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

16.2. Inversion in the Recherches 301
Thus, from ABEL’S own description of it, the idea of considering the inverse func-
tions appears quite natural. However, by this simple step, an entirely new class of
functions was introduced, and they certainly looked different from anything known
so far.
With ABEL’S choice of representation of the integral, the inversion became
α =
� x
0
dx
� (1 − c 2 x 2 ) (1 + e 2 x 2 ) � φ (α) = x. (16.2)
First, ABEL introduced a special name for the integral from 0 to 1 c :
“By thus letting
ω
2 =
� 1
c
0
∂x
√ [(1 − c 2 x 2 ) (1 + e 2 x 2 )] ,
it is evident that φ (α) is positive and increasing from α = 0 to α = ω
2 .”8
This remark seems to indicate that ABEL was well aware that for the inversion to be
meaningful, the integral had to be a strictly monotonous function.
ABEL’S next step consisted in the observation that the integral was an odd function
of x, and thus φ (−α) = −φ (α). At this point, ABEL had thus obtained the function φ
�
.
for a segment of the real axis � − ω 2 , ω 2
16.2.1 Going complex
ABEL’S study of the inverse functions of elliptic integrals relied importantly on the
extension of these inverse functions to allow for imaginary and complex arguments.
As discussed below, this aspect is extremely interesting in connection with the creation
of a (rigorous) theory of complex integration.
In analogy with the substitution of −x for x used above, ABEL observed:
“By inserting into (1) xi instead of x (where i for short represents the imaginary
quantity √ −1) and designating the value of α by βi, it gives
xi = φ (βi) and β =
� x
0
∂x
√ [(1 + c 2 x 2 ) (1 − e 2 x 2 )] .” 9
7 “Je me propose, dans ce mémoire, de considérer la fonction inverse, c’est-à-dire la fonction φα,
déterminée par les équations
(N. H. Abel, 1827b, 102).
8 “En faisant donc
�
α =
sin θ = φ (α) = x.”
ω
2 =
� 1
c
0
∂θ
√ � 1 − c 2 sin 2 θ � et
∂x
√ [(1 − c 2 x 2 ) (1 + e 2 x 2 )] ,
il est évident, que φα e[s]t positif et va en augmentant depuis α = 0 jusqu’à α = ω 2 [. . . ]” (ibid., 104).