Contour integration can be loads of fun: constructing assorted loops, contours or connected segments in z-space (complex space - see the blog posts on complex functions) then trying to graph them.
Since C is a contour then z’(t), i.e. dz/dt is piecewise continuous on the interval a < t < b so the existence of the integral is assured.
For which z = 3.8 exp (i q) (- p/2 < q < p/2)
= 14.44i òp/2 -p/2 dq
z = 0 + iy (0 < y < 1)
òC2 f(z) dz = ò 0 1 -i3x2( 1 +i) dx =
3(1-i) ò 0 1 x2dx = 1 – i