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.

Question

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.

tennispopj8 · Accepted Answer

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 · Accepted Answer

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

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

Answered question

Answer & Explanation

New Questions in Algebra II