Find the Maximum and Minimum values of $e^z$ when $z\le 1$.

Question

I need help finding the maximum and minimum values of $|e^z|$ on $|z|\le1$. I know we use the maximum modulus theorom but i cant seem to get an answer.

Answer 1

Note that $f(z)=e^z \ne 0$, so by the maximum principle applied to $f$ and to $1/f$, $|f|$ takes on both its maximum and its minimum on the boundary, i.e., on $|z|=1$. Since $|e^z| = e^x$ (where $z=x+iy$), the maximum value is $e$ (at $z=1$) and the minimum is $e^{-1}$ (at $z=-1$).