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Integrals with bessel functions

  1. Mar 4, 2009 #1
    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

    2*Pi*(int(sin(phi)*BesselI(0, a*sin(phi)), phi = 0 .. Pi)) (2)

    Can anyone solve (1) or (2)?
     
  2. jcsd
  3. Mar 5, 2009 #2

    Plugging it into Mathematica assuming a>0 gives 4*Pi*Sinh[a]/a.

    It doesn't tell the steps used, unfortunately :)
     
  4. Mar 5, 2009 #3
    Thanks!

    Apparently, Mathematica is the only program that can solve several of the integrals I am dealing with out of the box. Unfortunately, I do not have access to Mathematica.

    Would you please be so kind to try if Mathematica can solve

    int(exp(a*cos(phi))*(sin(phi))^2, phi = 0 .. Pi)

    with a a positive real constant?
     
  5. Mar 5, 2009 #4
    The integral

    assume(a > 0);
    int(exp(a*cos(phi))*sin(phi)^2, phi = 0 .. Pi);

    equals to

    (Pi/a)*BesselI(1,a)

    I have solved the integral using Mathematica, which seems to solve all these integrals out of the box.
     
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