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

206 Chapter 10. Toward rigorization of analysis
It all began when POISSON noticed that a peculiar situation arose from letting m =
1 3 and x = π in the binomial expansion of (2 cos x) m , 41
(2 cos x) m =
∞
∑
n=0
∏ n−1
k=0 (m − k)
cos ((m − 2n) x) .
n!
In a short paper, POISSON observed that the left hand side had the three values
− 3√ 2, 3√ �
1 +
2
√ �
−3
, and
2
3√ �
1 −
2
√ �
−3
2
although the right hand side was a single-valued function,
cos
�
π
� ∞ ∏
3 ∑
n=0
n−1
k=0
� �
13 − k
n!
= 1
2 × (1 + 1) 1 3 =
Thus, the sum of the series on the right hand side corresponded to neither of the values
of the expression on the left hand side but was the average of its two complex values.
Poisson’s example is a particular example of the kind of strange relations which
could result by interpreting formal equalities in situations outside the domain of nu-
3√ 2
2 .
merical equality. In this sense, it is similar to the peculiar formal equality
1
2
= 1 − 1 + 1 − 1 + . . .
which had puzzled mathematicians in the eighteenth century. However, Poisson’s ex-
ample was a convergent series and the problem was that it did not agree with its true
value.
Upon POISSON’S publication, mathematicians sought to understand how and why
this peculiarity emerged, and the debate also spread to Berlin. In Berlin, A. L. CRELLE
(1780–1855) and M. OHM (1792–1872) became interested in the explanation of this
phenomenon — as did the mysterious L. OLIVIER who will appear prominently in
chapter 13. 42 ABEL also became acquainted with the problem and it provoked him
into producing a new proof of the binomial theorem. Before attention is focused on
ABEL’S work in the theory of series, the following chapter is devoted to his greatest
inspiration in the field: CAUCHY’S Cours d’analyse.
41 (Poisson, 1811). POISSON wrote x = 200 ◦ and thus adhered to the new radian system.
42 See e.g. (Jahnke, 1987, 105–117).