Evaluate the following integral:
∫γ1ez−1dz where γ:[0,1]→C is a parameterization of the unit circle oriented counter clockwise.
Attempt at the solution:
Use the parameterization γ(t)=e2πit
∫γ1ez−1dz=∫γe−z1−e−zdz=∫102πie2πitee2πit1−ee2πitdt
let u(t)=e2πit so that du=2πie2πitdt
∫e2πi1e−u1−e−udu
let v(t)=1−e−u so that dv=e−udu
∫1−e2πi1−e−11vdv=log(1−e2πi)−log(1−e−1)
I think there is something fishy going on here. Since the log function is not defined the same way as in the case of the real numbers. Can we treat the upper and lower limits in the integral in the same way as with real numbers? Are the change of variables techniques still applicable?
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