RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

310 Chapter 16. The idea of inverting elliptic integrals
Next, JACOBI stated the addition formulae which were presented as well known re-
sults concerning elliptic integrals. Only then were complex values of the variable in-
troduced through a substitution sin φ = i tan ψ in the integrand:
�
dφ
1 − k2 sin2 = �
i dψ
φ cos2 ψ + k2 sin2 = �
i dψ
ψ 1 − k ′ k ′ sin2 .
ψ
Finally, JACOBI obtained the doubly periodic nature of the function sin amu from the
addition formulae.
As described, JACOBI’S inversion is quite similar to ABEL’S approach. Based on
two different elliptic integrals (corresponding to the complementary moduli k and k ′ ),
JACOBI could obtain the value of sin amiu on the imaginary axis. Then, by the addition
formulae which were apparently assumed to be valid for these complex values of u
and v, the two independent periods were deduced. JACOBI was aware that the doubly
periodic nature was a new and important feature of these new functions:
“elliptic functions have two periods, one real and one imaginary whenever
the modulus k is real. Both [periods] will be imaginary when the modulus itself is
imaginary. We call this the principle of double periodicities.” 30
JACOBI’S book became the corner stone of the research on elliptic functions in the
following generation, and his notation and ways of introducing elliptic functions be-
came standard for a while until he changed it by introducing elliptic functions by
certain infinite series (see chapter 20). In that respect, JACOBI’S works surpassed LEG-
ENDRE’S effort to update his monographs with the newest developments by ABEL and
JACOBI which resulted in a supplement to his Traité des fonctions elliptiques published
in 1828. 31
16.2.6 Comparison: An earlier idea on inversion
As indicated, ABEL’S inversion in the Recherches was the first inversion of elliptic inte-
grals into elliptic functions of a complex variable to appear in print. However, prior to
his departure on the European tour, ABEL had written a manuscript which also dealt
with the inversion of functions and which is interesting in the discussion of whether
ABEL used complex integration or not.
The result. In a manuscript which bears the lengthy but very accurate title Propriétés
remarquables de la fonction y = φx déterminée par l’équation f y.dx − f x � (a − y) (a1 − y) (a2 − y) . . . (am
0, f étant une fonction quelconque de y qui ne devient pas nulle ou infinie lorsque y =
30 “functiones ellipticas duplici gaudere periodo, altera reali, altera imaginaria, siquidem modulus k
est realis. Utraque fit imaginaria, ubi modulus et ipse est imaginarius. Quod principium duplicis
periodi nuncupabimus.” (C. G. J. Jacobi, 1829, 87).
31 (A. M. Legendre, 1825–1828, III).