Lack of existence of antiderivative for complex functionI have been asked to show that the function f ( z ) = z z 2 + 1 does not have an antiderivative on the subset { z ∈ C : 1 &lt; | z | }. However I don't see why this is the case, we can write z z 2 + 1 = 1 2 ( 1 z + i + 1 z − i ) whose naive antiderivative (using ln) has branch points at so taking a straight line between these points as our branch cut should allow us to make it continuous and single valued on our region. What am I missing?I know I am making a mistake somewhere as the residue theorem tells me the integral around a circle of radius 2 has value 2 π i and not 0 so no antiderivative exists. Any help is much appreciated.

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Lack of existence of antiderivative for complex functionI have been asked to show that the function   f  (  z  )  =      z                  z        2            +      1       does not have an antiderivative on the subset   {  z  ∈      C    :  1  &amp;lt;      |    z      |    }. However I don&#039;t see why this is the case, we can write       z                  z        2            +      1        =      1    2    (      1          z      +      i        +      1          z      −      i        ) whose naive antiderivative (using ln) has branch points at so taking a straight line between these points as our branch cut should allow us to make it continuous and single valued on our region. What am I missing?I know I am making a mistake somewhere as the residue theorem tells me the integral around a circle of radius 2 has value   2  π  i and not 0 so no antiderivative exists. Any help is much appreciated.

charringpq49u · Accepted Answer

Explanation:Consider a positively oriented circle   Γ of radius   &amp;gt;  1 centred at the origin.            1              2        π        i                  ∮    Γ              z                        z          2                +        1              d  zis the sum of the residues of   z      /    (      z    2    +  1  ), namely 1. But if your function had an antiderivative on that region, the result would have to be 0.

Lack of existence of antiderivative for complex function I have been asked to show that the functio

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