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

380 Chapter 20. General approaches to elliptic functions
Reiterating established knowledge. ABEL began his Précis — which he privately
called the “knockout of C. G. J. JACOBI’ (1804–1851)’ — by iterating some of his pre-
vious results pertaining to elliptic functions of the first kind. With the radical
�
∆ (x, c) = (1 − x2 ) (1 − c2x2 ),
he introduced his three kinds of elliptic integrals as
�
dx
˜ω (x, c) =
∆ (x) ,
� x 2 dx
˜ω0 (x, c) = , and
∆ (x)
�
dx
Π (x, c, a) = � � .
∆ (x, c)
1 − x2
a 2
One of the major tricks of the Précis was that once the elliptic integral of the first kind
had been inverted
θ =
� λ(θ)
0
dx
∆ (x, c) ,
this elliptic function could be used to describe the elliptic integrals of the second and
third kind,
�
˜ω0 (x, c) =
λ 2 �
(θ) dθ and Π (x, c, a) =
dθ
1 − λ2 θ
a 2
Thus, ABEL had essentially reduced the problem of inverting all three kinds of inte-
grals and he could combine the study of all elliptic functions in knowledge about the
elliptic functions of the first kind.
Obviously, among the key results which ABEL iterated for the elliptic function λ
was its two periods and the complete solution of the equation λ (θ ′ ) = λ (θ) which
we have already encountered multiple times. Moreover, ABEL also presented various
infinite representations of λ and investigated the conditions of transformations. Thus,
all the key components of his previous approaches were included in the Précis, albeit
in a more coherent and lucid form.
General properties of elliptic functions. The major new purpose of ABEL’S Précis
was to investigate a new program of representation for elliptic functions. In the pro-
cess, ABEL made important use of the insights which he had developed and presented
in relation with his Paris memoir.
With two polynomial functions f (even) and φ (odd), ABEL defined
ψ (x) = f (x) 2 − φ (x) 2 ∆ (x) 2
which was an even function and therefore could be split in factors as
ψ (x) = A
µ
∏
n=1
�
x 2 − x 2 �
n .
.
(20.1)