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Math Help - [SOLVED] showing a complex-valued function is entire

  1. #1
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    [SOLVED] showing a complex-valued function is entire

    f is continuous on \mathbb{C} and analytic on \mathbb{C} \setminus \mathbb{R}.
    Prove that f is an entire function.

    I know that I need to apply Morera's Theorem which states that if f is a continuous, complex-valued function defined on an open set D \in \mathbb{C}, satisfying

    \oint f(z)dz = 0
    for every closed curve C in D, then f must be holomorphic on D.

    I guess I'm not sure how to show that f satisfies \oint f(z)dz = 0. I thought I could use Cauchy's integral theorem to show that the function does satisfy this condition but I only know that f is holomorphic on \mathbb{C} \setminus \mathbb{R}. Any help would be much appreciated.
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  2. #2
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    split \oint f(z)dz = 0 into two integrals over two different closed contours: one that lies in the upper half plane and one in the lower. by the Cauchy theorem each of the integrals equals 0.
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  3. #3
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    Thank you for your help. I can see how to apply Cauchy's integral theorem in this manner, but in conjunction with Morera's theorem it seems as though I am only showing that f is holomorphic on \mathbb{C}\setminus\mathbb{R} but in order to show that f is entire I must show it is holomorphic on \mathbb{C} Is there something I am missing here?
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  4. #4
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    take *any* contout in the complex plane and divide it into two by splitting along the real line. you show that the integral is zero by Cauchy, and that implies f is entire
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