Mathematics · Complex Analysis

For a positive integer p, consider the holomorphic function f(z) = sin z z p for z &in; &complex; \ {0}. For which values of p does there exist a holomorphic function g : &complex; \ {0} → &complex; such that f(z) = g'(z) for z &in; &complex; \ {0}?

Verified Solution4 OptionsUpdated: October 2026👁️1 View

1All multiples of 4
2All multiples of 3
3All odd integers✓
4All even integers

Correct answer

All odd integers

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For a positive integer p, consider the holomorphic function f(z) = sin z z p for z &in; &complex; \ {0}. For which values of p does there exist a holomorphic function g : &complex; \ {0} → &complex; such that f(z) = g'(z) for z &in; &complex; \ {0}?

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For a positive integer p, consider the holomorphic function f(z) = sin z z p for z &in; &complex; \ {0}. For which values of p does there exist a holomorphic function g : &complex; \ {0} &rarr; &complex; such that f(z) = g'(z) for z &in; &complex; \ {0}?

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For a positive integer p, consider the holomorphic function f(z) = sin z z p for z &in; &complex; \ {0}. For which values of p does there exist a holomorphic function g : &complex; \ {0} → &complex; such that f(z) = g'(z) for z &in; &complex; \ {0}?