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Math Help - Complex Function Expansion

  1. #1
    Senior Member MaxJasper's Avatar
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    Question Complex Function Expansion

    If z is a complex variable z\in \mathbb{C}

    Prove that:

    e^{\frac{1}{2}a(z-1/z)}=\sum _{n=-\infty }^{\infty } J_n(a)z^n

    where:

    J_n(a)=\frac{1} {2\pi } \int _0^{2\pi }\text {cos} ({n*$\theta $} - a *{sin}(\theta )) d\theta
    Last edited by MaxJasper; October 19th 2012 at 10:59 AM.
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  2. #2
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    Re: Complex Function Expansion

    If you asked me that question 25 years ago, it would have been a cinch. All that I recall is that the function has an infinite (in both directions i.e. -inf to inf on the sum, as YOU have. One must apply the Residue Theorem judiciously. I shall try off line ans see if I can put it together. MAYBE, I''l be back!
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