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Math Help - Proof required for integration involving analytic function.

  1. #1
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    Question Proof required for integration involving analytic function.

    The question goes this way:

    (Q) The complex variable function f(z) is analytic in the complex plane, and |f(z)| < M, where M is a constant.

    Prove that

    I = ∫C f(z) / [(z-a)(z-b)] dz = 0,

    where a, b are two different complex number, and C is the circle |z| = R, (R > |a|, R> |b|).
    Let R→∞

    Using residual theorem, I have only managed to reach this far:
    I = 2πi {[f(a)-f(b)]/(a-b)}

    Now, how do I prove that f(a)=f(b)?

    Thanks.
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  2. #2
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    Quote Originally Posted by jalal0 View Post
    The question goes this way:

    (Q) The complex variable function f(z) is analytic in the complex plane, and |f(z)| < M, where M is a constant.

    Prove that

    I = ∫C f(z) / [(z-a)(z-b)] dz = 0,

    where a, b are two different complex number, and C is the circle |z| = R, (R > |a|, R> |b|).
    Let R→∞

    Using residual theorem, I have only managed to reach this far:
    I = 2πi {[f(a)-f(b)]/(a-b)}

    Now, how do I prove that f(a)=f(b)?

    Thanks.
    If |f(z)|\leq M for all points in the complex plane then by Liouville's theorem the function is constant and so f(a) = f(b) completing the proof.
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