Bessel Definition and 249 Threads

Hi all. I need an integral representation of z^{-\nu}K_{\nu} of a particular form. For K_{1/2} it looks like this: z^{-\frac{1}{4}}K_{1/2}(\sqrt{z}) \propto \int_{0}^{\infty}dt\exp^{-zt-1/t}t^{-1/2} How do I generalize this for arbitrary \nu? A hint is enough, maybe there's a generating...

Homework Statement Find the general solution to x'' + e^(-2t)x = 0, where '' = d2/dt2 Homework Equations - The Attempt at a Solution First I did a change of variables: Let u = e^(-t) Then du/dt = -e^(-t) dx/dt = dx/du*du/dt = -e^(-t)*dx/du d2x/dt2 = d/du(dx/dt)du/dt =...

Hi everyone Today during problem session we had this seemingly simple exercise, but I just can't crack it: We should give an example of an x \in \ell^2 with strict inequality in the Bessel inequality (that is an x for which \sum_{k=1}^\infty |<x,x_k>|^2 < ||x||^2, where (x_k) is an orthonormal...

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...

Can someone confirm that \int J_0(ax)xdx=\frac{J_1(ax)x}{a}? I can only find the solution if J(x) but i want J(ax) so what i did above makes logical sense to me but i can't find it anywhere. thanks

Hallo there. I m trying to integrate a bessel function but with no great success... I thing it can't be calculated.. I m trying to simulate the airy pattern of a certain aperture radius and wavelength in matlab. the integral is : int (besselj(1,16981.9*sin(x)))^2/ sin(x) dx where you can...

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...

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...

Not sure if this is the right place. Mathematica has a function BesselK[0,x] that returns the value of the modified Bessel function K_0 at x. Is there public documentation of how this algorithm works? If not, is there documentation regarding any algorithm of K_0? I am hoping it doesn't...

Hello, In my work, I have to solve the following integral: \int {exp(-aX^2)I_0(b\sqrt(cX^2+dX+e))}dX where I_0() is the modified Bessel function. I did not find the solution in any table of integral. Any help is appreciated. Thanks a lot in advance.

Homework Statement Hi, I need to integrate this: \int(J0(r))2rdr between 0<r<a It is for calculating the energy of a nondiffracting beam inside a radius of a. (the r is because of the jacobian in polar coordinates) The Attempt at a Solution I saw somewhere that said the integral was a...

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

I am trying to solve this equation in terms of Bessel functions. xy"-y'+(4x^3)y=0 I am sure how to do this. The first thing that comes to mind is to solve for a series solution. This solution can then be compared to the bessel function and from that I can determine the first solution and...

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...

For finding series expansion solution of problems like f(x) = h(x) for 0<x<1 f(x) = 0 for 1<x<2 0<x<2 Where the Fourier series expansion only integrate from x=0 to x=1 only and totally ignor the portion of x=1 to x=2. This is also true for Fourier bessel series expansion...

problem in Bessel equation help ... Homework Statement using the formula d\dx (x^n Jn(x))=x^n Jn-1(x) & 2n\x Jn(x)=Jn+1(x)+Jn-1(x) Homework Equations prove that integral from 0 to 1 (x(1-x^2)Jdot(x) dx = 4 J1(1) - 2 Jdot (1) The Attempt at a Solution it's difficult one i can not...

Hello, I'm trying to show that Integral[x*J0(a*x)*J0(a*x), from 0 to 1] = 1/2 * J1(a)^2 Here, (both) a's are the same and they are a root of J0(x). I.e., J0(a) = 0. I have found and can do the case where you have two different roots, a and b, and the integral evaluates to zero...

In Dodelson's cosmology book it is claimed that "For large x, J_0(x\theta)\rightarrow P_{x}(cos\theta)". Does anyone have any insight on how to begin proving this?

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...

Hi all can anyone help me to reduce following diff.Equ. to bessel eq. 4x^3*y''-y=0 thanks in advance . I am also still trying to show that it can be converted to bessel function.

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...

Hello PF, maybe you can help with this one! I need to show that the Laplace transform of J0(at) is (s^2 + a^2)^-1/2. My prof told me to start with the form: x2y'' + x y' + (x2 + p2)y = 0, where p = 0 ITC. What have I got so far?... Doing the Laplace transform on both sides, where Y...

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!

I am trying to solved a differential equation of Bessel type, X^2 Y('')+XY(')+(X^2-n^2)Y+YlogX=0, where Y(')=d/dx. Please help me that how to deal with such equation.

In solving a particular kind of integral I ended up with the following series \sum_{k=0}^\infty \frac{\Gamma[b+k]}{\Gamma[a+b+k]} \frac{(1-t^2)^k}{k!} \left(\frac{\omega}{2}\right)^k J_{a+b-\frac{1}{2} +k} (\omega) where 0<t<1, and a,b are small and positive. I tried looking it up in a...

I remember some of my linear algebra from my studies but can't wrap my head around this one. Homework Statement Say my solution to a DE is "f(x)" (happens to be bessel's equation), and it contains a constant variable "d" in the argument of the bessel's functions (i.,e. J(d*x) and Y(d*x)). So...

Hello Everyone trying to come up with a stratagey to solving this integral Int(x^3*J3(x),x) no limits Ive tried some integration by parts and tried breaking it down into J1 and J0's however i still get to a point where I have to integrate either : Int(x*J1(x),x) or Int(J6(x),x)

Hi, I'm stuck on this question from a calculus book; Show that y'' + ((1+2n)/x)y' + y = 0 is satisfied by x-nJn(x) Is it correct that when I differentiate that, I get these: y= x-nJn(x) y'=-x-nJn+1(x) y''=nx-n-1Jn+1(x) - x-n(dJn+1(x)/dx)? The Attempt at a Solution Equation in...

Homework Statement Show that y'' + ((1+2n)/x)y' + y = 0 is satisfied by x-nJn(x) Homework Equations y= x-nJn(x) y'=-x-nJn+1(x) y''=nx-n-1Jn+1(x) - x-n(dJn+1(x) /dx) The Attempt at a Solution Equation in question becomes: x-n(2(n/x)Jn+1 - Jn - ((1+2n)/x)Jn+1 + Jn) =...

A typical BVP of Bessel function is approximation of f(x) by a Bessel series expansion with y(0)=0 and y(a)=0, 0<x<a. For example if we use J_{\frac{1}{2}} to approximate f(x) on 0<x<1. Part of the answer contain J_{\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}sin(\alpha_{j}x), j=1,2,3... This...

Homework Statement I cannot get the answer given by the book. The question is: [SIZE="4"]Using Bessel function of order = 2 to represent f(x): [SIZE="4"]f(x)=0 for 0<x<1/2 and f(x)=1 for 1/2<x<1. The Answer given by the book is -2\sum_{j=1}^{\infty}...

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)...

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

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: [SIZE="5"]J[SIZE="2"]-3/2 [SIZE="5"](x)=\sqrt{\frac{2}{\pi x}}[SIZE="6"][\frac{-cos(x)}{x} [SIZE="5"]- [SIZE="4"]sin(x) [SIZE="6"]] Main reason is \Gamma[SIZE="4"](n-3/2+1) give a negative value for n=0 and possitive value...

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...

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

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 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...