Part BCSIR NET December 2023argument-principle-extra-factor

Argument principle extra factor

Let f be a meromorphic function on an open set containing the unit circle C and its interior. Suppose f has no zeros and no poles on C, and let n_p and $n_{0}$ denote the number of poles and zeros of f inside C. Which one of the following is true?

A. $(1/2 π i) \int_{C} ($ zf)′/(zf) dz $= n_{0} - n_{p} + 1$ .
B. $(1/2 π i) \int_{C} ($ zf)′/(zf) dz $= n_{0} - n_{p} - 1$ .
C. $(1/2 π i) \int_{C} ($ zf)′/(zf) dz $= n_{0} - n_{p}$ .
D. $(1/2 π i) \int_{C} ($ zf)′/(zf) dz $= n_{p} - n_{0}$ .

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Execution slip

The idea was right. The computation was not.

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Related counterexample: f′(z) ≠ 0 everywhere implies f is injective

Argument principle extra factor

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