I would like to understand this theorem better. If f(z) is analytic in a simply connected domain D, then for every simple closed path C in D we have [CONTOUR INTEGRAL]f(z)dz=0 The simply closed path is also called a contour. Now if we have a function which is entire in its domain (is analytic everywhere) say exp(z) or cos(z) or even sec(z), then the contour integral is zero. Right? So what is special about this? Is it that we are considering the 'complex' version of an INDEFINITE integral? I have done complex integration before with given endpoints (like the integral of z^n from 1+i to 0 as an example), but now Mr Cauchy is telling me that the integral of z^n is actually zero? Obviously I am not getting this idea of 'taking paths' through the complex plane and thus evaluating contour integrals. Could someone (probably lethe ) give me an explanation or forward me to a site. Cheers.