Showing that some Holomorphic Functions have AntiderivativesShow that there is a holomorphic function defined in the set Ω = { z ∈ C : | z | &gt; 4 }whose derivative is z ( z − 1 ) ( z − 2 ) ( z − 3 ) Is there a holomorphic function on Ω whose derivative is z 2 ( z − 1 ) ( z − 2 ) ( z − 3 ) ?I know that each holomorphic function on an open convex set has an antiderivative, but Ω isn't convex. Should I look for functions with these derivatives, such as f ( z ) = 1 2 ( log ⁡ ( 1 − z ) − 4 log ⁡ ( 2 − z ) + 3 log ⁡ ( 3 − z ) ) and g ( z ) = 1 2 ( 9 log ⁡ ( z − 3 ) − 8 log ⁡ ( z − 2 ) + log ⁡ ( z − 1 ) ), or are there any well-known propositions on finding antiderivatives in open, non-convex sets?

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Showing that some Holomorphic Functions have AntiderivativesShow that there is a holomorphic function defined in the set  Ω  =  {  z  ∈      C    :  |  z  |  &amp;gt;  4  }whose derivative is       z          (      z      −      1      )      (      z      −      2      )      (      z      −      3      )      Is there a holomorphic function on   Ω whose derivative is            z      2              (      z      −      1      )      (      z      −      2      )      (      z      −      3      )            ?I know that each holomorphic function on an open convex set has an antiderivative, but   Ω isn&#039;t convex. Should I look for functions with these derivatives, such as   f  (  z  )  =      1    2    (  log  ⁡  (  1  −  z  )  −  4  log  ⁡  (  2  −  z  )  +  3  log  ⁡  (  3  −  z  )  ) and   g  (  z  )  =      1    2    (  9  log  ⁡  (  z  −  3  )  −  8  log  ⁡  (  z  −  2  )  +  log  ⁡  (  z  −  1  )  ), or are there any well-known propositions on finding antiderivatives in open, non-convex sets?

partyjnopp9wa · Accepted Answer

Step 1We have that:      1          (      z      −      1      )      (      z      −      2      )        =      1          z      −      2        −      1          z      −      1        ,,hence:       1          (      z      −      1      )      (      z      −      2      )      (      z      −      3      )        =      1          2      (      z      −      1      )        −      1          z      −      2        +      1          2      (      z      −      3      )      and                     (1)                              z                      (            z            −            1            )            (            z            −            2            )            (            z            −            3            )                          =                  1                      2            (            z            −            1            )                          −                  2                      z            −            2                          +                  3                      2            (            z            −            3            )                              Step 2as well as:                     (2)                                          z            2                                (            z            −            1            )            (            z            −            2            )            (            z            −            3            )                          =                  1                      2            (            z            −            1            )                          −                  4                      z            −            2                          +                  9                      2            (            z            −            3            )                              where       1          z      −      1        ,      1          z      −      2        ,      1          z      −      3       are regular functions over the domain   D  =  {  z  ∈      C    :      |    z      |    ≥  4  }, so that primitives for the LHS of (1) or (2) over D can be written in terms of   log  ⁡  (  z  −  1  ),   log  ⁡  (  z  −  2  ),   log  ⁡  (  z  −  3  )

Showing that some Holomorphic Functions have Antiderivatives Show that there is a holomorphic funct

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