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Find all the holomorphic functions ##f: \mathbb C \to \mathbb C## such that ##f'(0)=1## and for all ##x,y \in \mathbb R##,

##f(x+iy)=e^xf(iy)##

I am completely stuck with this exercise, for the second condition, I know that ##f(iy)=g(y)+ih(y)##, but is there something that ##g## and ##h## must satisfy? I would appreciate any hints.