Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation
x
2
d
2
y
d
x
2
+
x
d
y
d
x
+
(
x
2
−
α
2
)
y
=
0
{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0}
for an arbitrary complex number α, the order of the Bessel function. Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α.
The most important cases are when α is an integer or half-integer. Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer α are obtained when the Helmholtz equation is solved in spherical coordinates.
Hello,
While reading Sakurai (scattering theory/Eikonal approximation section), I encountered a referenced integral
##
\int_0^\infty J_1(x)^2\frac{dx}{x}=1/2
##
I also see this integral from a few places (wolfram, DLMF, etc), so I tried to prove this from various angles (recurrence relations...
Hello all
I am trying to solve the following integral with Mathematica and I'm having some issues with it.
where Jo is a Bessel Function of first kind and order 0. Notice that k is a complex number given by
Where delta is a coefficient.
Due to the complex arguments I'm integrating the...
Hello.
Does anybody know a proof of this formula?
$$J_{2}(e)\equiv\frac{1}{e}\sum_{i=1}^{\infty}\frac{J_{i}(i\cdot e)}{i}\cdot\frac{J_{i+1}((i+1)\cdot e)}{i+1}$$with$$0<e<1$$
We ran into this formula in a project, and think that it is correct. It can be checked successfully with numeric...
I edited this to remove some details/attempts that I no longer think are correct or helpful.
But my core issue is I have never seen this approach to approximating integrals that is used in the attached textbook image. Any more details on what is happening here, or advice on where to learn more...
In Wikipedia article on Bessel functions there is an integral definition of “non-integer order” a (“alpha”). For imaginary order ia I get that Jia* = J-ia, where * is complex conjugate and ia and -ia are subscripts. Then in same article there is a definition of Neumann function, again for...
Hey all,
I wanted to know if anyone knew somewhere I could find the asymptotic behavior for small x (i.e x approaching 0) limit of the modified Bessel equations with complex order. The wikipedia page for Bessel functions...
Hi
When we find integrals of Bessel function we use recurrence relations.
But this requires that we have the variable X raised to some power and multiplied with the function .
But how about when we have Bessel function of first order and without multiplication?
How should we integrate it ?
Hi PF!
Do you know of any examples of the Ritz method which use Bessel functions as trial functions? I’ve seen examples with polynomials, Legendre polynomials, Fourier modes. However, all of these are orthogonal with weight 1. Bessel functions are different in this way.
Any advice on an...
Integral
\int^{\pi}_0\sin^3xdx=\int^{\pi}_0\sin x \sin^2xdx=\int^{\pi}_0\sin x (1-\cos^2 x)dx=\frac{4 \pi}{3}
Is it possible to write integral ##\int^{\pi}_0\sin^3xdx## in form of Beta function, or even Bessel function?
I learned about Bessel functions and steady-state temperature distributions in the past. Recently, I was searching online for some example problems on the topic and found the "original question" along with the solution online as a PDF file.
While I am unsure will it be appropriate for me to...
From my understanding of diffraction pattern is supposed to result in something like this
However when I plot it I get the central peak without the ripples (even when broadening the view). My result
My code is as follows
%1) Define the grid. Define vectors so that they include 0...
I tried integration by parts with both ##u = x^2, dv = J_0 dx## and ##u = J_0, du = -J_1 dx, dv = x^2 dx.## But neither gets me in a very good place at all. With the first, I begin to get integrals within integrals, and with the second my powers of ##x## in the integral would keep growing...
I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m + 1)...
Hello everyone.
I try to plot a figure from a journal article. I gave the equations in the inserted image. I wrote the script given below for that. I expect to obtain a plot like the one given on the left but I end up with something totally different. So, the values of ##I_{0}## and ##I_{1}##...
I can only find a solution to \int_{0}^{r} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho
with the Lommel's integral . On my last thread (here), I got an idea about how to execute this when m = n (Bessel functions with the same order) using Lommel's integrals (Using some properties of Bessel...
Hello,
For my homework I am supposed to get-
into the form of a Bessel equation using variable substitution. I am just not sure what substitution to use.
Thanks in advance.
Hi everyone,
I'm working through the boundary conditions and I could not figure out what to do with the last boundary condition (when z=L)
I know that the values for K are:
How so?
1. Homework Statement
A hollow right angle cylinder of radius a and length l. The sides and bottom are...
Homework Statement
[/B]
From the Rodrigues’ formulae, I want to derive nature of the spherical Bessel and Neumann functions at small values of p.
Homework Equations
[/B]
I'm going to post an image of the Bessel function where we're using a Taylor expansion, which I'm happy with and is as far...
