State and prove Cauchys integral theorem?

Cauchy's Integral Theorem states that if ( f ) is a holomorphic function on a simply connected domain ( D ), then for any closed curve ( C ) within ( D ), the integral of ( f ) over ( C ) is zero:

[ \oint_C f(z) , dz = 0. ]

Proof Outline: Let ( f ) be holomorphic in ( D ) and ( C ) a closed curve in ( D ). Since ( f ) is holomorphic, it is differentiable everywhere in ( D ), and we can apply Green's Theorem in the plane, which relates the line integral around a closed curve to a double integral over the region ( R ) enclosed by ( C ). Since the partial derivatives of ( f ) are continuous, the integral of the derivatives over ( R ) is zero, thus confirming the result ( \oint_C f(z) , dz = 0 ).