Need help with a proof concerning zero-free holomorphic functions. Suppose f ( z ) i

Emanuel Keith

Emanuel Keith

Answered question

2022-06-05

Need help with a proof concerning zero-free holomorphic functions.
Suppose f ( z ) is holomorphic and zero-free in a simply connected domain, and that g ( z ) for which f ( z ) = exp ( g ( z ) )
The question I am answering is the following:
Let t 0 be a complex number. Prove that h ( z ) holomorphic such that f ( z ) = ( h ( z ) ) t
I see that the idea makes sense, but a nudge in the right direction would be appreciated.

Answer & Explanation

tennispopj8

tennispopj8

Beginner2022-06-06Added 20 answers

If f is zero free and it is defined in a simply-connected domain, you can define a logarithm of f as
r ( z ) = z 0 z f ( w ) d w f ( w ) .
This integral is univalent, as f / f is holomorphic in a simply-connected domain.
Clearly f ( z ) = f ( z 0 ) exp ( r ( z ) )
Then h ( z ) = w 0 exp ( r ( z ) / t ) , where w 0 t = f ( z 0 )
April Bush

April Bush

Beginner2022-06-07Added 6 answers

Nudge: Remember that for real numbers r > 0, you can define r t = exp ( t ln r ). Maybe you can do something similar for holomorphic functions?

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