Hello,
i am trying to solve this equation for x
besselj(0,0.5*x)*bessely(0,4.5*x)-besselj(0,4.5*x)*bessely(0,0.5*x) ==0;
I tried vpasolve, but it gave me answer x=0 only. fzero function didnt work, too.
What function can solve this equation?
Thanks
Homework Statement
i have been trying to learn bessel function for some time now but to not much help
firstly, i don't even understand why frobenius method works why does adding a factor of x^r help to fix the singularity problem. i saw answers on google like as not all function can be...
show that
(a^2-b^2)\int_{0}^{P} J_{v}(ax)J_{v}(bx)x\,dx=P\left\{bJ_{v}(aP)J^{'}_{v}(bP)-aJ^{'}_{v}(ap)J_{v}(bP)\right\}
when J^{'}_{v}(aP)=\d{J_{v}(ax)}{(ax)},(x=P)
I don, have idea
Bessel function
using g(x,t)=g(u+v,t)=g(u,t)g(v,t)
to show that J_{0}(u+v)=J_{0}(u)J_{0}(v)+2\sum_{s=1}^{\infty}J_{s}(u)J_{-s}(v)
___________________________________________________________________________________________
my solution
g(u+v,t)=e^{\frac{u+v}{2}(t-\frac{1}{t})}...
[Mentors' note: Moved from the technical forums, so no template]
Hi,
I have to find energy levels of an electron in a cylindrical shape. I know how to derive the formula below:
However, I'm not sure which zero value and what intger p I need to use in order to find the lowest energy.
If these...
Generating function for Bessel function is defined by
G(x,t)=e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n
Why here we have Laurent series, even in case when functions are of real variables?
I have question regarding gamma function. It is concerning ##\Gamma## function of negative integer arguments.
Is it ##\Gamma(-1)=\infty## or ##\displaystyle \lim_{x \to -1}\Gamma(x)=\infty##? So is it ##\Gamma(-1)## defined or it is ##\infty##? This question is mainly because of definition of...
Homework Statement
In a article I have found this transformation (exp to bessel function) . I have two questions.
Homework EquationsThe Attempt at a Solution
a)where did the Cos go after setting n=1 and n=-1 ? in the third equations ( it is equal to -wmt-pi/2)? why?)
b)how did the writer...
Hi PF!
I'm trying to put the first derivative of the modified Bessel function of the first kind evaluated at some point say ##\alpha## in a sum where the ##ith## function is part of the index. What I have so far is
n=3;
alpha = 2;
DBesselI[L_, x_] := D[BesselI[L, x], {x, 1}]
Sum[BesselI[L...
Homework Statement
I need to simplify the following integral
$$f(r, \theta, z) =\frac{1}{j\lambda z} e^{jkr^2/2z} \int^{d/2}_0 \int^{2\pi}_0 \exp \left( -\frac{j2\pi r_0 r}{z\lambda} \cos \theta_0 \right) r_0 \ d\theta_0 dr_0 \tag{1}$$
Using the following integrals:
$$\int^{2\pi}_0 \cos (z...
Hi, I'm no expert in math so I'm struggling with solving these integrals, I believe there's an analytical solution (maybe in http://www.hfa1.physics.msstate.edu/046.pdf).
$$V_{1234}=\int_{x=0}^{\infty}\int_{y=0}^{\infty}d^3\pmb{x}d^3\pmb{y}\...
Hello all,
I am trying to take the inversion of this function that is in Laplace domain. I've tried using a wolfram alpha solver, and I know I can probably use stehfest algorithm to numerically solve it but wanted to know if there was an exact solution.
the function is...
Homework Statement
For the FM modulation, the amplitudes of the side bands can be predicted from
v(t)=ΣAJn(I)sin(ωt)
Where is a sideband frequency and Jn(I) is the Bessel function of the first kind and the nth order evaluated at the modulation index .Given the table of Bessel functions...
Homework Statement
Given the bessel equation $$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} -(1-x)y=0$$ show that when changing the variable to ##u = 2\sqrt{x}## the equation becomes $$u^2\frac{d^2y}{du^2}+u\frac{dy}{du}+(u^2-4)y = 0$$
Homework Equations
The Attempt at a Solution
##u=2\sqrt{x}##...
Homework Statement
How to integrate this?
##\int_{0}^{A} x e^{-a x^2}~ I_0(x) dx##
where ##I_0## is modified Bessel function of first kind?
I'm trying per partes and looking trough tables of integrals for 2 days now, and I would really really appreciate some help.
This is a part of a...
Homework Statement
I want to prove this relation
##J_{n-1}(x) + J_{n+1}(x)=\frac{2n}{x}J_{n}(x))##
from the generating function. The same question was asked in this page with solution.
http://www.edaboard.com/thread47250.html
My problem is the part with comparing the coefficient. I don't...
