Prove. Let C be a simple closed curve containing point a in its interior. Then
We prove the first part first.
I Prove: If C be a simple closed curve containing point a in its interior, then
Proof. Construct a circle Γ of radius ε with center at a of such size that Γ lies entirely inside C (since a is an interior point it can be done). See Fig. 1. By the Principle of the Deformation of Contours we have
Now on Γ, |z - a| = ε and z - a = εeiθ, where z - a represents a vector from point a to point z on Γ with an amplitude of θ. Thus z = a + εeiθ , 0 θ 2π. Since dz = iεeiθdθ, 1) becomes
We now prove Part 2.
II Prove: If C be a simple closed curve containing point a in its interior, then
Proof. Again we construct circle Γ as in part I and by reasoning similar to the above we have