Bessel Definition and 276 Threads

I am trying to evaluate\int J_{2}(x)dx I have been trying to use all the identities involving Bessel function to no prevail. The ones I used are: \frac{d}{dx}[x^{-p}J_{p}(x)]=-x^{-p}J_{p+1}(x) (1) \frac{d}{dx}[x^{p}J_{p}(x)]=-x^{p}J_{p-1}(x) (2)...

I need to convertx^{2}y''+2xy'+[kx^{2}-n(n+1)]y=0 using y=x^{-\frac{1}{2}}w to a normal Modified Bessel Equation and I cannot get to that. I check many times and I must be having a blind spot! This is my work: y=x^{-\frac{1}{2}} w \Rightarrow...

I am almost certain I understand the Bessel function expension correctly, but I just want to verify with you guys to be sure: 1) J_{p}(\alpha_{j}x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}\alpha_{j}^{2n+p}x^{2n+p}}{n!\Gamma(n+p+1)2^{2n+p}} 2)...

Hello guys, I'm new here. i was working on a mathematical methods in physics book and there is a part that i don't understand. so i want to ask if anyone knows... while finding a second solution for bessel diff. eq.(for m=0) the book used wronskian method. in the method there is J^2 bin the...

Homework Statement Known formula:J_0(k\sqrt{\rho^2+\rho'^2-\rho\rho'\cos\phi})=\sum e^{im\phi}J_m(k\rho)J_m(k\rho') I can't derive to next equation which is e^{ik\rho\cos\phi}=\sum i^me^{im\phi}J_m(k\rho) Homework Equations Can anyone help me? Thanks a lot! The Attempt at a Solution

Homework Statement In Jackson 3.16 we have to prove the expansion \frac{1}{\left{|}\vec{x}-\vec{x'}\right{|}}=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}dke^{im(\phi-\phi')}J_m(k\rho)J_m(k\rho')e^{-k(z_{>}-z_{<})} Homework Equations The Attempt at a Solution I tried to use the...

What are the approximations for Bessel functions J_n with small arguments? I've had a very hard time finding this online. Thanks! -Matt

I am working on some numerical works. I use the computer language: Fortran language. Here I have a problem about the Bessel functon. Now I know the value of Bessel[v,x], where v is positive and real. I want to know the value of Bessel[-v,x]. I don't know their relation. Can you help me...

I have been working on this for a few days and cannot prove this: J-3/2 (x)=\sqrt{\frac{2}{\pi x}}[\frac{-cos(x)}{x} - sin(x) ] Main reason is \Gamma(n-3/2+1) give a negative value for n=0 and possitive value for n=1,2,3... I cannot find a series representation of this gamma...

Homework Statement By appropriate limiting procedures prove the following expansion: J_0 (k\sqrt {\rho ^2 + \rho '^2 - 2\rho \rho '\cos (\phi )} ) = \sum\limits_{m = - \infty }^\infty {e^{im\phi } J_m (k\rho )J_m (k\rho ')} Homework Equations...

In the solution to a recent problem set, my prof referenced a "general Bessel ODE" which he gave in the form: x^{2}\frac{d^{2}y}{dx^{2}}+x\left(a+2bx^{q}\right)\frac{dy}{dx}+\left[c+dx^{2s}-b\left(1-a-q\right)x^{q}+b^{2}x^{2q}\right]y=0 The only format of the Bessel ODE that appears in the...

Homework Statement u''-bc (x^m) u =0Homework Equations How can I write the general solution in terms of Bessel function?The Attempt at a Solution This form is a transformed vresion of y'+by^2=cx^m with dummy variable by=1/u *du/dx

Jackson 3.16 has one derive the orthonormality of the bessel functions, that is: \int\limits_0^\infty {\rho J_v (kp)J_v (k'p)d\rho } = \frac{{\delta (k - k')}}{k} Now, I was able to show that infact, they are orthogonal, but I haven't been able to figure out the 1/k term. Basically...

Homework Statement Prove J(-m) = [(-1)^m][J(m)] (Note: by "J(-m)" I mean "subscript (-m)") Homework Equations J(-m) = sum [((-1)^n) * (x/2)^(2n-m)]/[n! \Gamma(n - m + 1)] J(m) should be obvious. The Attempt at a Solution I tried just plugging in the above formulas hoping to get a...

Jackson 3.12: An infinite, thin, plane sheet of conducting material has a circular hole of radius a cut in it. A thin, flat disc of the same material and slightly smaller radius lies in the plane, filling the hole, but separted from the sheet by a very narrow insulating ring. The disc is...

