Stein 책의 증명을 웬만해서는 좋아하지만, 몇몇 개의 증명은 너무 말로만 넘긴 느낌이 들어서, 논리를 채워야 하는 경우가 종종 발생한다.

복소변수 함수를 공부하다보면, 실변수 함수와는 다른 성질을 갖는 경우가 종종 발생한다. 대표적인 예가 다음과 같은 경우가 아닐까 싶다.

Theorem.If is holomorphic and injective, then for all .

*Proof.* Suppose for some . Then note that and are analytic at . So there is such that and for all .

Let . Then for , we note that

By definition of , for all with , we have

So by Rouché’s theorem, and have same number of zeros counting with multiplicities inside .

Write in . Then in . Since , this shows that

where and is holomorphic at and .

So has zeros by definition of and the previous observation. Since and for all , the zeros of are all simple. But is injective, a contradiction.