Number Theory: An Introduction to Mathematics

1 Elliptic Integrals 497

In general, supposeg(x)=(x−α)h(x),wherehis a cubic. If

h(x)=h 0 (x−α)^3 +h 1 (x−α)^2 +h 2 (x−α)+h 3

and we make the change of variablesx=α+ 1 /y,theng(x)=g∗(y)/y^4 ,where

g∗(y)=h 0 +h 1 y+h 2 y^2 +h 3 y^3 ,

and
∫
R(x,w)dx=

∫

R∗(y,v)dy,

whereR∗(y,v)is a rational function ofyandv,andv^2 =g∗(y).
Since any even power ofwis a polynomial inx, the integrand can be written in
the formR(x,w)=(A+Bw)/(C+Dw),whereA,B,C,Dare polynomials inx.
Multiplying numerator and denominator by(C−Dw)w, we obtain

R(x,w)=N/L+M/Lw,

whereL,M,Nare polynomials inx. By decomposing the rational functionN/Linto
partial fractions its integral can be evaluated in terms of rational functions and (real or
complex) logarithms. By similarly decomposing the rational functionM/Linto partial
fractions, we are reduced to evaluating the integrals

I 0 =

∫

dx/w, In=

∫

xndx/w, Jn(γ)=

∫

(x−γ)−ndx/w,

wheren∈Nandγ∈C.
The argument of the preceding paragraph is actually valid ifw^2 =gis any poly-
nomial. Suppose now thatgis a cubic without repeated roots, say

g(x)=a 0 x^3 +a 1 x^2 +a 2 x+a 3.

By differentiation we obtain, for any integerm≥0,

(xmw)′=mxm−^1 w+xmg′/ 2 w=( 2 mxm−^1 g+xmg′)/ 2 w.

Since the numerator on the right is the polynomial

( 2 m+ 3 )a 0 xm+^2 +( 2 m+ 2 )a 1 xm+^1 +( 2 m+ 1 )a 2 xm+ 2 ma 3 xm−^1 ,

it follows on integration that

2 xmw=( 2 m+ 3 )a 0 Im+ 2 +( 2 m+ 2 )a 1 Im+ 1 +( 2 m+ 1 )a 2 Im+ 2 ma 3 Im− 1.

It follows by induction that, for each integern>1,

In=pn(x)w+cnI 0 +c′nI 1 ,

wherepn(x)is a polynomial of degreen−2andcn,cn′are constants. Thus the evalu-
ation ofInforn>1 reduces to the evaluation ofI 0 andI 1.