## 2008年11月16日 星期日

### Analytic function on a disk

Suppose f is a holomorphic function on the unit disk that is continuous up to the boundary. If f vanishes on an arc of the boundary circle, show that it is identically zero.

This is a standard question from complex analysis. I just learn today that there is a really cute solution to it. You may want to think about it before continuing.

The classical method is to use Schwartz reflection principle and argue that you can analytically continue the function outside the disk a little bit; then the continued function vanishes on a segment, so it must vanish identically.

The cute solution that I was referring to is the following: take (finitely many) copies of f, rotate each of them suitably and multiply the rotated functions altogether. Then the product is going to vanish identically on the boundary of the unit disk, and of course the product is holomorphic inside the disk. Hence the product is identically zero, and thus f is identically zero. (Just argue that all derivatives of f vanishes at the origin by differentiating the product.)