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

260 Chapter 12. ABEL’s reading of CAUCHY’s new rigor and the binomial theorem
The solution to Poisson’s example. The ultimate result in ABEL’S binomial paper
concerned the series which had probably inspired him to work on the binomial theo-
rem in the first place. As a result of applying his characterization of the convergence
of the binomial series, ABEL found precise conditions for the convergence of the bino-
mial series corresponding to (2 cos x) m . The result — an analogy of which ABEL also
communicated to HOLMBOE in a letter (see page 226) — was that the identity
(2 cos x) m = cos mx + m
1
cos (m − 2) x + m (m − 1)
1 · 2
cos (m − 4) x + . . .
was valid when m was positive and x belonged to the interval � − π 2 , π 2
� . Thus, ABEL
ruled out validity of this formula in the situation of Poisson’s example which had in-
volved setting x = π. For general values of x, ABEL obtained an identity which in-
volved a correction term,
� (2ρ−1)π
2
(2 cos x) m cos 2ρmπ =
∞
∑
k=0
� �
m
cos (m − 2k) x
k
, (2ρ+1)π
�
2 . Thus, ABEL’S resolution to Poisson’s example consisted of
for x ∈
two steps deriving from his general proof of the binomial theorem. First, he divided
the values of x into smaller intervals in which the value of cos x had a constant sign.
And second, he considered all expressions as single valued and introduced an addi-
tional term to provide the correction.
12.10 Aspects of ABEL’s binomial paper
Having presented and investigated the contents of ABEL’S binomial paper, I believe
that three slightly broader aspects of it also merit attention: ABEL’S use of complex
numbers, his use of functional equations, and the style of the binomial paper.
12.10.1 ABEL’s understanding of complex numbers
Compared with CAUCHY’S Cours d’analyse, ABEL’S proof of the binomial theorem ex-
celled by including complex values of the exponent. For complex values of the ar-
gument x, CAUCHY had reduced the study of the binomial series corresponding to
(1 + x) m to the study of two real series by writing out the real and imaginary parts.
By diligent use of polar representations of complex numbers, ABEL succeeded in re-
ducing the functional equations for complex exponents m to the simple, additive one.
Thus, in the binomial paper, ABEL worked with complex numbers which he always
reduced to pairs of reals either as real and imaginary parts or in polar representation.
In his inversion of elliptic integrals into elliptic functions (see chapter 16, below), ABEL
also worked with complex numbers as arguments of functions. Again, complex num-
bers were reduced to real and imaginary parts. From the rather scarce evidence, it