Thread: Complex Analysis: find one complex number w such that w is not in the range f(z)= e^z

Find one complex number w such that w is not in the range of f(z)= e^z, and explain why this number is not in the range of f(z).

Take w=0.

The exponential function is never 0. Why? Write x = x+iy. Then e^z = e^x(cos(y)+isin(y)). Here e^x is the usual real exponential, which is well-known never to be 0. And cosine and sine are also well-known never to be 0 simultaneously, hence e^z is non-zero for all z.