Use Cauchy Integral Formula to evaluate the integral

[DanniHuang]

Homework Statement

The question is needed to be done by using an appropriate substitution and the Cauchy Integral Formula.

Homework Equations

Evaluate the complex integral: ∫e^(e^it) dt, from 0 to 2∏

The Attempt at a Solution

I cannot find an appropriate substitution for the integral.

 

[Kreizhn]

Perhaps the best way to see how to do this question is that we somehow need to make this into a contour integral. The easiest things around which to integrate are circles right? For example, if f(z) is a complex function and we want to integrate around the unit circle \{ z \in \mathbb C: |z|^2 = 1 \}, it may be convenient to parameterize the circe as z = e^{it}. Thus if somebody asks you to integrate e^{e^{it}} for t \in [0,2\pi), what function is being integrated about what contour?