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Prove That Ind (Gamma) Is Always an integer

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Let gamma be a continuously differentiable closed curve in the complex plane, with parameter interval [a,b], and assume that gamma (t) is not equal to 0 for every t in [a,b]. Define the index of gamma to be Ind (gamma) = 1/2pii integral from a to b of gamma prime (t) over gamma (t) dt. Prove that Ind (gamma) is always an integer.

Hint: There exist Rho on [a,b] with Rho prime = gamma prime over gamma, Rho (a) = 0. Hence gamma exp (- Rho) is constant. Since gamma (a) = gamma (b) it follows that exp Rho (b) = exp Rho (a) = 1. Note that Rho (b) = 2pi i Ind (gamma).

Compute Ind (gamma) when gamma (t) = e^ int, a = 0, b = 2 pi. Explain why Ind (gamma) is often called the winding number of gamma around 0.

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The solution proves whether Ind (gamma) is always an integer.

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