I encountered a question in complex analysis. Given an analytic function on the complex plane, except for the positive real ray, and continuous everywhere - show it is entire.
My idea is to use Poisson formula on different disks around the origin, and get harmonic functions for both the real and imaginary parts, and then use the uniqueness of analytic functions if the agree on the boundary.
I wonder if there are different, more intuitive and elementary approaches to solve this problem.