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

382 Chapter 20. General approaches to elliptic functions
elliptic functions (of the first kind) by a formal inversion of the corresponding elliptic
integral and an extension to the complex domain. However, his contemporaries and
successors soon found other ways of introducing elliptic functions more preferably. In
the present context, it suffices to consider three different approaches.
JACOBI. With ABEL out of the competition, JACOBI’S influence on the theory of el-
liptic functions toward the end of the 1820s was overwhelming. JACOBI’S notation
and means of introducing the functions became standardized through a number of
publications starting with his Fundamenta nova which was the first monograph de-
voted to the study of the elliptic functions. 3 In the Fundamenta nova, JACOBI de-
fined elliptic functions as inverses of elliptic integrals but beginning with a course
taught in 1838, the changed the foundation to the so-called theta-functions. These
were a particular set of four exponential series the simplest of which can be written
as ϑ (x) = ∑n∈Z (−1) n qn2e2nix . 4 Based on these series, JACOBI could introduce and
investigate all parts of the theory of elliptic functions. Thus, JACOBI introduced series
as the basic objects upon which everything else should be built. Other definitions by
series, e.g. as ratios of power series were also being suggested and adopted.
J. LIOUVILLE (1809–1882). A completely different approach to the introduction of
elliptic functions was taken by LIOUVILLE who chose to develop an entire theory for
doubly periodic functions. Of such functions, he was able to deduce a number of re-
sults and eventually prove that they could be used to represent the inverses of elliptic
integrals. 5 Thus, LIOUVILLE circumvented the approach of ABEL and JACOBI and in-
vestigated the concept of functions defined by what ABEL had deduced as a property.
The strength of the approach was that eventually, the classes of elliptic functions and
doubly periodic, meromorphic functions were found to coincide.
K. T. W. WEIERSTRASS (1815–1897). The final approach which I wish to mention
was through differential equations. For instance, WEIERSTRASS introduced his func-
tion ℘ (u) as the solution to the differential equation
� �2 dy
= 4y
du
3 − g2y − g3 for g2, g3 constants
which had a pole at u = 0. 6 Subsequently, WEIERSTRASS found means of obtaining
more direct representations of his elliptic functions, for instance in the form
σ (u + v) σ (u − v)
℘ (v) − ℘ (u) =
σ2u · σ2v where the function σ could be expanded in an infinite product. 7
3 (C. G. J. Jacobi, 1829).
4 (Houzel, 1986, 304–305).
5 (Lützen, 1990, 555).
6 (Houzel, 1986, 298).
7 (ibid., 306).