Complex Exponential/Trigonometric Functions

Question

I'm having trouble on proving the following state of a Lemma using the power series of $\exp z$ centered at $0$:

For all $z \in C$:

$\exp(z + 2\pi i) = \exp(z)$

and

$\exp(z) \neq 0$

All help would be very appreciated.

Answer 1

Let $\phi(z) = e^{z+2 \pi i} - e^z$. Note that $\phi'(z) = \phi(z)$, $\phi(0) = 0$ and so $\phi^{(k)} (0) = 0$ for all $k$. Hence $\phi(z) = \sum_{n=0}^\infty {\phi^{(n)} (0) \over n!} z^n = 0$ for all $z$.

Let $\eta(z) = e^z e^{-z}$, and note that $\eta'(z) = 0$ and $\eta(0) = 1$. hence $\eta(z) = 1$ for all $z$, and so $e^z \neq 0$ for all $z$.