I want to ask if you how to compute such integral like: int(t**2*BesselJ(1,a*t)*BesselJ(1,b*t)*BesselJ(1,c*t), t=1..w) or int(t**3*BesselJ(1,a*t)*BesselJ(1,b*t)*BesselJ(1,c*t)*BesselJ(1,d*t), t=1..w) The same question if any BesselJ is replaced by BesselY. Thanks

Homework Statement Find domain of \sum_{n= 0}^\infty \frac{(-1)^{n}x^{2n}}{2^{2n}(n!)^{2}} Homework Equations The Attempt at a Solution I set it all up but I can't really seem to simplify it. \frac{(-1)^{n+1}x^{2(n+1)}}{2^{2n+2}(n+1)!^{2}}\bullet\frac{2^{2n}(n!)^{2}}{(-1)^{n}x^{2n}}

Hi guys, Does anyone have any ideas about an analytical solution for the following integral? \int_{0}^{2\pi}J_{m}\left(z_{1}\cos\theta\right)J_{n}\left(z_{2}\sin\theta\right)d\theta J_{m}\left(\right) is a Bessel function of the first kind of order m. Thanks.

Hi there, I am calculating the Fourier transform of the bessel function J_0^2(bx) by using Maple. I tried two equations and get two results. \int J_0^2(bx)e^{-j2\pi fx}dx=G^{2, 1}_{2, 2}\left(-1/4\,{\frac {{w}^{2}}{{b}^{2}}}\, \Big\vert\,^{1/2, 1/2}_{0, 0}\right) {\pi }^{-1}{b}^{-1}...

Hi there, I am starting with the Bessel functions and have some problems with it. I am getting stuck with this equation. I could not find this kind of integral in the handbooks. 1. \int_0^aJ_0^2(bx)dx Besides of this, I have other equations in similar form but I think this integral...

Hello, What is \frac{d}{dx}K_v\left(f(x)\right)=? Thanks in advance

Hello, When I write: BesselK[1,2] in the Mathematica editor, the output is the same as the input. But I want to evaluate it numerically. In other words, I want the output be a number. How can I do that? Regards

The problem is to prove the following: \sum_{m>0}J_{j+m}(x)J_{j+m+n}(x) = \frac{x}{2n}\left(J_{j+1}(x)J_{j+n}(x) - J_{j}(x)J_{j+n+1}(x)\right). Now for the rambling... I've been reading for a while, but this is my first post. Did a quick search, but I didn't find anything relevant. I could...

Hi everyone, I have a question concerning the derivation of the J_0(t). In my book, it states that the inverse laplace transform of (s^2+1)^-1/2 is this function. It gives me a contour to integrate around and derive it. The problem is this: I always get an extra I in the answer. This is...

Hi everybody ! Maybe this post should go under partial differential equations but I'm not sure... I have the following problem and I would like to know if someone could give me some hints or something to read related to this. I'm studying multiple reflections of acoustics waves in a...

I really have no idea. I started with the frobenius method. Until the recurrence formula. I got that already. But I just don't know where to plug in the 1/2 into the equation. Can anyone help? I just need to know where to put in the 1/2? Or can i use the normal bessel function which in...

Differntial equation involving bessel functions - pls help! 1. I am trying to simplify the expression in the attachment below to extract some data: https://www.physicsforums.com/attachment.php?attachmentid=18352&d=1239157280 2. the relevant equation for beta is given by...

Hi everyone, I need some help solving a bessel function of the 1st order. The equation is used to calculate the mutual inductance between two inductors. The equation is: M=(1.45*10^-8)*integral [J1(1.36x)J1(0.735x)exp(-13.6x)]dx the integral is from zero to infinity. Can someone help...

Hi, I am en electrical engineering grad student and I have to solve an equation to calculate the mutual inductance between an antenna and a micro-inductor. I think it is a Bessel equations but I don't know how to solve. M(a,b,d)=(1.45x10^-8)*integral(J1(1.36x)*J1(0.735x)*exp(-x-13.6))dx...

I am trying to solve int(int(exp(a*cos(theta)*sin(phi))*sin(phi), phi = 0 .. Pi), theta = 0 .. 2*Pi) (1) with a a constant. Using the second last definite integral on http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions the integral (1) reduces to...

Hello, What is the value of the following derivavtive: \frac{d}{d\gamma}\left[ 1-\frac{2\gamma}{\sqrt{p}}e^{-\gamma \sigma/p} K_1\left(\frac{2\gamma}{\sqrt{p}} \right) \right]where K_1(.) is the modified Bessel function of the second kind and order 1? Some Paper shows that the result is...

Homework Statement By appropriate limiting procedures prove the following expansion \frac{1}{\left(\rho^2+z^2\right)^{1/2}}=\int^{\infty}_{0} e^{-k\left|z\right|}J_{0}(k\rho)dk Homework Equations The Attempt at a Solution I tried to implicate the fourier-bessel series but it...

