Complex analysis question.

Regarding the Cauchy integral theorem which states the following: "if two different paths connect the same two points, and a function is holomorphic everywhere "in between" the two paths, then the two path integrals of the function will be the same."

- will the integral of f(z) always be 0 if the C is defined as |z|=constant (which means a circle thus simply connected) and f is holomorphic ?

- how can one tell if C is simply connected ?

- is there a faster way to prove a function f(z) is holomorphic without going through the Cauchy–Riemann equations ?

Question

Complex analysis question. 

Regarding the Cauchy integral theorem which states the following: "if two different paths connect the same two points, and a function is holomorphic everywhere "in between" the two paths, then the two path integrals of the function will be the same."

- will the integral of f(z) always be 0 if the C is defined as |z|=constant (which means a circle thus simply connected) and f is holomorphic ?

- how can one tell if C is simply connected ?

- is there a faster way to prove a function f(z) is holomorphic without going through the Cauchy–Riemann equations ?