these functions are analytic, then why integration is zero?

Question

for both integrals below, consider c to be ccw unit circle.
I used Cauchy integral formula and I calculated the integral below:
$$\int \frac{{e^z}^2}{z^2} dz = 0$$ I used unit circle parametrization and I calculated the integral below:
$$\int Re(z).\bar{z} dz = 0$$ But does the results of integrals being zero mean, functions inside integrals are analytic on and in c? ( bc, Cauchy Goursat says integration would yield a zero if function inside integral is analytic every where in and on c) . They are not analytic I believe, so why are the integrals become zero?

Answer 1

if f is continuous on G (G is simply connected)and the integral of f = 0 then f is analytic on G.

(converse of Cauchy theorem is true if f is continuous)