- #1

Apogee

- 45

- 1

Let C be the unit circle in the complex plane (|z| = 1). If you calculate the contour integral of (1/z)dz over C using Cauchy's Integral Formula, you get 2*pi*i. If you calculate it using the path z(t)=e^(it), t in [0,2pi], you also receive 2*pi*i.

However, if I do the second method, but make the substitution u = e^(it), the limits of integration (specifically 0 and 2pi) both become 1. Hence, after substitution, the integral is 0. Why is this the case? I imagine that u-substitutions work differently with complex integrals, but I'm trying to understand what is wrong with making this substitution.