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Prove an integral representation of the zero-order Bessel function

  1. Jul 4, 2014 #1
    1. The problem statement, all variables and given/known data
    In section 7.15 of this book: Milonni, P. W. and J. H. Eberly (2010). Laser Physics.
    there is an equation (7.15.9) which is an integral representation of the zero-order Bessel function:

    [itex]J_0(\alpha\rho)=\frac{1}{2\pi}\int^{2\pi}_{0}e^{i[\alpha(xcos{\phi}+ysin{\phi})]}d\phi[/itex]

    This equation could also be found in this paper:
    Durnin, J. (1987). "Exact solutions for nondiffracting beams. I. The scalar theory." Journal of the Optical Society of America A 4(4): 651.

    2. Relevant equations
    here [itex]x=\rho cos{\phi}, y=\rho sin{\phi}[/itex].


    3. The attempt at a solution
    Rerwite it as:
    [itex]J_0(\alpha\rho)=\frac{1}{2\pi}\int^{2\pi}_{0}e^{i[\alpha(\rho cos^2{\phi}+\rho sin^2{\phi})]}d\phi[/itex]
    this lead to:
    [itex]J_0(\alpha\rho)=\frac{1}{2\pi}\int^{2\pi}_{0}e^{i[\alpha\rho]}d\phi[/itex]
    and after integral of \phi, this becomes
    [itex]J_0(\alpha\rho)=e^{i[\alpha\rho]}?[/itex]

    Also I tried to look it up in the handbook of mathematics by Abramowitz, M. but failed to find this equation, except one like this:
    [itex]J_0(t)=\frac{1}{\pi}\int^{\pi}_0 e^{itcos{\phi}}d\phi[/itex]
    this integral from 0 to [itex]\pi[/itex] could be rewritten to [itex]2\pi[/itex]

    [itex]J_0(t)=\frac{1}{2\pi}\int^{2\pi}_0 e^{-itcos{\phi}}d\phi[/itex]

    as http://math.stackexchange.com/quest...ic-integral-int-02-pi-e-2-pi-i-lambda-cost-dt
    describes.

    yet, this is not what I want.

    Still, this equation is not found in some wiki pages:
    http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
    http://en.wikipedia.org/wiki/Bessel_function

    Thanks for any reply!
     
    Last edited: Jul 5, 2014
  2. jcsd
  3. Jul 22, 2014 #2
    Well, I have found that the error appears in step 2.
    It should be phi' instead of phi.
    after that, it should be rho*cos(phi'-phi) above exp.
    and then the integral would be J0.

    Thanks anyway!
     
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