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

330 Chapter 17. Steps in the process of coming to “know” elliptic functions
clear that the preferred definitions and representation vary over time and between
mathematical traditions.
In ABEL’S theory of elliptic functions, the objects were introduced as inversions of
elliptic integrals. The largely Eulerian tradition to which ABEL’S Recherches generally
belongs emphasized representations of transcendental functions by infinite (power)
series and infinite products. On the way to obtaining these representations, ABEL also
came across the functional equations (17.8) and proved that elliptic functions were
doubly periodic and could be written as the ratio of power series. All these represen-
tations and key results were later assumed as definitions of the concept elliptic function
depending on the setting and context in which they were introduced.
17.4 Conclusion
One major achievement of the search for representations was, of course, that based on
formulae such as (17.5), approximations to φ (α) could be computed with any degree
of accuracy. Another, and equally interesting — but less anticipated — result was that
infinite expressions, themselves, could play a role in the development of the theory. In
the Recherches, this aspect remained little cultivated but in subsequent papers, ABEL
occasionally applied infinite expressions, even to answer algebraic questions (see next
chapter).
In the Recherches, ABEL did more than solve the division problem for the lemnis-
cate. While the lemniscate provided a clear question to which he produced a clear
answer, the other part of the paper dealt with problems of more intrinsic nature and
provided answers which are now hardly recognizable as answers because the ques-
tions which they answer have faded in importance.