Bessel function Definition and 145 Threads

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...

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...

Homework Statement I'm trying to convert s(t) = sin(2 \pi f_c t + I sin[2 \pi f_1 t + I_2 sin\{2 \pi f_2 t\}]) into Bessel functions of the form s(t) = \sum_k J_k(I_1) \times J_n (k I_2) sin(2\pi [f_c + k_1 f_1 + n f_2]t) Homework Equations Standard trigonometric equation for sin...

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...

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...

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 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

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...

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 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 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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 ...

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...

To calculate a p.d.f. of a r.v., I need to integral a product of two bessel function as \mathcal{L}^{-1} \left( abs^2 K_n( \sqrt{as}) K_n( \sqrt{bs} ) \right) where \mathcal{L}^{-1} is the inverse Laplace transform. I think some properties about the bessel function can solve this...

Homework Statement Show that Jn(x+y) = ∑ Jr(x)Jn-r(y) ; where (Jn)= bessel function , ∑ varies from (-to+)infinity for r Jo(x+y) = Jo(x)Jo(y) +2 ∑ Jr(x)J-r(y) ∑ varies from (1 to infinity) for r Homework Equations The Attempt at a Solution I have solved the first...

Hi This is one of the problems for my take home final exam on differential equations. I have been looking for a solution for this problem intensely for the last two days. This problem comes from Calculus vol 2 by Apostol section 6.24 ex 7. here it is Homework Statement Use the identities...

If we want to find x giving J_m(x)=0 where m=any constants, how can we analytically get x? Thank you

bessel function please explain 1. Homework Statement summation limits (n=j to infinity) (-a/4)**n/n!(2n_ n+j) =(-1)**j e**(-a/2) I(a/2) where j>=1 the rest are constants and I is summation index i was just solving a SHM problem involving Fourier transform in which this happens to be one...

bessel function please explain this step Homework Statement summation limits (n=j to infinity) (-a/4)**n/n!(2n_ n+j) =(-1)**j e**(-a/2) I(a/2) where j>=1 the rest are constants and I is summation index i was just...

(Repost of thread, wrong forum). Hi all, I'm writing a simulation of Chladni plates in Max/MSP and hope to use it in granular synthesis. I have found two formulas on the web; square and circular plate. I understand the square but the circular is quite confusing as I'm not a mathematician...

why does the v parameter in the bessel function x^2y``+xy`+(x^2-v^2)y=0 have to be real and nonegative?

Hello, I am a geologist working on a fluid mechanics problem. Solving the PDE for my problem, this Bessel integral arises: \int_{0}^{R} x^3 J_0 (ax) dx where J_0 is the Bessel function of first kind, and a is a constant. I haven't found the solution in any table or book, and due to...

I'm trying to show that the Bessel function of the first kind satisfies the Bessel differential equation for m greater of equal to 1. The Bessel function of the first kind of order m is defined by J_m(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{2^{m+2n}n!(n+m)!}x^{m+2n} = x^m...

Hello, I hope someone can show me where I got stuck/wrong. Verify that the Bessel function of index 0 is a solution to the differential equation xy" + y' + xy = 0. Note that my "<= 1" DOES NOT mean less than or equal to 1 but an arrow pointing to the left... it is said to be "equation 1"...

Hello guys, i had a little chat with a teacher of mine and he asked me how can someone plot the zero order Bessel function. Here is what I've done.. using the integral expresion for J_{0}(r) J_{0}(r)=\frac {1}{\pi}\int_0^\pi \cos(r\cos\theta)d\theta i can calculate the first order...