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

17.3. Characterization of ABEL’s representations 329
Convergence of infinite expressions. As LIE commented in the notes, ABEL’S “meth-
ods in obtaining expressions for the functions φ (α), f (α), F (α) in series and infinite
products do not seem to us to be satisfactory in all details.” 10 In particular, it is re-
markable that the ardent rigorist of the binomial paper never even mentioned the
word convergence in the Recherches. In the Précis, however, the convergence of the se-
ries (17.6) was stated as a fact without further explanation.
ABEL’S way of obtaining series expansions of elliptic functions essentially let the
number of terms in a finite expression grow to infinity. ABEL’S concern for the con-
vergence of the process was dealt with by the method of writing the expression as one
part which was independent of n and another part which vanished with increasing n.
Thus, it appears, two different standards of rigor in the theory of series were in-
volved in ABEL’S research using infinite series. In foundational issues, a strict adher-
ence to A.-L. CAUCHY’S (1789–1857) program and the associated theoretical complex
was advocated by ABEL. However, when it came to research on new groundbreaking
objects, ABEL used the methods which he had learned from EULER and was content
with observing that his results were sound.
17.3.2 The need for multiple representations
A final aspect which is revealing of the role played by representations of elliptic func-
tions in ABEL’S works is the necessity of obtaining multiple representations. Under-
standably, functions introduced in such an indirect way as the inversion of a non-
elementary integral needed some other means of numerical determination and this is
one of the roles played by representations in ABEL’S theory.
Convergence. As already noticed repeatedly, ABEL was not very explicit about the
convergence of his infinite representations. This may have been a reason for not rely-
ing on any single representation but deriving multiple and various representations in
the hope that at least some of them would prove adequate in particular instances.
Applications. A connected motivation for multiple representations could also be the
ambition of multiple applications (within pure mathematics). In the following chapter,
an instance where infinite representations play a central role in the proof of a theorem
will be described. ABEL and his contemporaries would have hoped and expected to
find many similar instances.
Aesthetics. The third aspect of the discussion is less technical and more of a personal
and contextual nature which can only be appreciated in a broader time scale. In section
21.2, where the discussion of representations is taken up again, it will also become
10 (N. H. Abel, 1881, II, 306).