For a positive integer p, consider the holomorphic function
\(f(z)=\frac{\sin z}{z^p}\) for \(z ∈ \mathbb{C} \backslash\{0\}\)

For which values of p does there exist a holomorphic function g ∶ \(\mathbb{C}\) \{0} → \(\mathbb{C}\) such that f(z) = g'(z) for z ∈ \(\mathbb{C}\) \{0}?

Concept:

A function f(z) is said to be holomorphic in a domain D if f(z) has no singularities in D.

Explanation:

g'(z) =\(f(z)=\frac{\sin z}{z^p}\) z ∈ \(\mathbb{C}\) \{0}

⇒ g'(z) = \(\frac{1}{z^p}(z-\frac{z^3}{3!}+\frac{z^5}{5!}-...)\)

⇒ g'(z) = \((\frac{z^{1-p}}{1!}-\frac{z^{3-p}}{3!}+\frac{z^{5-p}}{5!}-...)\)

Integrating both sides we get

g(z) = \((\frac{z^{2-p}}{1!(2-p)}-\frac{z^{4-p}}{3!(4-p)}+\frac{z^{6-p}}{5!(6-p)}-...)\)

So g(z) can not be holomorphic if p is a multiple of 2, 3 and 4.

∴ Options (1), (3) and (4) are not correct.

Hence option (2) is correct