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

302 Chapter 16. The idea of inverting elliptic integrals
The formal substitution of an imaginary value thus seemed to preserve the form
of the function with the sole exception that the roles of the quantities c 2 and e 2 were
interchanged. To ABEL this was fully sufficient, as he simple wrote:
“thus, one sees by supposing c instead of e and e instead of c,
φ (αi)
i
changes into φ (α) .” 10
Thus, when I let φ (c,e) (α) denote the function in (16.2), ABEL’S formal imaginary
substitution gave
φ (c,e) (αi) = iφ (e,c) (α)
and he had found the function φ (c,e) for a section of the imaginary axis � − ¯ω 2 , ¯ω �
2 i, 11 in
which
¯ω
2 =
� 1
e
0
16.2.2 Addition theorems
dx
� (1 + c 2 x 2 ) (1 − e 2 x 2 ) .
Of central importance to ABEL’S approach to the inversion was the use which he made
of addition formulae for elliptic functions.
Auxiliary functions f and F. ABEL introduced two auxiliary functions which he
named f and F, derived from φ (α), which played central parts in his deductions and
were treated analogous to φ,
f (α) =
�
1 − c2φ2 �
(α) and F (α) = 1 + e2φ2 (α).
Obviously, the product of these functions equals φ ′ (α) and the functions f and F were,
themselves, doubly periodic functions corresponding to JACOBI’S cn and dn, respec-
tively, which will be introduced and discussed later.
9 “En mettant dans (1.) xi au lieu de x (ou i, pour abréger, représente la quantité imaginaire √ − 1) et
désignant la valeur de α par βi, il viendra
xi = φ (βi) et β =
� x
0
∂x
√ [(1 + c 2 x 2 ) (1 − e 2 x 2 )] .”
(N. H. Abel, 1827b, 104).
10 “[. . . ] donc on voit, qu’en supposant c au lieu de e et e au lieu de c,
φ (αi)
i
se changera en φα.”
(ibid., 104).
11 I write [a, b] i as a short-hand for the segment of the imaginary axis which can also be written as
{xi : x ∈ [a, b]}.