536 XII Elliptic Functions
=( 1 −e−πi/^12 )/( 1 +e−πi/^12 )=itanπ/ 24.Thus||≤tanπ/ 24 < 2 /15 and|/ 2 |^4 < 2 × 10 −^5 .SinceIτ≥
√
3 /2forτin the
region 5, for the solutionqwe have
|q|≤e−π√3 / (^2) < 1 / 15.
Having determinedq, we may calculateK(τ),snu,...from their representations by
theta functions.
7 FurtherRemarks
Numerous references to the older literature on elliptic integrals and elliptic functions
are given by Fricke [12]. The more important original contributions are readily avail-
able in Euler [10], Lagrange [21], Legendre [22], Gauss [13], Abel [1] and Jacobi [16],
which includes his lecture course of 1838.
It was shown by Landen (1775) that the length of arc of a hyperbola could be ex-
pressed as the difference of the lengths of two elliptic arcs. The change of variables
involved is equivalent to that used by Lagrange (1784/5) in his application of theAGM
algorithm. However, Lagrange used the transformation in much greater generality,
and it was his idea that elliptic integrals could be calculated numerically by iterat-
ing the transformation. The connection with the result of Landen was made explicit by
Legendre (1786).
By bringing together his own results and those of others the treatise of
Legendre [22], and his earlierExercices de calcul integral(1811/19), contributed sub-
stantially to the discoveries of Abel and Jacobi. The supplementary third volume of his
treatise, published in 1828 when he was 76, contains the first account of their work in
book form.
The most important contribution of Abel (1827) was not the replacement of ellip-
tic integrals by elliptic functions, but the study of the latter in the complex domain.
In this way he established their double periodicity, determined their zeros and poles
and (besides much else) showed that they could be represented as quotients of infinite
products.
The triple product formula of Jacobi (1829) identified these infinite products with
infinite series, whose rapid convergence made them well suited for numerical compu-
tation. Infinite series of this type had in fact already appeared in theTheorie analy- ́
tique de la Chaleurof Fourier (1822), and Proposition 3 had essentially been proved
by Poisson (1827). Remarkable generalizations of the Jacobi triple product formula
to affine Lie algebras have recently beenobtained by Macdonald [23] and Kac and
Peterson [17]. For an introductory account, see Neher [24].
It is difficult to understand the glee with which some authors attribute to Gauss
results on elliptic functions, since the world owes its knowledge of these results not to
him, but to others. Gauss’s work was undoubtedly independent and in most cases ear-
lier, although not in the case of the arithmetic-geometric mean. The remark, in§335 of
hisDisquisitiones Arithmeticae(1801), that his results on the division of the circle into
nequal parts applied also to the lemniscate, was one of the motivations for Abel, who