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Esther Hoffman

Esther Hoffman

Answered question

2022-04-30

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 equations
a) 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?

Answer & Explanation

Cristal Roth

Cristal Roth

Beginner2022-05-01Added 13 answers

Step 1
If 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's differentiation formula, and implies that F is infinitely often complex differentiable.)
Step 2
You 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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