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19 Elliptic IntegralsSymmetric Integrals

§19.18 Derivatives and Differential Equations

  1. §19.18(i) Derivatives
  2. §19.18(ii) Differential Equations

§19.18(i) Derivatives

19.18.1 RF(x,y,z)z=16RD(x,y,z),
19.18.2 ddxRG(x+a,x+b,x+c)=12RF(x+a,x+b,x+c).

Let j=/zj, and 𝐞j be an n-tuple with 1 in the jth place and 0’s elsewhere. Also define

19.18.3 wj =bj/j=1nbj,
a =a+j=1nbj.

The next two equations apply to (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23).

19.18.4 jRa(𝐛;𝐳)=awjRa1(𝐛+𝐞j;𝐳),
19.18.5 (zjj+bj)Ra(𝐛;𝐳)=wjaRa(𝐛+𝐞j;𝐳).

§19.18(ii) Differential Equations

19.18.6 (x+y+z)RF(x,y,z)=12xyz,
19.18.7 (x+y+z)RG(x,y,z)=12RF(x,y,z).
19.18.8 j=1njRa(𝐛;𝐳)=aRa1(𝐛;𝐳).
19.18.9 (xx+yy+zz)RF(x,y,z)=12RF(x,y,z),
19.18.10 ((xy)2xy+12(yx))RF(x,y,z)=0,

and two similar equations obtained by permuting x,y,z in (19.18.10).

More concisely, if v=Ra(𝐛;𝐳), then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation:

19.18.11 j=1nzjjv=av,

and also a system of n(n1)/2 Euler–Poisson differential equations (of which only n1 are independent):

19.18.12 (zjj+bj)lv=(zll+bl)jv,

or equivalently,

19.18.13 ((zjzl)jl+bjlblj)v=0.

Here j,l=1,2,,n and jl. For group-theoretical aspects of this system see Carlson (1963, §VI). If n=2, then elimination of 2v between (19.18.11) and (19.18.12), followed by the substitution (b1,b2,z1,z2)=(b,cb,1z,1), produces the Gauss hypergeometric equation (15.10.1).

The next four differential equations apply to the complete case of RF and RG in the form Ra(12,12;z1,z2) (see (19.16.20) and (19.16.23)).

The function w=Ra(12,12;x+y,xy) satisfies an Euler–Poisson–Darboux equation:

19.18.14 2wx2=2wy2+1ywy.

Also W=Ra(12,12;t+r,tr), with r=x2+y2, satisfies a wave equation:

19.18.15 2Wt2=2Wx2+2Wy2.

Similarly, the function u=Ra(12,12;x+iy,xiy) satisfies an equation of axially symmetric potential theory:

19.18.16 2ux2+2uy2+1yuy=0,

and U=Ra(12,12;z+iρ,ziρ), with ρ=x2+y2, satisfies Laplace’s equation:

19.18.17 2Ux2+2Uy2+2Uz2=0.