Hi, I am working on the derivation of an equation on electrokinetic flow in microfluidic. I am stuck at a point that need me to do an integration in the form of r * cosh (Io(r)) where r = variable to be integrated I0 = zero order modified bessel function of the first kind Is there...

hallo, i now spent an hour looking for a formula connecting the modified bessel functions I_n and K_n to the hypergeometrical series F(a,b;c;z). has somedoby an idea? thank you

Homework Statement The Bessel function generating function is e^{\frac{t}{2}(z-\frac{1}{z})} = \sum_{n=-\infty}^\infty J_n(t)z^n Show J_n(t) = \frac{1}{\pi} \int_0^\pi cos(tsin(\vartheta)-n\vartheta)d\vartheta Homework Equations The Attempt at a Solution So far I...

Homework Statement Bessels equation of order n is given as the following: y'' + \frac{1}{x}y' + (1 - \frac{n^2}{x^2})y = 0 In a previous question I proved that Bessels equation of order n=0 has the following property: J_0'(x) = -J_1(x) Where J(x) are Bessel functions of...

I am calculating the Poynting Vector along a circular waveguide (a simplest circular waveguide which only contains a dielectric medium inside and is coverd by perfect conductor). When calculating this, two Bessel function integration appear which state: 1. ∫ {Jn'[(tnl/a)r]}^2 rdr, in [0...

Hello, I am in the process of showing that the modified Bessel function, I_v(x), is a solution to the modified Bessel equation, x^2*y''+x*y'-(x^2+v^2)*y=0 I have differentiated the MBF twice and plugged it into show that the left hand side is in fact 0. After a good amount of work...

I really need to prove eq. 10.1.45 and 10.1.46 of Abramowitz and Stegun Handbook on Mathematical functions. Is an expansion of e^(aR)/R in terms of Special Functions! Any help will be appreciated.

Hi everybody... i would like to seek help for the problem below. the assumptions I've considered is that transfer is radial only since it is a very long cylinder (infinitely long) that transfer in z direction is negligible, thermal radiation is zero, and wood properties are constant. Starting...

Hello, I am a researcher working on electromagnetic field. when solving the PDE equation, this integral about Bessel funtion arises: \int_{R1}^{R2} x J_1 (sx) dx where J_1 is the 1th order Bessel function of first kind, and s is a constant, R1 and R2 is integral interval. I have not...

Homework Statement I am attempting to solve the 2nd order ODE as follows using the generalized solution to the Bessel's equation Homework Equations original ODE: xd^{2}y/dx^{2}-3dy/dx+xy=0 The Attempt at a Solution My first thought is to bring out an x^-1 outside of the function so...

Hi, Do you have any idea to solve this integral? \int^{\phi_{1}}_{\phi_{2}} exp[j cos(x)] dx where \phi_{1} and \phi_{2} are an arbitrary angles. If \phi_{1}=\pi and \phi_{2}=0, the answer for this integral is a Bessel function. Thanks, Viet.

Homework Statement so, without typing the whole thing (because I do not know how to use any LaTeX or similar program) what is the domain for the Bessel function J(sub 1)(x) = ... Homework Equations I am to understand that taking the derivative of this monster will give me some kind...

Hi Guys, I'm an undergrad student...and i have a difficulty trying to solve 4xy" + 4y' + y = 0, and express the solution in term of Bessel function. I have tried Frobenius method...then...it didn't work..and I'm really confused Could anyone please help me with this?...i'd would really...

The following link: http://electron6.phys.utk.edu/QM1/modules/m1/free_particle.htm mentions something about the Bessel-Parseval relation... could someone explain what this is exactly and how it works?

Homework Statement Show that: \int_0^xJ_0(t)dt=2\sum_{n=0}^{\infty}J_{2n+1}(x) Homework Equations I know that J_0(t)=\sum_{s=0}^{\infty}\frac{(-1)^s}{s!s!}\frac{t^{2s}}{2^{2s}} The Attempt at a Solution I tried to calculate the integral and i get ...

hello every body ... I am a new member in this forums ..:smile: and i need ur help in telling me what's the perfect way to study legendre and bessel function for someone doesn't know anything about them and having a hard time in trying to understand ... i`ll be thankful if u...

I have a problem in electromagnetism giving a DE that looks something like a Lapacian or a Bessel function, I'm told. It derives from cylindrical coordinates. .\ \ \ \ \ \ \ \ \left( \partial_{r} ^2 + \frac{1}{r}\partial_{r} - \frac{1}{r^2}\right)E = \frac{1}{c^2}\partial_{t}^2 E\ \ \ \ \ \ \...

hello, while working on a problem i encountered the following integral :(limits are zero and infinity) Integral[J1(kR)dk] J1 is the first order bessel function..cudnt put 1 in subscripts.. Is there an analytical solution for this?? also is it possible to integrate it numerically...