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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 n0n_{0} denote the number of poles and zeros of f inside C. Which one of the following is true?

  1. A.(1/2πi)∫C((1/2\pi{}i) \int_C (zf)′/(zf) dz =n0−np+1= n_{0} - n_p + 1.✓ Correct
  2. B.(1/2πi)∫C((1/2\pi{}i) \int_C (zf)′/(zf) dz =n0−np−1= n_{0} - n_p - 1.
  3. C.(1/2πi)∫C((1/2\pi{}i) \int_C (zf)′/(zf) dz =n0−np= n_{0} - n_p.
  4. D.(1/2πi)∫C((1/2\pi{}i) \int_C (zf)′/(zf) dz =np−n0= n_p - n_{0}.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

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

More on this topic

The chapter behind this: Argument principle, Rouché and zero counting — free to read

From Zeros and Mappings › Argument principle, Rouché's theorem, open mapping

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