7 Further Remarks 537carried out this extension. (For a modern account, see Rosen [25].) However, Gauss’s
claim in a letter to Schumacher of 30 May 1828, quoted in Krazer [20], that Abel had
anticipated about a third of his own research is quite unjustified, and not only because
of his inability to bring his work to a form in which it could be presented to the world.
It wasprovedby Liouville (1834) that elliptic integrals of the first and second
kinds are always ‘nonelementary’. For anintroductory account of Liouville’s theory,
see Kasper [18]. (But elliptic integrals of the third kind may be ‘elementary’; see
Chapter IV,§7.)
The three kinds of elliptic integral may also be characterized function-theoretically.
On the Riemann surface of the algebraic functionw^2 =g(z),wheregis a cubic
without repeated roots, the differentialdz/wis everywhere holomorphic, the differen-
tialzdz/wis holomorphic except for a double pole at∞with zero residue, and the
differential [w(z)+w(a)]dz/ 2 (z−a)w(z)is holomorphic except for two simple poles
ataand∞with residues 1 and−1 respectively.
Many integrals which are not visibly elliptic may be reduced to elliptic integrals by
a change of variables. A compilation is given by Byrd and Friedman [8], pp. 254–271.
The arithmetic-geometric mean may also be defined for pairs of complex numbers;
a thorough discussion is given by Cox [9]. For the application of theAGMalgorithm
to integrals which are not strictly elliptic, see Bartky [4].
The differential equation (6) is a special case of the hypergeometric differential
equation. In fact, if|λ|<1, then by expanding( 1 −λx)−^1 /^2 ,resp.( 1 −λx)^1 /^2 ,bythe
binomial theorem and integrating term by term, the complete elliptic integralsK(λ)=∫ 1
0[4x( 1 −x)( 1 −λx)]−^1 /^2 dx, E(λ)=∫ 1
0[( 1 −λx)/ 4 x( 1 −x)]^1 /^2 dx,may be identified with the hypergeometric functions(π/ 2 )F( 1 / 2 , 1 / 2 ; 1 ;λ), (π/ 2 )F(− 1 / 2 , 1 / 2 ; 1 ;λ),whereF(α,β;γ;z)= 1 +αβz/ 1 ·γ+α(α+ 1 )β(β+ 1 )z^2 / 1 · 2 ·γ(γ+ 1 )+···.Many transformation formulas for the complete elliptic integrals may be regarded as
special cases of more general transformation formulas for the hypergeometric function.
The proof in§3thatK( 1 −λ)/K(λ)has positive real part is due to Falk [11].
It follows from (12)–(13) by induction thatS(nu)andS′(nu)/S′(u)are rational
functions ofS(u)for every integern. The elliptic functionS(u)is said to admitcom-
plex multiplicationifS(μu)is a rational function ofS(u)for some complex number
μwhich is not an integer. It may be shown thatS(u)admits complex multiplication if
and only ifλ= 0 ,1 and the period ratioiK( 1 −λ)/K(λ)is a quadratic irrational, in
the sense of Chapter IV. This condition is obviously satisfied ifλ= 1 /2, the case of
the lemniscate.
A functionf(u)is said to possess analgebraic addition theoremif there is a poly-
nomialp(x,y,z), not identically zero and with coefficients independent ofu,v,such
that
p(f(u+v),f(u),f(v))=0forallu,v.