Number Theory: An Introduction to Mathematics

4 Theta Functions 517

as the standard elliptic integral of the second kind, and

Π(u,a):=(λ/ 2 )

∫u

0

S′(a)S(v)dv/[1−λS(a)S(v)] (25)

as the standard elliptic integral of the third kind.
Many properties of these functions may be obtained by integration from corre-
sponding properties of the functionS(u). By way of example, we show that

E(u+a)−E(u−a)− 2 E(a)=−λS′(a)S(u)/[1−λS(a)S(u)]. (26)

Indeed it is evident that both sides vanish whenu=0, and it follows from (12) that
they have the same derivative with respect tou. Integrating (26) with respect tou,we
further obtain

Π(u,a)=uE(a)−( 1 / 2 )

∫u+a

u−a

E(v)dv. (27)

Thus the functionΠ(u,a), which depends on two variables (as well as the parame-
terλ) can be expressed in terms of functions of only one variable. Furthermore, we
have theinterchange property(due, in other notation, to Legendre)

Π(u,a)−uE(a)=Π(a,u)−aE(u). (28)

If we takeu= 2 K= 2 K(λ), thenS′(u)=0 and henceΠ(a,u)=0. Thus

Π( 2 K,a)= 2 KE(a)−aE( 2 K), (29)

which shows that the complete elliptic integral of the third kind can be expressed in
terms of complete and incomplete elliptic integrals of the first and second kinds.
In order to justify takingΠ(u,a)as the standard elliptic integral of the third kind,
we show finally thatS(a)takes all complex values. Otherwise, ifS(u)=cfor all
u∈C,thenc=0and

f(u)=S(u)/[S(u)−c]

is holomorphic in the whole complex plane. Furthermore, it is doubly-periodic with
two periodsω 1 ,ω 2 whose ratio is not real. Since it is bounded in the parallelogram with
vertices 0,ω 1 ,ω 2 ,ω 1 +ω 2 , it follows that it is bounded inC. Hence, by Liouville’s
theorem,fis a constant. SinceSis not constant andc=0, this is a contradiction.

4 ThetaFunctions.............................................

Theta functions arise not only in connection with elliptic functions (as we will see),
but also in problems of heat conduction, statistical mechanics and number theory.
Consider the bi-infinite series
∑∞

n=−∞

qn

2

zn= 1 +

∑∞

n= 1

qn

2

zn+

∑∞

n= 1

qn

2

z−n,