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

12.10. Aspects of ABEL’s binomial paper 261
seems justified to say that ABEL held a strictly algebraic view of complex numbers
and that he considered these numbers rather unproblematic.
Because some of ABEL’S initial theorems — in particular Lehrsatz IV — dealt with
power series, they have subsequently been interpreted as pertaining to complex vari-
ables. However, there is no absolute indication that this was the interpretation which
ABEL held. ABEL’S original formulations were not very explicit about these issues
and often neglected taking numerical values into consideration. However, I find very
little reason to suspect that ABEL would develop his theorems for complex variables
and afterwards go through rather cumbersome arguments to reduce complex series to
series of real terms.
Going through ABEL’S loans from the Christiania University library, V. BRUN (1885–
1978) discovered that ABEL had — in 1822 — borrowed the volume of the transactions
of the Danish Academy in which his fellow Norwegian C. WESSEL (1745–1818) had
published his geometric interpretation of complex numbers as directed line segments
in the plane. 64 However, as Ø. ORE (1899–1968) also pointed out, ABEL was prob-
ably much more interested in a paper on equations which C. F. DEGEN (1766–1825)
published in the same volume. 65 Even if ABEL read WESSEL’S paper — which seems a
reasonable assumption given the limited amount of Danish mathematical literature —
he certainly never did anything to adopt its idea or promote it in any other way. This
just supports K. ANDERSEN’S (⋆1941) hypothesis that the geometrical interpretation
of complex numbers was not a hot topic in the first decades of the nineteenth century. 66
12.10.2 ABEL on functional equations
As had been the case in both EULER’S and CAUCHY’S proof of the binomial theorem,
functional equations played a central part in ABEL’S proof. In his Cours d’analyse,
CAUCHY had developed the topic into a theory of its own and he studied multiple
types of functional equations. 67
In a paper published in 1827 in CRELLE’S Journal, 68 ABEL presented some results
on functional equations, which when applied to the functional equation
φ (x) + φ (y) = φ (x + y) (12.27)
gave the solution φ (x) = Ax as the unique (continuous) solution to this equation.
It was precisely this functional equation which was central to ABEL’S proof of the
binomial theorem. Whereas EULER and CAUCHY had used the equation
f (x) f (y) = f (x + y)
64 (Brun, 1962, 110–111). For an analysis of WESSEL’S work, see (K. Andersen, 1999).
65 (C. F. Degen, 1799).
66 (K. Andersen, 1999, 94).
67 (J. Dhombres, 1992).
68 (N. H. Abel, 1827c).