What is Bessel functions: Definition and 83 Discussions
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
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{\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.
I've been studying a few books on PDE's, specifically the heat equation. I have one book that covers this topic in cylindrical coordinates. All the examples are applied to a solid cylinder and result in a general Fourier Bessel series for 3 common cases that can be found easily with an online...
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...
I have to find 2 solutions of this Bessel's function using a power series.
##x^2 d^2y/dx^2 + x dy/dx+ (x^2 -9/4)y = 0##
I'm using Frobenius method.
What I did so far
I put the function in the standard form and we have a singularity at x=0. Then using ##y(x) = (x-x_0)^p \sum(a_n)(x-x_0)^n##...
From a previous post about the Relationship between the angular and 3D power spectra , I have got a demonstration making the link between the Angular power spectrum ##C_{\ell}## and the 3D Matter power spectrum ##P(k)## :
1) For example, I have the following demonstration,
##
C_{\ell}\left(z...
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...
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...
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...
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
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...
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...
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...
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...
Where can I find and how can I derive the orthogonality relations for Hankel's functions defined as follows:
H^{(1)}_{m}(z) \equiv J_{n}(z) +i Y_{n}(z)
H^{(2)}_{m}(z) \equiv J_{n}(z) - i Y_{n}(z)
Any help is greatly appreciated.
Thanks
Hi,
I have previously made a post in order to gain some insight in my rather out of control project. Long story short I am investigating vibration of a circular plate and its standing waves. After consultation at this forum I have been guided in the direction of acoustics and bessel functions...
Hi PF!
I was wondering if anyone could shed some light on my understanding of arriving at the coefficients of Bessel Equations? Namely, why do we use the indicial equation to determine coefficients?
As an example, if we have to solve $$s^2 \alpha'' + 2 s \alpha ' - \frac{1}{4} \gamma^2 s^2...
Hello all,As an exercise my research mentor assigned me to solve the following set of equations for the constants a, b, and c at the bottom. The function f(r) should be a basis function for a cylindrical geometry with boundary conditions such that the value of J is 0 at the ends of the cylinder...
Homework Statement
Calculate:
a) ##\frac{d}{dx}(xJ_1(x)-\int _0^xtJ_0(t)dt)##
b) ##xJ_1(x)-\int _0^xtJ_0(t)dt##
c) let ##\xi _{k0} ## be the ##k## zero of a function ##J_0##. Determine ##c_k## so that ##1=\sum _{k=1}^{\infty }c_kJ_0(\frac{x\xi _{k0}}{2})##.Homework Equations
The Attempt at a...
I have derived these pair of coupled diff equations for U_1 (r) and U_2 (r):
r^2 \dfrac{d^2 U_1 (r)}{dr^2} + r \dfrac{d U_1 (r)}{dr} + r^2 U_2(r) = 0
and r^2 \dfrac{d^2 U_2 (r)}{dr^2} + r \dfrac{d U_2 (r)}{dr} - r^2 U_1(r) = 0
Or written in matrix form
(r^2 \dfrac{d^2}{dr^2} + r...
Before stating the main question,which section should the special functions' questions be asked?
Now consider the Bessel differential equation:
\rho \frac{d^2}{d\rho^2}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})+\frac{d}{d\rho}J_{\nu}(\alpha_{\nu m} \frac{\rho}{a})+(\frac{\alpha_{\nu m}^2...
When solving a differential equation for Bessel Functions, how do you know when to use the 1st kind or Neumann functions. How do you know which order of the bessel function to use?
As per orthogonality condition this equation is valid:
\int_0^b xJ_0(\lambda_nx)J_0(\lambda_mx)dx = 0 for m\not=n
I want to know the outcome of the following:
\int_0^b xJ_0(\lambda_nx)Y_0(\lambda_mx)dx = 0
for two cases:
m\not=n
m=n
Hi
I have proved (through educated guess-work and checking analytically) the following identity
\int\limits_0^\infty\int\limits_0^\infty s_1 \exp\left(-\gamma s_1\right) s_2 \exp\left(-\gamma s_2\right) J_0\left(s_1r_1\right) J_0\left(s_2r_2\right) ds_1ds_2 =...
Hello All.
