1 Elliptic Integrals 501Suppose now thatgis a real cubic or quartic with a pair of conjugate complex
roots. Then we can write
g(x)=Q 1 Q 2 =(a 1 x^2 + 2 b 1 x+c 1 )(a 2 x^2 + 2 b 2 x+c 2 ),where the coefficients are real,a 1 c 1 −b^21 >0anda 2 c 2 −b^22 =0, buta 2 may be zero.
Consider first the case wherea 2 =0andb 1 =b 2 a 1 /a 2 .Then
Q 1 =a 1 (x+b 1 /a 1 )^2 +b′ 1 , Q 2 =a 2 (x+b 1 /a 1 )^2 +b′ 2 ,where
b′ 1 =(a 1 c 1 −b^21 )/a 1 , b′ 2 =(a 2 c 2 −b^22 )/a 2.If we puty=(x+b 1 /a 1 )^2 ,then
R(x)=R 1 (y)+R 2 (y)y^1 /^2 ,where the rational functionsR 1 ,R 2 are determined by the rational functionR,and
dx/g(x)^1 /^2 =±dy/2[y(a 1 y+b′ 1 )(a 2 y+b′ 2 )]^1 /^2.Thus we are reduced to the case of a cubic with 3 distinct real roots.
In the remaining cases there exist distinct real valuess 1 ,s 2 ofssuch that the poly-
nomialQ 1 +sQ 2 is proportional to a perfect square. ForQ 1 +sQ 2 is proportional to
a perfect square if
D(s):=(a 1 +sa 2 )(c 1 +sc 2 )−(b 1 +sb 2 )^2 = 0.We h av eD( 0 )=a 1 c 1 −b^21 >0. Ifa 2 =0, thenb 2 =0andD(±∞)=−∞.Onthe
other hand, ifa 2 =0, thenD(−a 1 /a 2 )<0, sinceb 1 =b 2 a 1 /a 2 ,andD(s)has the
sign ofa 2 c 2 −b^22 for both large positive and large negatives. Thus the quadraticD(s)
has distinct real rootss 1 ,s 2. Hence
Q 1 +s 1 Q 2 =(a 1 +s 1 a 2 )(x+d 1 )^2 , Q 1 +s 2 Q 2 =(a 1 +s 2 a 2 )(x+d 2 )^2 ,wherea 1 +sja 2 = 0 (j= 1 , 2 )and
d 1 =(b 1 +s 1 b 2 )/(a 1 +s 1 a 2 ), d 2 =(b 1 +s 2 b 2 )/(a 1 +s 2 a 2 ).Consequently
Q 1 =A 1 (x+d 1 )^2 +B 1 (x+d 2 )^2 , Q 2 =A 2 (x+d 1 )^2 +B 2 (x+d 2 )^2 ,where
A 1 =−s 2 (a 1 +s 1 a 2 )/(s 1 −s 2 ), B 1 =s 1 (a 1 +s 2 a 2 )/(s 1 −s 2 ),
A 2 =(a 1 +s 1 a 2 )/(s 1 −s 2 ), B 2 =−(a 1 +s 2 a 2 )/(s 1 −s 2 ).