Problem evaluating a contour integral using parametrization

You are assuming that there is a differentiable function $\ln$ from $\mathbb{C}\setminus \{0\}$ into $\mathbb C$ such that $\ln'(z)=\frac1z$. There isn't.


Because there is no branch of logarithm in $\mathbb{C}\setminus \{0\}$, so your function has no antiderivative. That's a big difference between real and complex analysis-you can't take the elementary real functions and just use them in the complex plane. Logarithm is a function which requires a branch.

Tags:

Complex Analysis

Logarithms

Contour Integration

Complex Integration

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