Number Theory: An Introduction to Mathematics

498 XII Elliptic Functions

Consider now the integralJn(γ). In the same way as before, for any integerm≥1,

d{(x−γ)−mw}/dx=−m(x−γ)−m−^1 w+(x−γ)−mg′/ 2 w

={− 2 mg+(x−γ)g′}/ 2 w(x−γ)m+^1.

We can write

g(x)=b 0 +b 1 (x−γ)+b 2 (x−γ)^2 +b 3 (x−γ)^3

and the numerator on the right of the previous equation is then

− 2 mb 0 +( 1 − 2 m)b 1 (x−γ)+( 2 − 2 m)b 2 (x−γ)^2 +( 3 − 2 m)b 3 (x−γ)^3.

It follows on integration that

2 (x−γ)−mw=− 2 mb 0 Jm+ 1 (γ)+( 1 − 2 m)b 1 Jm(γ)

+( 2 − 2 m)b 2 Jm− 1 (γ)+( 3 − 2 m)b 3 Jm− 2 (γ),

whereJ− 1 (γ)=

∫

(x−γ)dx/wis a constant linear combination ofI 0 andI 1 .Since
gdoes not have repeated roots,b 1 =0ifb 0 =0.
It follows by induction that ifg(γ)=b 0 =0 then, for anyn>1,

Jn(γ)=qn((x−γ)−^1 )w+dnJ 1 (γ)+dn′I 0 +dn′′I 1 ,

whereqn(t)is a polynomial of degreen−1anddn,dn′,d′′nare constants. On the other
hand, ifg(γ)=0theng′(γ)=b 1 =0 and, for anyn≥1,

Jn(γ)=rn((x−γ)−^1 )w+enI 0 +e′nI 1 ,

wherern(t)is a polynomial of degreenanden,en′are constants.
Thus the evaluation of an arbitrary elliptic integral can be reduced to the evalua-
tion of

I 0 =

∫

dx/w, I 1 =

∫

xdx/w, J 1 (γ)=

∫

(x−γ)−^1 dx/w,

wherew^2 =gis a cubic without repeated roots,γ ∈Candg(γ)=0. Following
Legendre (1793), to whom this reduction is due, integrals of these types are called
respectivelyelliptic integrals of the first, second and third kinds.
The cubicgcan itself be simplified. Ifαis a root ofgthen, by replacingxby
x−α, we may assume thatg( 0 )=0. Ifβis now another root ofgthen, by replacing
xbyx/β, we may further assume thatg( 1 )=0. Thus the evaluation of an arbitrary
elliptic integral may be reduced to one for whichghas the form

gλ(x):= 4 x( 1 −x)( 1 −λx),

whereλ∈Candλ= 0 ,1. This normal form, which was used by Riemann (1858)
in lectures, is obtained from the normal form of Legendre by the change of variables
x=sin^2 θ. To draw attention to the difference, it is convenient to call itRiemann’s
normal form.