Question

If c is a simple closed curve enclosing a region of area a, prove that a = zdz. 2i

Answer 1

Here's what I've done:

∫Cf(z) dz=∫C(u dx−v dy)+i ∫C(v dx+u dy)

=∫CP dx+Q dy

=∬R(Qx−Py) dA

=∬R(−vx−uy) dA +i∬R(ux−vy) dA

Here: f(z)=z¯ =x−iy =u+iv

Cauchy-Riemann: ux=1, vy=−1, and uy=0=vx

then,

∫Cz¯ dz=∬R(0) dA +i∬R(1−(−1)) dA

=2i ∬R1 dA

-- and I guess from here, the integration of has me a confused because the final answer is

12i∫z¯ dz