 
Summary: 1. More on the exponential function.
Proposition 1.1. We have
(i) exp(z) = exp(z) for z C;
(ii)  exp(iy) = 1 for y R.
Proof. Exercise for the reader.
Definition 1.1. We define
cos : C C and sin : C C
by letting
cos(z) =
exp(iz) + exp(iz)
2
and sin(z) =
exp(iz)  exp(iz)
2i
whenever z C. The reader may want to derive the addition laws for cos and sin
from the addition law for the exponential map. Note that exp R, cos R and sin R
are all real valued. Note that cos is even and that sin is odd.
Theorem 1.1. We have
(i) exp = exp, cos =  sin, and sin = cos;
(ii) cos(z)2
