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

10.4. New types of series 205
Figure 10.2: JEAN BAPTISTE JOSEPH FOURIER (1768–1830)
FOURIER multiplied both sides of the equation by sin nx and integrated from 0 to π,
� π
0
φ (x) sin nx dx =
∞ �
∑ ai i=1
sin ix sin nx dx,
where the integration of the sum was carried out term-wise. By the orthogonality,
⎧
�
⎨
π
if n = i,
sin ix sin nx dx = 2
⎩
0otherwise,
FOURIER found the coefficients of the expansion (10.2) to be
π
2 a � π
i = φ (x) sin ix dx.
0
The interchange of summation and integration would soon become a point of objec-
tion against FOURIER’S rigor.
SIMÉON-DENIS POISSON’S peculiar example. Almost simultaneous with FOURIER’S
first investigations, a problem arose which also involved series which were not power-
series. The problem was raised by SIMÉON-DENIS POISSON in 1811 and was inten-
sively debated for the next decades, in particular in the French journal Annales de
mathématiques pures et appliquées. 40
40 This is well described in (Jahnke, 1987, 105–117) and (Bottazzini, 1990, lx–lxiii).