Bessel function Definition and 133 Threads

  1. Y

    I need to verify Bessel function expension.

    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)...
  2. S

    Deriving Bessel Function Equation with Basic Relation

    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
  3. U

    Bessel FUnction small arguments

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

    How to Find Bessel[-v,x] Given Bessel[v,x] in Fortran?

    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...
  5. Pengwuino

    How can Bessel functions be used to prove the expansion of a specific function?

    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...
  6. S

    Finding the Domain of the Bessel Function Series

    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}}
  7. V

    Integral of square of Bessel function

    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...
  8. E

    The Derivative of Bessel Function of the Second Kind

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

    Mathematica Bessel Function Evaluation Problem in Mathematica

    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
  10. J

    Derviation of bessel function of first kind via contour integration

    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...
  11. L

    How to prove bessel function J1/2(x) = sqrt(2/πx)sinx;

    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...
  12. S

    Bessel Function First Kind: Can Someone Help with C and RK Method?

    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...
  13. S

    Mutual inductance using bessel function

    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...
  14. C

    Integration of cosh ( bessel function )

    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...
  15. M

    Exploring Bessel Function Generating Function

    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...
  16. W

    Showing That the Modified Bessel Function of the First Kind is a Solution

    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...
  17. R

    Converting 2nd order ODE to Bessel Function

    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...
  18. V

    What is the domain of Bessel function J1(x)?

    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...
  19. K

    Express solution as bessel function

    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...
  20. Phrak

    Solving a Bessel Function DE in Electromagnetism | Cylindrical Coordinates

    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\ \ \ \ \ \ \...
  21. P

    Integral of first order (first kind) bessel function

    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...
  22. J

    A problem about integral of modified bessel function

    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...
  23. M

    Bessel Function Summation: Jo(x+y)

    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...
  24. G

    Prove a sum identity for bessel function

    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...
  25. M

    Analytically Solving Bessel Functions for x Giving J_m(x)=0

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

    How Do Bessel Functions Relate to Fourier Transforms in SHM Problems?

    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...
  27. R

    Bessel function explain this step

    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...
  28. I

    Bessel function for a 2D circular plate

    (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...
  29. A

    Bessel Function: Real v Parameter for y=0

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

    How Do You Integrate x^3 J_0(ax) Over 0 to R?

    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...
  31. quasar987

    Bessel function and Bessel D.E.

    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...
  32. I

    Solutions of D.E - Bessel Function

    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"...
  33. R

    Ploting zero order Bessel function

    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...
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