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I have a question about complex analysis. I am trying to get some intuition about residues and integration.

Let $c(t) = e^{2\pi it}$ be the unit circle of the complex plane.

I consider the following functions :
$f(z) = z$
$g(z) = 1/z$

Integrating $f$ around the $c(t)$ circle yields $0$ because f has no pole inside the unit disk.
Integrating $g$ around the $c(t)$ circle yields $2\pi i$ (residues theorem).

But $g(c(t)) = f(c(-t))$ : intuitively, I should be integrating the same function around the same circle, but in opposite directions. I would expect the integral values to be opposite, but I don't understand why one is null and the other not!

There is surely an obvious reasoning or calculus error, and I can't see where.

Thanks Yuufo

2 Answers 2