∫γ (ze^(z^2))dz when γ is the unit circle

∫γ (ze^(z^2))dz when γ is the unit circle

Since the integrand is analytic at any complex number, this integral equals 0 by Cauchy's Theorem.

Alternately, the integrand has antiderivative (1/2)e^(z^2) at any z in C.
Since the contour is a closed loop, its beginning and end points are equal, and thus the integral equals 0.

I hope this helps!