Prove Euler Identity without using Euler Formula

[mathwonk]

if you integrate the form dlog(z) = dz/z around the upper half of the unit circle, and get ipi, it seems to me you have proved that e^(ipi) = -1.

i.e. here e^z is defined as the inverse of the integral of dz/z.

 

[morphism]

How are you evaluating that integral?