Results 1 to 4 of 4

Math Help - Laplace transform of Bessel functions

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    2

    Laplace transform of Bessel functions

    Dear all,

    I would like to ask if anyone of you knows where to get the Laplace transform of the following function:

    f(t) = I_n(a*t) * J_m(b*sqrt{t})

    With

    I_n() = Modified Bessel function of the first kind, of integer order n
    J_m() = Bessel function of the first kind, of integer order m
    a,b = real numbers



    I was trying to look up in the book

    A.P. Prudnikov, "Integrals and series, vol. 4 Laplace transforms"

    but is not available in my library until march 8th!


    I thank your kind help,
    Home-made-bread.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by homemadebread View Post
    Dear all,

    I would like to ask if anyone of you knows where to get the Laplace transform of the following function:

    f(t) = I_n(a*t) * J_m(b*sqrt{t})

    With

    I_n() = Modified Bessel function of the first kind, of integer order n
    J_m() = Bessel function of the first kind, of integer order m
    a,b = real numbers



    I was trying to look up in the book

    A.P. Prudnikov, "Integrals and series, vol. 4 Laplace transforms"

    but is not available in my library until march 8th!


    I thank your kind help,
    Home-made-bread.
    I don't know the legal status of Library Genesis but they have a copy of volume 4 of Prudnikov in djvu format (for which you can download a free reader).

    So google for Library Genesis and take it from there.

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2010
    Posts
    2
    Quote Originally Posted by CaptainBlack View Post
    I don't know the legal status of Library Genesis but they have a copy of volume 4 of Prudnikov in djvu format (for which you can download a free reader).

    So google for Library Genesis and take it from there.

    CB
    Hello everybody,

    I first thank CaptainBlack so much for answering, I even did not know about Library Genesis. I think I will use it a lot in the future!


    However, Prudnikov's book does not give the result I am looking for. Any other suggestion?


    Thanks in advance.
    My best regards, Home-made-bread
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Quote Originally Posted by homemadebread View Post
    . Any other suggestion?


    Thanks in advance.
    My best regards, Home-made-bread
    Yeah, numerically. Ok, suppose I wanted to know what the Laplace Transform of that was at n=2, m=3, a=1,b=1/2,s=2 then I'd do the following in Mathematia:

    Code:
    In[3]:= n = 2;
    m = 3;
    a = 1;
    b = 1/2;
    s = 2;
    NIntegrate[
     BesselI[n, a t] BesselJ[m, b Sqrt[t]] Exp[-s t],
     {t, 0, \[Infinity]}]
    
    Out[8]= 0.000288939
    Bingo. Then I claim:

    \mathcal{L}\left\{I_2(t) J_3(1/2\sqrt{t})\right\}\biggr|_{s=2}\approx 0.00029

    and if I needed to, I could do this for any values of the parameters and as long as the integration is well-behaved, I could obtain the value of the transform pretty accurately and if necessary, run a fit on the data points to obtain a least-square fit function f(s) even if s were complex.
    Last edited by shawsend; January 23rd 2010 at 05:14 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Laplace transform of a product of functions
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: November 13th 2011, 12:48 PM
  2. Laplace transform involving Heaviside functions
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: August 14th 2011, 06:50 AM
  3. Laplace transform and Fourier transform what is the different?
    Posted in the Advanced Applied Math Forum
    Replies: 8
    Last Post: December 29th 2010, 10:51 PM
  4. fourier transform of the bessel function
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 4th 2009, 12:18 AM
  5. Fourier Transform of a Bessel Function
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: June 26th 2008, 08:42 PM

Search Tags


/mathhelpforum @mathhelpforum