Complex Analysis, holomorphic in circle.

[kidsasd987]

Hi, I have a question regarding corollary 2.3. in the uploaded image.

it looks very trivial to me becauese Cauchy's theorem states "if f(z) is holomorphic, its closed loop integral
will be always 0". Is this what the author is trying to say? what's the necesseity of the larger disk D' at here?
Why do we use D'?

 

[wrobel]

You understand right, just do not bother about that. Usually such a type argument (about D') is already contained in the proof of the Cauchy theorem. Look above how they formulate Cauchy theorem.

"if f(z) is holomorphic, its closed loop integral
will be always 0".

the loop must be in the domain where f is holomorphic and this loop must be shrinkable in this domain. and the loop must be contained compactly in the domain