Theorem 1(Cauchy-Green formula, Pompeiu formula)
Suppose that
We can prove this by first consider the domain
in
Finally, let
Theorem 2 (Cauchy Integral Formula)
Suppose that
This is because
Theorem 3 (Cauchy Integral Theorem)
Suppose that
Note that Theorem 2 and Theorem 3 are equivalent.
Cauchy-Goursat Theorem
It is often too strong to have smooth boundary. We can refine the condition and get the following theorems.
Theorem 2’
Suppose that
Theorem 3’
Suppose that
Note that Theorem 2’ and Theorem 3’ are also equivalent. And the theorems are true for multi-connected fields.