I have an example I want to clearify,
Let C be a semicircle from 2 to -2 which passes through 2i and let T be a semicircle from 2 to -2 which passes through -2i.
If you take the integral of z^2 dz around both paths the is the same as the function is analytic so the integral is independant of the path,
However if you take the integral of the conjugate of z over these to paths you get
So they equal, but the f(z)=conjugate of z isnt an analytic function, is it just a coincidence?