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

262 Chapter 12. ABEL’s reading of CAUCHY’s new rigor and the binomial theorem
as the foundation for their proofs, ABEL chose to focus on the other equation (12.27)
because it was better suited for his investigations of complex exponents. When he had
to address the multiplicative functional equation expressing the modulus of the series,
he transformed it into the form (12.27) by way of exponentiation.
The additive functional equation (12.27) had also been studied by CAUCHY in his
Cours d’analyse, and it can be interpreted as testimony to ABEL’S familiarity with that
book that he was able to replace the basic tool of the previous proofs with one more
suited for his slightly more general situation.
Two general problems concerning functional equations. As mentioned, ABEL pub-
lished two papers in 1826 and 1827 in which he addressed questions concerning func-
tional equations. The problems which he attacked were within the immediate scope
of CAUCHY’S approach to the theory although they may appear a little odd. Thus,
ABEL’S results can be seen as contributing to the early growth of the theory of func-
tional equations.
In paper published in 1826, 69 ABEL dealt with functions f such that f (z, f (x, y))
was symmetric in x, y, z. For such functions, ABEL obtained the characterization:
“Whenever a function f (x, y) of two independent variable quantities x and
y has the property that f (z, f (x, y)) is a symmetric function of x, y, z, there will
always be a function ψ for which
ψ ( f (x, y)) = ψx + ψy.” 70
Furthermore, ABEL found that the stipulated function ψ could be determined by the
differential equation
ψ (x) = ψ ′ � ∂ f
∂x (x, y)
(y)
dx.
∂ f
∂y (x, y)
In the process, ABEL also integrated the equation
to find
∂r (x, y)
φ (y) =
∂x
��
r = ψ
�
φ (x) dx +
∂r (x, y)
φ (x)
∂y
�
φ (y) dy ,
a result which will resurface in section 16.2.2 where ABEL’S deduction of the addition
theorems of elliptic functions are described.
69 (N. H. Abel, 1826e).
70 “Sobald eine Function f (x, y) zweier unabhängig veränderlichen Größen x und y die Eigenschaft
hat, daß f (z, f (x, y)) eine symmetrische Function von x, y und z ist, so muß es allemal eine Function
ψ geben, für welche
ist.” (ibid., 13).
ψ f (x, y) = ψ (x) + ψ (y)