Number Theory: An Introduction to Mathematics

46 I The Expanding Universe of Numbers

and the relationeize−iz=1 implies that

cos^2 z+sin^2 z= 1.

From the power series forezwe obtain, for everyz∈C,

cosz=

∑

n≥ 0

(− 1 )nz^2 n/( 2 n)!= 1 −z^2 /2!+z^4 /4!−···,

sinz=

∑

n≥ 0

(− 1 )nz^2 n+^1 /( 2 n+ 1 )!=z−z^3 /3!+z^5 /5!−···.

From the differential equation we obtain, for everyz∈C,

d(cosz)/dz=−sinz, d(sinz)/dz=cosz.

From the addition theorem we obtain, for allz,ζ∈C,

cos(z+ζ)=coszcosζ−sinzsinζ,

sin(z+ζ)=sinzcosζ+coszsinζ.

We now consider periodicity properties. By the addition theorem for the exponen-
tial function,ez+h=ezif and only ifeh=1. Thus the exponential function has period
hif and only ifeh=1. Sinceeh=1 impliesh=ixfor some realx, and since cosx
and sinxare real for realx, the periods correspond to those real values ofxfor which

cosx= 1 , sinx= 0.

In fact, the second relation follows from the first, since cos^2 x+sin^2 x=1.
By bracketing the power series for cosxin the form

cosx=( 1 −x^2 /2!+x^4 /4!)−( 1 −x^2 / 7 · 8 )x^6 /6!−( 1 −x^2 / 11 · 12 )x^10 /10!−···

and takingx=2, we see that cos 2<0. Since cos 0=1andcosxis a continuous
function ofx, there is a least positive valueξofxsuch that cosξ=0. Then sin^2 ξ=1.
In fact sinξ=1, since sin 0=0andsin′x=cosx>0for0≤x<ξ. Thus

0 <sinx<1for0<x<ξ

and

eiξ=cosξ+isinξ=i.

As usual, we now writeπ= 2 ξ.Fromeπi/^2 =i, we obtain

e^2 πi=i^4 =(− 1 )^2 = 1.

Thus the exponential function has period 2πi. It follows that it also has period 2nπi,
for everyn∈Z. We will show that there are no other periods.