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Find the first 3 terms of the asymptotic expansion of Jn(x)

  1. Feb 24, 2016 #1
    1. The problem statement, all variables and given/known data
    The bessel function Jn(x) is defined by the integral
    Jn(x)=1/(inπ)∫0πeixcosφcos(nφ)dφ

    From this formula, find the first 3 terms of the asymptotic expansion of Jn(x) when x=n and n is a large positive integer.

    2. Relevant equations


    3. The attempt at a solution
    I tried combining the cos and exp term together
    Jn(x) = ∫π ei(xcosφ+nφ)
     
    Last edited by a moderator: Feb 24, 2016
  2. jcsd
  3. Feb 24, 2016 #2

    Mark44

    Staff: Mentor

    How is this justified? ##e^{\cos(mx)} \cdot \cos(ny) \ne e^{\cos(mx + ny)}##.
     
  4. Feb 24, 2016 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Aside from the missing factor outside the integral, and a possible factor of 2 or 1/2, what he wrote is OK because he switched from ##\int_{\phi=0}^{\pi} \cdots## to ##\int_{\phi=-\pi}^{\pi} \cdots##, and used the evenness of the ##\cos(\phi)## in the exponential.
     
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