Mathematics · Complex Analysis
Let D = {z ∈ ⊆ : |z| < 1} be the open unit disc and C be its positively oriented boundary {|z| = 1}. Fix a finite set {z1, z2, ..., zn} ⊆ D of distinct points and consider the polynomial g(z) = (z - z1)(z - z2)...(z - zn) of degree n. Let f be a holomorphic function in an open neighbourhood of the closure of D, and define P(z) = 12πi ∫C f(ζ) g(ζ) - g(z)(ζ - z)g(ζ) dζ. Which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1P(zj) = 0 for all j = 1, 2, ..., n.
- 2P is a rational function on ⊆ with poles at z1, z2, ..., zn.
- 3P is a polynomial of degree at most n - 1.✓
- 4P is a polynomial of degree n.
Correct answer
P is a polynomial of degree at most n - 1.
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