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

12.10. Aspects of ABEL’s binomial paper 263
The following year, ABEL treated another problem concerning the form of func-
tions satisfying a functional relation. 71 In that paper, he studied the form of functions
φ which satisfy
φ (x) + φ (y) = ψ (x f (y) + y f (x)) . (12.28)
ABEL found his main result which stated that the most general way of satisfying the
equation (12.28) was with
φ (x) = φ ′ �
(0) f (0)
dx
f (x) f ′ � �
x
and ψ (x) = φ (0) + φ .
(0) x f (0)
One very simple application of this result is particularly interesting and a reconstruction
of it is given below. If we let f (x) = 1, we obtain α = f (0) = 1 and α ′ = f ′ (0) =
0. Therefore, the function φ which satisfies the equation (ψ = φ)
φ (x) + φ (y) = φ (x + y) (12.29)
must satisfy the requirement
�
dx
φ (x) = φ (0) = xφ (0) + C,
1
where C = 0 by (12.29). Consequently, ABEL’S result leads to the result that the only
(continuous) solution to the functional equation
φ (x) + φ (y) = φ (x + y)
is the linear function φ (x) = Ax for some constant A. As observed, this paper —
which was published in 1827 — therefore contains a generalization of the additive
functional equation (12.27) which had been so important to his proof of the binomial
theorem. However, in the binomial paper, ABEL probably relied directly on CAUCHY’S
Cours d’analyse.
12.10.3 Concepts and calculations in the binomial paper
ABEL’S paper on the binomial series shows a remarkable blend of concepts and ex-
plicit calculations. A superficial, textual analysis of ABEL’S paper reveals a division
of the paper: First, six preliminary theorems were presented which were applicable
to classes of series. These were cast in a strictly Euclidean presentational style with
definitions of convergence and continuity, statements of the six theorems and proofs
following each theorem. Second, detailed and explicit considerations of the conver-
gence of the binomial series as well as formulae for its sum were given. These in-
vestigations relied extensively on explicit manipulations of the formulae in forms as
described above. When analyzed from this perspective, ABEL’S binomial paper shows
traits of both concept based and formula based mathematics as discussed in chapter
21, below. Thus, the binomial paper is an example of the transitional status of ABEL’S
mathematics which exhibited similarities with both paradigms.
71 (N. H. Abel, 1827c).