Bessel functions Definition and 83 Threads

Hi, as part of my maths course i am learning about bessel functions. But this is something that I am not fully comfortable with - there seems to be a lot of tricks. There is a statement in my notes that when \alpha_n>>1...

Hi, I posted this on the homework forum, but I haven't gotten any responses there. I thought there might be a better chance here. 1. Homework Statement I have the ODE h'' + h'/r + λ2h = 1, where h = h(r), and I want to find h(r). 2. Homework Equations The corresponding...

Homework Statement I have the ODE h'' + h'/r + λ2h = 1, where h = h(r), and I want to find h(r). Homework Equations The corresponding homogeneous equation is a Bessel equation that has the solution hh = c1J0(λr) + c2Y0(λr), where J0 and Y0 are Bessel functions. Now I was planning on using...

Hi, I work in a computational neuroscience lab, where we study human perception using Bayesian models. In our models we often have to compute products and ratios of Bessel functions (specifically, zeroth-order modified Bessel functions of the first kind). Our computations could speedup...

Hi, This is a question about modes in step index fibers, however its just the math in the following equations that I'm having trouble with, so you don't need to know the question. basically we have the following 2 equations: J1/uJ0 + K1/wK0 = 0 k12J1/uJ0 + k22K1/wK0 = 0 where Jn is a first...

Homework Statement Can anyone tell me if: \frac{d}{dx}J_k(ax)=aJ'_k(x) where a is a real positive constant and J_k(x) is the Bessel function of the first kind. Regards John Homework Equations The Attempt at a Solution

Can we integrate double integrals involving bessel functions and sinusoids in maple. Also, the overlap of sine and cosine over the range of 0 to 2 * Pi must be exactly zero, but, in maple, it gives some value (of the order of -129). Is there any software, which can compute the exact double...

Homework Statement Find the expansion of 1 - x^2 on the interval 0 < x < 1 in terms of the Eigenfunctions J_0 ( \sqrt{ \lambda_k ^{(0)}} x) (where \lambda_k ^{(0)} denotes the kth root > 0 of J_0) of (x u')' + \lambda x u = 0 u(1) = 0 u and u' bounded.Homework Equations Hint from the...

Hi, I have been trying to solve this differential equation for a while now. Now I get to the point where I have the solution, but it includes an integral. The integral is \int x J_{1/4}(ax) J_{1/4}(bx) e^{-x^2t}dx , where a and b are constants, and the integral is from zero to...

Homework Statement I'm given a standard form of Bessel's equation, namely x^2y\prime\prime + xy\prime + (\lambda x^2-\nu^2)y = 0 with \nu = \frac{1}{3} and \lambda some unknown constant, and asked to find its eigenvalues and eigenfunctions. The initial conditions are y(0)=0 and...

I seek a way to integrate J0, bessel function. I try to use some of the identities I can find, but it takes me no were. Please help!

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...

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

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}...

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 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...

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...

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...

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 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.

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...

Homework Statement I'm wondering if the bessel functions are pure real. What I really want to know is that if the bessel funtions are J and Y (i.e. first and second kinds), and the Hankel functions are H_1=J+iY and H_2=J-iY, then can we say that H_1=H_{2}^{*} where the * denotes complex...

Hi all, I am trying to find an expression for the values of the derivates of the Bessel-J_1 functions at two. The function is defined by J_1(x)=\sum_{k=0}^\infty{\frac{(-1)^k}{(k+1)!k!}\left(\frac{x}{2}\right)^{2k+1}} this I can differentiate term by term, finding for the n^th derivative at...

Hello Trying to calculate and simulate with Matlab the Steady State Temperature in the circular cylinder I came to the book of Dennis G. Zill Differential Equations with Boundary-Value Problems 4th edition pages 521 and 522 The temperature in the cylinder is given in cylindrical...

Hi all, I was just wondering if anyone knew how to differentiate Bessel functions of the second kind? I've looked all over the net and in books and no literature seems to address this problem. I don't know if its just my poor search techniques but any assistance would be appreciated.

The Bessel function can be written as a generalised power series: J_m(x) = \sum_{n=0}^\infty \frac{(-1)^n}{ \Gamma(n+1) \Gamma(n+m+1)} ( \frac{x}{2})^{2n+m} Using this show that: \sqrt{\frac{ \pi x}{2}} J_{1/2}(x)=\sin{x} where...

I want to solve the partial differential equation \Delta f(r,z) = f(r,z) - e^{-(\alpha r^2 + \beta z^2)} where \Delta is the laplacian operator and \alpha, \beta > 0 In full cylindrical symmetry, this becomes \frac{\partial_r f}{r} + \partial^2_rf + \partial^2_z f = f - e^{-(\alpha r^2 +...

Hi there ; I wanted you to help me with a problem. Well, I'm now studying griffiths' quantum book and now I'm trying the three dimensional schrodinger equation. I just wanted to know more about bessel functions. Can anyone give me a link for it? Some useful book will be good too. Thanks a...

I have the solution to a particular D.E. (Airy's D.E.) which is in terms of Airy functions, namely a linear combination of Ai(x) and Bi(x), to which I have to fit to the boundary conditions. Both Ai(x) and Bi(x) can be cast into a form which involves both modified Bessel functions of the first...

Hey guys I was wondering if you could help me out with a proof of the recursion relations of Bessel functions on my homework: Show by direct differentiation that J_{\nu}(x)=\sum_{s=0}^{\infty} \frac{(-1)^{s}}{s!(s + \nu)!} \left (\frac{x}{2}\right)^{\nu+2s} obeys the...

What role do Bessel functions play in frequency modulation theory?