Results 1 to 2 of 2

Math Help - Airy Functions problem

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    36

    Airy Functions problem

    Show that  y(z)=\frac{1}{2\pi i}\int_{C}{exp(zt-\frac{t^{3}}{3})dt} is a solution of  y''(z)=zy(z)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    I think you need an i in there huh? Like:

    A(s)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(z^3/3+sz)}dz

    then A''-sA=0

    Differentiate under the integral sign and use Cauchy's Theorem to express the Airy integral in terms of a contour integral over a square region bordering the real axis of height say i. Then since the integrand is analytic, we can express the Airy integral in terms of the horizontal leg at z=x+i since the integral over the two vertical legs tend to zero. Then:

    Ai(s)=\frac{1}{2\pi}\int_{\infty}^{-\infty} e^{i(z^3/3+sz)}dz,\quad z=x+i

    Now:

    A''-sA=-\frac{1}{2\pi}\int_C (z^2+s)e^{i(z^3/3+sz)}dz;\quad z=x+i

    but that integrand is the differential of f(z)=ie^{i(z^3/3+sz)} and since the integrand is analytic, we can just evaluate it at the endpoints:

    \frac{1}{2\pi}\int_{\infty}^{-\infty}df

    and note if you substitute z=x+i into f, it's order is O(e^{-x^2}) so we have:

    O(e^{-x^2})\biggr|_{\infty}^{-\infty}

    which is zero.
    Last edited by shawsend; January 21st 2010 at 04:53 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Functions problem
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: September 11th 2011, 04:45 PM
  2. Functions problem
    Posted in the Algebra Forum
    Replies: 5
    Last Post: August 5th 2011, 05:34 AM
  3. Airy Function problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 27th 2008, 05:39 AM
  4. Airy function
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 14th 2008, 02:01 PM
  5. pde involving airy function
    Posted in the Calculus Forum
    Replies: 0
    Last Post: October 19th 2007, 02:47 AM

Search Tags


/mathhelpforum @mathhelpforum