Complex number question

Let f be an analytic function in C. Assume that f satisfies f(z + 2π) = f(z), f(z + 2πi) = f(z), for all z ∈ C. Show that f is constant.

Let Ω ⊂ C be a domain and f : Ω → C be analytic. Assume that the function cos(f(z)) is constant. Show that f is constant.

Sample Solution