500 XII Elliptic Functions
Ifgis a cubic or quartic with only real roots, this can be achieved by a linear frac-
tional transformation, mapping roots ofgto roots ofgλ. Appropriate transformations
are listed in Tables 1 and 2. It should be noted thatλis always across-ratioof the four
roots ofgin Table 2, and thatλis always a cross-ratio of the three roots ofgand the
point ‘∞’inTable1.
Table 1.Reduction to Riemann’s normal form,ga cubic with all roots real
dx/g(x)^1 /^2 =dy/μgλ(y)^1 /^2
g(x)=A(x−α 1 )(x−α 2 )(x−α 3 ), whereα 1 >α 2 >α 3 ; αjk=αj−αk
gλ(y)= 4 y( 1 −y)( 1 −λy), where 0<λ< 1 , y∈( 0 , 1 )
μ=(α 13 )^1 /^2 / 2 ,λ 0 =α 23 /α 13 , 1 −λ 0 =α 12 /α 13.A λ Range Transformation Corresponding values
+ 1 λ 0 x≥α 1 y=(x−α 1 )/(x−α 2 ) x=∞ y= 1
α 1 0
− 11 −λ 0 α 2 ≤x≤α 1 =α 13 (x−α 2 )/α 12 (x−α 3 )α 1 1
α 2 0
+ 1 λ 0 α 3 ≤x≤α 2 =(x−α 3 )/α 23 α 2 1
α 3 0
− 11 −λ 0 x≤α 3 =α 13 /(α 1 −x)α 3 1
−∞ 0Table 2.Reduction to Riemann’s normal form,ga quartic with all roots real
dx/g(x)^1 /^2 =dy/μgλ(y)^1 /^2
g(x)=A(x−α 1 )(x−α 2 )(x−α 3 )(x−α 4 ), whereα 1 >α 2 >α 3 >α 4 ;αjk=αj−αk
gλ(y)= 4 y( 1 −y)( 1 −λy), where 0<λ< 1 , y∈( 0 , 1 )
μ=(α 13 α 24 )^1 /^2 / 2 ,λ 0 =α 23 α 14 /α 13 α 24 , 1 −λ 0 =α 12 α 34 /α 13 α 24.A λ Range Transformation Corresponding values
+ 1 λ 0 x≥α 1 y=α 24 (x−α 1 )/α 14 (x−α 2 ) x=∞ y=α 24 /α 14
α 1 0
− 11 −λ 0 α 2 ≤x≤α 1 =α 13 (x−α 2 )/α 12 (x−α 3 )α 1 1
α 2 0
+ 1 λ 0 α 3 ≤x≤α 2 =α 24 (x−α 3 )/α 23 (x−α 4 )α 2 1
α 3 0
− 11 −λ 0 α 4 ≤x≤α 3 =α 13 (x−α 4 )/α 34 (α 1 −x)α 3 1
α 4 0
+ 1 λ 0 x≤α 4 =α 24 (x−α 1 )/α 14 (x−α 2 )α 4 1
−∞ α 24 /α 14