos \theta } d\theta = 2\pi I_0(x)$$ where $I_0$ is the modified Bessel function. Is there a way/trick to find an analytical expression for $$\int_{0}^{2\pi} e^{x\...
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3answers
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Closed form of integral over fractional part $\int_0^1 \left\{\frac{1}{2}\left(x+\frac{1}{x}\right)\right\}\,dx$

Recenly, several interesting questions have been posted asking for closed forms of integrals over the fractional part of certain functions. For me the story started with Evaluation of $\int_{0}^{1}\...
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2answers
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Finding the particular solution to $ \phi''(r) + \frac{1}{r}\phi'(r)+C_1\phi(r) = C_2 J_0(\alpha r/a) $

My question is similar to this one, but it's slightly different: $$ \phi''(r) + \frac{1}{r}\phi'(r)+C_1\phi(r) = C_2 J_0(\alpha r/a) $$ where $C_1$, $C_2$ and $a$ are constants. For the particular ...
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Faster Fourier - Bessel Coefficient Calculations on Python

So I'm trying to get the fourier-bessel coefficients of a very large array of numbers that are around 1 million points in size, but I'm coming across some speed issues with calculating $J_{o}(x)$ for ...
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1answer
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Proving $\left(\frac{1}{z}\frac{d}{dz}\right)(z^{-v}J_v(z))=-z^{-v-1}J_{v+1}(z)$

Use the series definition $$\sum_{k=0}^{\infty} \frac{(-1)^k(z/2)^{2k+v}}{k!\Gamma (1+k+v)}$$ to show that $$\left(\frac{1}{z}\frac{d}{dz}\right)(z^{-v}J_v(z))=-z^{-v-1}J_{v+1}(z)$$ I have shown that ...
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Approximate Solution for $\int_{E_T}^\infty \frac{2 E b^\nu}{\Gamma (\nu)} \int_0^\infty x^{\nu -2}\exp(-b x)\exp(-(A^2 + E^2)/x) I_0(2EA/x) dx dE$

(This question is more about solving the integral than what the integral represents.) The homodyned K distribution has the following probability density function (PDF): $$ f(E | A, \nu, b) = \frac{2 ...
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Convert product of Bessel functions of the first and second kind into a Meijer G-function

There exists such an equality which converts the product of Bessel functions of the first and second kind into a Meijer G-funtion, $$x^\mu J_\nu(x)Y_\nu(x)=-\pi^{-1/2}G_{13}^{20}\left(x^2\left| \...
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2answers
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How to evaluate the integral $\int_{0}^{2\pi}e^{-iA(x\cos\varphi+y\sin\varphi)}\cos(l\varphi)\,d\varphi$?

$$\int_{0}^{2\pi}\exp\left(-iA(x\cos\varphi+y\sin\varphi)\right)\cos(l\varphi)\,d\varphi$$ I'm trying to evaluate the integral for an interference problem in Physics. When $y=0$, this reduces to the ...
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1answer
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Finding the Fourier-Bessel Series For $f(x) = x$, $0 < x < 2$, With Respect to the Orthogonal Set $\{ J_1 (k_n(x)) \}$

I am trying to find the Fourier-Bessel series for $f(x) = x$, $0 < x < 2$, with respect to the orthogonal set $\{ J_1 (k_n(x)) \}$, where $k_n$ is the $n^{th}$ positive root of the equation $J_1(...
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Integral of the bessel function

i have a qustion on the integration of a modified bessel function. According to the reference: Werner Rosenheinrich,"TABLES OF SOME INDEFINITE INTEGRAL OF BESSEL FUNCTIONS OF INTEGER ORDER", 2017 $\...
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Looking for bounds on $|I_n(z)|$ for $z \in \mathbb{C}$ where $I_n(z)$ is modified Bessel function

I am looking for some bounds on the magnitude of the modified Bessel function in the complex setting. That is let $I_v(z), z \in \mathbb{C}$ be modified Bessel function of order $v$. Are there any ...
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1answer
28 views

Why do I have to use Frobenius method in Bessel's equation?

I just learnt how to apply the power series method in differential equations and I'm now trying to understand the extended method of Frobenius. As an example, the textbook gives me Bessel's equation ...
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Integration of $\int_0^l\int_0 ^{2\pi} \frac {\exp (i k\sqrt{ (r^2+r (vt) \sin \phi)})}{r^2 + r (vt) \sin \phi+d^2} \, r \,dr \, d\phi$.

I have faced some difficulties to do the following integral $$\int_0^l\int_0 ^{2\pi} \frac {\exp (i k \sqrt{(r^2 + r vt \sin \phi)})}{r^2 + r vt \sin \phi +d^2} \, r \,dr \, d\phi$$ Where, $vt$ ...
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1answer
40 views

How to find global/local maxima and minima of $\frac{J_1(kx)}{x}$?

Suppose we have a function of the form $$f(x) = \frac{J_1(kx)}{x}$$ where $J_1$ is a Bessel function of the first kind, and k is some constant, and we would like to to find $x$ such that this function ...
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1answer
86 views

Fourier transform of product of Bessel functions

I need help finding the Fourier transform of the function $$ \rho(\vec{r}) = \alpha \delta_{\vec{r},0} \left(\lambda\lambda' J_1 (\beta |\vec{r}|)Y_1(\beta |\vec{r}|) - \pi^2 J_0 (\beta |\vec{r}|)Y_0(...
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0answers
33 views

Inverse Laplace transform related to modified Bessel function of the second kind

To solve a fluid diffusion problem, I need to calculate an inverse Laplace transform. The integral, according to the Inverse Laplace theorem, has the form: \begin{equation} \label{i_laplace_1} p(r,t)...
3
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1answer
141 views

Integral of Modified Bessel Function

Evaluate: $$\int\limits_{x=0}^\infty e^{-\alpha x^2}I_0(x) \ln(I_0(x))x \, dx$$ where $I_0(x)$ is the modified Bessel function of the first kind and zeroth order, and $\alpha>0$. I can find ...
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2answers
96 views

Inverse Laplace transform of $K_0 \left(r \sqrt{s^2-1}\right)$

This question is about inverse Laplace transform $\mathscr{L}^{-1}:s\rightarrow t$. Although I was not able to find appropriate contour to invert $K_0 \left(r s\right)$, I somehow know that $$\...
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0answers
50 views

Integrating the exponential of a periodic function

I am wondering whether the following integral $$ Y = \int_{x_0}^{x_1} \exp{\left(a\cos x+b\sin x\right)}\,dx,$$ has a known closed form for given $a,b,x_0$ and $x_1$. Since the ...
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0answers
41 views

Multiple integral involving trigonometric functions over a hypercube.

Let $d\ge 1$ be an integer and $\vec{A} := \left(A_j\right)_{j=1}^d \in {\mathbb R}^d$ subject to $\sum\limits_{j=1}^d A_j^2 \le 1$. Define a following integral: \begin{equation} {\mathfrak I}^{(d)}(\...