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For a positive integer p, consider the holomorphic function
 for 

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

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  1. All even integers
  2. All odd integers
  3. All multiples of 3
  4. All multiples of 4

Answer (Detailed Solution Below)

Option 2 : All odd integers
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Detailed Solution

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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) = z ∈  \{0}

⇒ g'(z) = 

⇒ g'(z) = 

Integrating both sides we get

g(z) = 

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

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