About the Project
19 Elliptic IntegralsSymmetric Integrals

§19.22 Quadratic Transformations

  1. §19.22(i) Complete Integrals
  2. §19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)
  3. §19.22(iii) Incomplete Integrals

§19.22(i) Complete Integrals

Let x>0, y>0, a=(x+y)/2, and p0. Then

19.22.1 RF(0,x2,y2) =RF(0,xy,a2),
19.22.2 2RG(0,x2,y2) =4RG(0,xy,a2)xyRF(0,xy,a2),
19.22.3 2y2RD(0,x2,y2) =14(y2x2)RD(0,xy,a2)+3RF(0,xy,a2).
19.22.4 (p±2p2)RJ(0,x2,y2,p2)=2(p±2a2)RJ(0,xy,a2,p±2)3RF(0,xy,a2)+3π/(2p),


19.22.5 2p±=(p+x)(p+y)±(px)(py),

and hence

19.22.6 p+p =pa,
p+2+p2 =p2+xy,
p+2p2 =(p2x2)(p2y2),
4(p±2a2) =(p2x2±p2y2)2.

Bartky’s Transformation

19.22.7 2p2RJ(0,x2,y2,p2)=v+vRJ(0,xy,a2,v+2)+3RF(0,xy,a2),

If p=y, then (19.22.7) reduces to (19.22.3), but if p=x or p=y, then both sides of (19.22.4) are 0 by (19.20.9). If x<p<y or y<p<x, then p+ and p are complex conjugates.

§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)

The AGM, M(a0,g0), of two positive numbers a0 and g0 is defined in §19.8(i). Again, we assume that a0g0 (except in (19.22.10)), and define cn=an2gn2. Then

19.22.8 2πRF(0,a02,g02)=1M(a0,g0),
19.22.9 4πRG(0,a02,g02)=1M(a0,g0)(a02n=02n1cn2)=1M(a0,g0)(a12n=22n1cn2),


19.22.10 RD(0,g02,a02)=3π4M(a0,g0)a02n=0Qn,


19.22.11 Q0 =1,
Qn+1 =12Qnangnan+gn.

Qn has the same sign as a0g0 for n1.

19.22.12 RJ(0,g02,a02,p02)=3π4M(a0,g0)p02n=0Qn,

where p0>0 and

19.22.13 pn+1 =pn2+angn2pn,
εn =pn2angnpn2+angn,
Q0 =1,
Qn+1 =12Qnεn.

(If p0=a0, then pn=an and (19.22.13) reduces to (19.22.11).) As n, pn and εn converge quadratically to M(a0,g0) and 0, respectively, and Qn converges to 0 faster than quadratically. If the last variable of RJ is negative, then the Cauchy principal value is

19.22.14 RJ(0,g02,a02,q02)=3π4M(a0,g0)(q02+a02)(2+a02g02q02+g02n=0Qn),

and (19.22.13) still applies, provided that

19.22.15 p02=a02(q02+g02)/(q02+a02).

§19.22(iii) Incomplete Integrals

Let x, y, and z have positive real parts, assume p0, and retain (19.22.5) and (19.22.6). Define

19.22.16 a =(x+y)/2,
2z± =(z+x)(z+y)±(zx)(zy),

so that

19.22.17 z+z =za,
z+2+z2 =z2+xy,
z+2z2 =(z2x2)(z2y2),
4(z±2a2) =(z2x2±z2y2)2.


19.22.18 RF(x2,y2,z2)=RF(a2,z2,z+2),
19.22.19 (z±2z2)RD(x2,y2,z2)=2(z±2a2)RD(a2,z2,z±2)3RF(x2,y2,z2)+(3/z),
19.22.20 (p±2p2)RJ(x2,y2,z2,p2)=2(p±2a2)RJ(a2,z+2,z2,p±2)3RF(x2,y2,z2)+3RC(z2,p2),
19.22.21 2RG(x2,y2,z2)=4RG(a2,z+2,z2)xyRF(x2,y2,z2)z,
19.22.22 RC(x2,y2)=RC(a2,ay).

If x,y,z are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when x,y<z (implying a<z<z+), and descending Gauss transformations when z<x,y (implying z+<z<a). Ascent and descent correspond respectively to increase and decrease of k in Legendre’s notation. Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not.

If p=x or p=y, then (19.22.20) reduces to 0=0 by (19.20.13), and if z=x or z=y then (19.22.19) reduces to 0=0 by (19.20.20) and (19.22.22). If x<z<y or y<z<x, then z+ and z are complex conjugates. However, if x and y are complex conjugates and z and p are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and p±2p2=±|p2x2|0.

The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. The equations inverse to (19.22.5) and (19.22.16) are given by

19.22.23 x+y =2a,
xy =(2/a)(a2z+2)(a2z2),
z =z+z/a,

and the corresponding equations with z, z+, and z replaced by p, p+, and p, respectively. These relations need to be used with caution because y is negative when 0<a<z+z(z+2+z2)1/2.