Results 1 to 10 of 10

Math Help - bessel function-legendre polynomials

  1. #1
    Newbie
    Joined
    Jun 2011
    Posts
    5

    Unhappy bessel function-legendre polynomials


    Can anyone help me to solve integral of function exp(i alfa x)P_n(x)dx
    in limits form -1 to 1
    where P_n(x) is legendre polynomial.
    Solution sholud be
    sqr(2pi/alfa)iJ_n+1/2(alfa)
    where J_n+1/2(alfa)
    is modifided bessel function
    pls help!
    and sorry that i written it like this but i forgot how to use latex
    Attached Files Attached Files
    • File Type: pdf 1.pdf (25.2 KB, 24 views)
    Last edited by hannaa18; June 16th 2011 at 03:40 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    May 2010
    From
    Los Angeles, California
    Posts
    274
    Thanks
    1

    Re: bessel function-legendre polynomials

    One way to prove this is to start with the spherical harmonic expansion of a plane wave. Alternatively, you could verify directly that J_{n+1/2} satifies the approproaite d.e. with initial conditions.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jun 2011
    Posts
    5

    Re: bessel function-legendre polynomials

    can you please show me how to verify this.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    May 2010
    From
    Los Angeles, California
    Posts
    274
    Thanks
    1

    Re: bessel function-legendre polynomials

    What text are you using?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jun 2011
    Posts
    5

    Re: bessel function-legendre polynomials

    an introduction to the teory of functions of a complex variable by e.t.copson
    page 341.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jun 2011
    Posts
    5

    Re: bessel function-legendre polynomials

    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    May 2010
    From
    Los Angeles, California
    Posts
    274
    Thanks
    1

    Re: bessel function-legendre polynomials

    I'll take a look at your text tomorrow. In the meantime, here's the plane wave expansion

    e^{ikr\cos \gamma}=\sum_{n=0}^\infty a_nj_n(kr)P_n(\cos \gamma).

    where j_n is the spherical Bessel function. From this you can deduce the integral representation of j_n.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    May 2010
    From
    Los Angeles, California
    Posts
    274
    Thanks
    1

    Re: bessel function-legendre polynomials

    OK, I have his book and I located the question. It seems he wants to deduce Bauer's formula from the result and so we can't start with the plane wave expansion. Give me a little time to think of the argument intended by the author.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Senior Member
    Joined
    May 2010
    From
    Los Angeles, California
    Posts
    274
    Thanks
    1

    Re: bessel function-legendre polynomials

    Here's the argument I think Copson is expecting. From example 2 on page 320 we have

    J_\nu (z)=\frac{(\frac{1}{2}z)^\nu}{\Gamma(\nu+\frac{1}{  2})\Gamma(\frac{1}{2})}\int_0^\pi\cos(z\cos \theta)\sin^{2\nu}\theta\, d\theta .

    Now making the substitution t=\cos \theta gives

    J_{n+\frac{1}{2}}=\frac{(\frac{1}{2}z)^{n+\frac{1}  {2}}}{\Gamma(n+1)\Gamma(\frac{1}{2})}\int_{-1}^1\cos(zt)(1-t^2)^n\, dt .

    Now integrate n times by parts and use Rodrigues's formula

    P_n(t)=\frac{1}{2^nn!}\frac{d^n}{dt^n}(t^2-1)^n.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Jun 2011
    Posts
    5

    Re: bessel function-legendre polynomials

    Thank you very much for your help!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Zeros of Legendre polynomials
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 7th 2011, 04:55 AM
  2. Using the generating function for the Legendre polynomials
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: May 17th 2011, 11:28 AM
  3. Quadrature Using Legendre Polynomials
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: December 29th 2010, 07:36 AM
  4. Legendre Polynomials
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 23rd 2008, 09:18 AM
  5. legendre polynomials
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 8th 2007, 07:23 AM

Search Tags


/mathhelpforum @mathhelpforum