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

17.2. Elliptic functions as ratios of power series 325
17.2 Elliptic functions as ratios of power series
In the Précis, 4 listed among the established facts of elliptic functions, ABEL observed
that elliptic functions of the first kind (which he now denoted λ, see below) were
expressible as the ratio of two convergent power series,
With the notation
λ (θ) = θ + a1θ3 + a2θ5 + . . .
1 + b2θ4 + b3θ6 . (17.6)
+ . . .
φ (θ) = θ + a1θ 3 + a2θ 5 + . . . and
f (θ) = 1 + b2θ 4 + b3θ 6 + . . . ,
(17.7)
ABEL claimed that the functions φ and f (which are not to be confused with the func-
tions of the same names in the Recherches) satisfied the linked functional equations
φ � θ ′ + θ � φ � θ ′ − θ � = φ 2 (θ) f 2 � θ ′� − φ 2 � θ ′� f 2 (θ) and
f � θ ′ + θ � f � θ ′ − θ � = f 2 (θ) f 2 � θ ′� − c 2 φ 2 (θ) φ 2 � θ ′� .
(17.8)
The sign of the first equation is actually wrong, see below. ABEL also mentioned the
same result in his letter to LEGENDRE but he never published any demonstration of
it. 5
In order to see how ABEL came to this expression, the following reconstruction
may be suggested based on ABEL’S sparse hints.
In his comments published in the second volume of the Œuvres, M. S. LIE (1842–
1899) has presented a reconstruction of ABEL’S reasoning based on the same sources,
papers and manuscripts. LIE indicated how ABEL’S manuscript notes — given the
power series expansion (17.7) — could be interpreted as steps toward determining the
remaining coefficients a1, a2, . . . , b2, b3, . . . . However, I interpret the notes slightly dif-
ferently and infer from them a suggestion of how ABEL came to claim the series ex-
pansion (17.7), itself, by use of the expansion in two Maclaurin series. I am confident
that the following reconstruction is close to ABEL’S original argument.
Derivation of functional equations. In the Précis, ABEL had presented the following
consequence of the addition formulae,
Supposing that λ (θ) was written as
λ � θ ′ + θ � λ � θ ′ − θ � = λ2 (θ ′ ) − λ 2 (θ)
1 − c 2 λ 2 (θ) λ 2 (θ ′ ) .
λ (θ) =
φ (θ)
f (θ)
4 (N. H. Abel, 1829d).
5 (Abel→Legendre, Christiania, 1828/11/25. N. H. Abel, 1902a, 82).
(17.9)