Note:

The properties of the exponential function used here must be established using power series expansions. Specifically prove using power series the following:

(i)

ex$(z_1+z_2) =$   ex$(z_1) \cdot$   ex$(z_2)$

(ii)

There exists a unique positive real root of $\cos(x) =0$ in $[0,2]$ (via the real power series for the cosine function) and we denote this root by $\pi/2$.

(iii)

$\cos(2\pi +x) = \cos x,\; \sin (2 \pi +x)= \sin x$ (using addition formula for $\sin$ and $\cos$ following (i) )

(iv)

If $\cos x=\cos y,\; \sin x=\sin y$ then there exists $k \in \mathbb{Z}$ such that $x-y = 2\pi ik.$