Hi,
I have recently studied about basis for wavelet function which is helpful to design any function. Likewise, what is the basis for bessel function and how can it be implemented for an image ( because image is also a function). Specifically, I am interested to know how bessel function can be...
Hi, i want to know , can we deduce the bessel function of ist kind from second kind by using conditions as i read second kind is more generalized solution. thanks
at what value of k should the following integral function peak when plotted against k?
I_{\ell}(k,k_{i}) \propto k_{i}\int^{\infty}_{0}yj_{\ell}(k_{i}y)dy\int^{y}_{0}\frac{y-x}{x}j_{\ell}(kx)\frac{dx}{k^{2}}
This doesn't look like any orthogonality relationship that I know, it's a 2D...
Orthogonality condition for the 1st-kind Bessel function J_m
$$\int_0^R J_m(\alpha_{mp})J_m(\alpha_{mq})rdr=\delta_{pq}\frac{R^2}{2}J_{m \pm 1}^2(\alpha_{mn}),$$
where α_{mn} is the n^{th} positive root of J_m(r), suggests that an original function f(r) could be decomposed into a series of 1-st...
I know that the limit for the spherical bessel function of the first kind when $x<<1$ is:
j_{n}(x<<1)=\frac{x^n}{(2n+1)!}
I can see this from the formula for $j_{n}(x)$ (taken from wolfram's webpage):
j_{n}(x)=2^{n}x^{n}\sum_{k=0}^{\infty}\frac{(-1)^{n}(k+n)!}{k!(2k+2n+1)!}x^{2k}
and...
Homework Statement
Find the scalar product of diracs delta function ##\delta(\bar{x})## and the bessel function ##J_0## in polar coordinates. I need to do this since I want the orthogonal projection of some function onto the Bessel function and this is a key step towards that solution. I only...
Homework Statement
I have run into a number of problems while working through problems regarding Bessel and Modified Bessel Functions. At one point I run into i^{m}e^{\frac{im\pi}{2}} and it needs to equal (-1)^m but I'm not sure how it does. This came up while trying to solve an identity for...
<<Moderator note: Missing template due to move from other forum.>>
Good afternoon. I'm trying to solve a differential equation with bessel function solutions. I am trying to solve
\begin{equation*}
y''(x)+e^{2x}y(x)=0
\end{equation*}
using the substitution ##z=e^x##. The textbook this problem...
Homework Statement
Show that the Bessel functions Jn(x) (where n is an integer) have a very nice generating function, namely,
G(x,t) := ∑ from -∞ to ∞ of tn Jn(x) = exp {(x/2)((t-T1/t))},
Hint. Starting from the recurrence relation
Jn+1(x) + Jn-1(x) = (2n/x)Jn(x),
show that G(x,t)...
I am struggling to find the antiderivative of the following function:
f(x)=\frac{J_{0}(ax)J_{1}(bx) }{x+x^{4} }
\\
J_{0},{~}J_{1} : Bessel{~}functions{~}of{~}the{~}first{~}kind\\
a, b: constants
\\
F(x)=\int_{}^{} \! f(x) \, dx =?
Who can help?
This is a helpful document I got from one of my DE's teachers in graduate school, and I've toted it around with me. I will type it up here, as well as attach a pdf you can download.
Bessel Functions
$$J_{\nu}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{\nu+2m}}{2^{\nu+2m} \, m! \,\Gamma(\nu+m+1)}$$...
Hi Physics Forums.
I am wondering if I can be so lucky that any of you would know, if these two functions -- defined by the bellow integrals -- have a "name"/are well known. I have sporadically sought through the entire Abramowitz and Stegun without any luck.
f(x,a) = \int_0^\infty\frac{t\cdot...
Hello,
I am trying to find Fourier Bessel Transform (i.e. Hankel transform of order zero) for Yukuwa potential of the form
f(r) = - e1*e2*exp(-kappa*r)/(r) (e1, e2 and kappa are constants). I am using the discrete sine transform routine from FFTW ( with dst routine). For this potential...
Hi, everybody. Mathematic handbooks have given a sum formula for the modified Bessel function of the second kind as follows
I have tried to evaluate this formula. When z is a real number, it gives a result identical to that computed by the 'besselk ' function in MATLAB. However, when z is a...
I'm trying to show that a function defined with the following recurence relations
$$\frac{dZ_m(x)}{dx}=\frac{1}{2}(Z_{m-1}-Z_{m+1})$$ and $$\frac{2m}{x}Z_m=Z_{m+1}+Z_{m-1}$$ satisfies the Bessel differential equation
$$\frac{d^2}{dx^2}Z_m+\frac{1}{x}\frac{d}{dx}Z_m+(1-\frac{m^2}{x^2})Z_m=0$$...