Mathematics · Complex Analysis
Let S be a finite subset of ℂ containing 0. Let f : ℂ ∖ S → ℂ be a holomorphic function which has a simple pole at 0. For R > 0, let γR denote the path γR(t) = R e2πi t for t ∈ [0, 1]. Which of the following statements is necessarily true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1If S is not a singleton set, then ∫γR f(z)dz = 0 for some R > 0 such that S ∩ { z ∈ ℂ : |z| = R } = ∅.
- 2If ∫γR f(z)dz = 0 for some R > 0, then S is not a singleton set.✓
- 3The function f(1/z) has an essential singularity at 0.
- 4The function f(1/z) has a removable singularity at 0.
Correct answer
If ∫γR f(z)dz = 0 for some R > 0, then S is not a singleton set.
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