State an antiderivative F : D ⟶ C or explain why an antiderivative does not exist.State an antiderivative F : D ⟶ C or explain why an antiderivative does not exist.a) f ( z ) = z 2 , D = C b) f ( z ) = z sin ⁡ z , D = C c) f ( z ) = | z | 2 , D = C d) f ( z ) = z ¯ , D = C For a) and b) I just performed simple integration for the equationsa) z 3 / 3 + C and b) − z cos ⁡ z + sin ⁡ z + C but I am stuck on c) and d). It seems like an antiderivative does not exist but how do i explain that?

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State an antiderivative   F  :  D  ⟶      C   or explain why an antiderivative does not exist.State an antiderivative   F  :  D  ⟶      C   or explain why an antiderivative does not exist.a)   f  (  z  )  =      z    2    ,  D  =      C  b)   f  (  z  )  =  z  sin  ⁡  z  ,  D  =      C  c)   f  (  z  )  =      |    z            |        2    ,  D  =      C  d)   f  (  z  )  =      z    ¯    ,  D  =      C  For a) and b) I just performed simple integration for the equationsa)       z    3        /    3  +  C and b)   −  z  cos  ⁡  z  +  sin  ⁡  z  +  C but I am stuck on c) and d). It seems like an antiderivative does not exist but how do i explain that?

Cristal Roth · Accepted Answer

Step 1If   F  :  D  →      C   is complex differentiable at all points of D then it is a holomorphic function in D, and its derivative F′ is holomorphic in D as well. (That is a consequence of, e.g., Cauchy&#039;s differentiation formula, and implies that F is infinitely often complex differentiable.)Step 2You can verify (e.g. with the Cauchy-Riemann equations) that the functions f in (c) and (d) are not holomorphic, and therefore not the derivative of a holomorphic function F.The restriction   D  =      C   is irrelevant in these cases,   f  (  z  )  =      |    z            |        2   and   f  (  z  )  =      z    ¯   are not holomophic in any open subset   D  ⊂      C

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