I'm currently in a crash course on X-ray Diffraction and Scattering Theory, and I've reached a point where I have to learn about Bessel Functions, and how they can be used as solutions to integrals of certain functions which have no solution. Or at least, that's as much as I...
Hello.
I'm not terribly proficient with Bessel functions, but I know that those of the first kind are given by
\begin{eqnarray}
J_n(x) & = & \left(\frac{x}{2}\right)^n\,\sum_{\ell=0}^\infty\frac{(-1)^\ell}{\ell!\,\Gamma(n+\ell+1)}\,\left(\frac{x}{2}\right)^{2\ell},
\end{eqnarray}
where...
Homework Statement
My teacher gave us a problem as an open question:
To calculate an integral involving Bessel Functions.
Homework Equations
The Attempt at a Solution
I've tried to convert this integral to one in which the Bessel function is in the numerator but failed. Does anyone know how to...
Homework Statement
x d2y(x)/dx2 + dy(x)/dx + 1/4 y(x)
Show that the solution can be obtained in terms of Bessel functions J0.
Homework Equations
Hint: set u = xa where a is not necessarily an integer. Judiciously select a to get y(u).
The Attempt at a Solution
I tried just...
My textbook states
J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = -\frac{2 \sin v \pi}{\pi x}
My textbook derives this by showing that
J_v(x) J'_{-v}(x) - J'_v(x) J_{-v}(x) = \frac{C}{x}
where C is a constant. C is then ascertained by taking x to be very small and using only the first order of...
Homework Statement
Solve equations 1) and 2) for J_{p+1}(x) and J_{p-1}(x). Add and subtract these two equations to get 3) and 4).
Homework Equations
1) \frac{d}{dx}[x^{p}J_{p}(x)] = x^{p}J_{p-1}(x)
2) \frac{d}{dx}[x^{-p}J_{p}(x)] = -x^{-p}J_{p+1}(x)
3) J_{p-1}(x) + J_{p+1}(x) =...
Homework Statement
I want to make sure that a solution to a differrential equation given by bessel functions of the first kind and second kind meet at a border(r=a), and it to be differenitable. So i shall determine the constants c_1 and c_2
I use notation from Schaums outlines
Homework...
Hi,
I need to solve one problem like this:
(a+b)*J_{1}[x(a+b)]-(a-b)*J_{1}[x(a-b)]=c
J_{1} denotes the first order Bessel function. Do you think that it is possible to solve this function in an analytical way?
Thanks,
Viet.
Homework Statement
How do I integrate \int_0^1 xJ_0(ax)J_0(bx)dx where J_0 is the zeroth order Bessel function?Homework Equations
See above.
Also, the zeroth order Bessel equation is (xy')'+xy=0The Attempt at a Solution
Surely we must use the fact that J_0 is a Bessel function, since we can't...
I am aware that Bessel functions of any order p are zero in the limit where x approaches infinity. From the formula of Bessel functions, I can't see why this is. The formula is:
J_p\left(x\right)=\sum_{n=0}^{\infty}...
Hi,
I use scilab 5.2.2
Ik have a problem to find the zeros or roots of the bessel functions J0,J1...
Whel I write besselj(0,3) I get the value of the bessel function Jo(3)=0,2600520.
Can someone help me how to find the zeros of these bessel functions in scilab.
Thank you
kind...
Homework Statement
Prove that J_{n}, Y_{n} satisfy
x^{2}*y''(x)+x*y'(x)+(x^{2}-n^{2})*y(x)=0
where n\inZ and x\in(R_{>0}
Homework Equations
The standard definitions of the bessel integrals as given here:
http://en.wikipedia.org/wiki/Bessel_Functions
The Attempt at a Solution...
Homework Statement
This is part of a vibrating circular membrane problem, so if I need to post more details please let me know. Everything is pretty straight forward with the information I'll provide but you never know.
We haven't really learned what these are, just that they are complicated...
I need to solve this general problem. Let's consider the following vector field in cylindrical coordinates:
\vec{A}=-J'_m(kr)\cos(\phi)\hat{\rho}+\frac{m^2}{k}\frac{J_m(kr)}{r}\sin(\phi)\hat{\phi}+0\hat{z}
where m is an integer, and k could satisfy to:
J_m(ka)=0 or J_m'(ka)=0 with a real.
(the...