Let $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ denote a function that satisfies the following equation: $$f(w,z+1)=w^{f(w,z)}$$ Is there a unique holomorphic function $f$ that satisfies this equation? If so, what is it? If not, what additional specifications are needed to find one?
Furthermore, $$f(w,1)=w$$