Complex integration along a curve

Question

I have to calculate this integral: $$ \int_C e^z\,dz $$ where $C$ is the circle $|z - jπ/2| = π/2$ from the point $z = 0$ to the point $z = jπ$.

I know how to calculate these with circles which central point is at $(0,0)$ but I don't know how it works with this.. So I will be very glad if I got any tips or help. Thanks!

Answer 1

Use the complex version of the fundamental theorem of calculus: $F(z) = e^z$ is an anti-derivative of your $f$, so $$ \int_C e^z\,dz = F(i\pi) - F(0) = -2. $$

Answer 2

$e^z$ is an entire function so by Cauchy integral theorem, your integral is 0.