Let U be a simply connected domain and let f be a meromorphic function on U with only finitely many zeroes and poles. Prove that there is a holomorphic function g : U → C and a rational function q, such that f ( z ) = e g ( z ) q ( z )

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Let U be a simply connected domain and let   f be a meromorphic function on U with only finitely many zeroes and poles. Prove that there is a holomorphic function g : U   →      C   and a rational function   q, such that  f  (  z  )  =      e          g      (      z      )        q  (  z  )

Danube2w · Accepted Answer

We at first prove the case were   f has neither poles nor zeros: We want to prove that if   f has no pole nor zeroes, it holds that   ∃    g  :  U  →      C   such that   ∀    z  ∈  U  :  f  (  z  )  =      e          g      (      z      )      Proof:We know that since   f has no zeroes in   U the function   h  (  z  )  =                    f        ′            (      z      )              f      (      z      )       is well defined in   U.Futhermore   U is simply connected therefore   ∃    F  :  U  →      C    :      F    ′    =  h  =                    f        ′            (      z      )              f      (      z      )      We now define   ϕ  :  U  →      C   and   ϕ  (  z  )  :=      e          −      F      (      z      )        f  (  z  )We see that ϕ is constant in   U since      ϕ    ′    (  z  )  =      e          −      F      (      z      )        (      f    ′    (  z  )  −      F    ′    (  z  )  f  (  z  )  )  =      e          −      F      (      z      )        (      f    ′    (  z  )  −  h  (  z  )  f  (  z  )  )  =      e          −      F      (      z      )        (      f    ′    (  z  )  −                    f        ′            (      z      )              f      (      z      )        f  (  z  )  )  =  0The last little definition, which will define our searched   g, needs that we pick a point       z    0    ∈  U and another       w    0   (depending on       z    0  ) such that       e                  w        0              =  f  (      z    0    )Now  g  :  U  →      C                g  (  z  )  :=  F  (  z  )  −  F  (      z    0    )  +      w    0  is holomorphic since   F is and it holds for all   z  ∈  U  ::      e          g      (      z      )        =      e          F      (      z      )            e          −      F      (              z        0            )            e                  w        0              =      e          F      (      z      )        ϕ  (      z    0    )  =      e          F      (      z      )        ϕ  (  z  )  =  f  (  z  )If   f has poles and zeroes:Let       z    1    ,  .  .  .  ,      z    k   be the poles (multiplicity       m                  z        1              ,  .  .  .  ,      m                  z        k            ) and       w    1    ,  .  .  .  ,      w    l   (multiplicity       m                  w        1              ,  .  .  .  ,      m                  w        l            ) the zeroes of   f then we just have to write   f as  f  (  z  )  =      e          g      (      z      )                          ∏                  j          =          1                          l                    (      w      −              w        j                    )                              m                                          w                j                                                                        ∏                  m          =          1                          k                    (      z      −              z        m                    )                              m                                          z                m

Let U be a simply connected domain and let f be a meromorphic function on U with only finitely

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