Complex Integration: Int C z*e^(z^2)dz where C is line segme

[lastlydreaming]

How would one evaluate the following:

Integral C z*e^(z^2)dz where C is the line segment that joins (1+i) to (3+2i)

From 0<=t<=1, I find z=(1+i)(1-t) +(3+2i)(t)
to be z=(2t+1)+ i(t+1).
Then dz is (2+i) dt.

How do I take the integral of the function z*e^(z^2)dz ?
I tried integration by parts but I managed to go into circles. Thanks.

 

[pka]

This is absolutely trivial. An entire function has a path-independent integral.

\(\displaystyle \L
f(z) = e^{z^2 } \quad \Rightarrow \quad f'(z) = 2ze^{z^2 }\)