§2. Uniform convergence
213
(6.3)
1/2 = cos x - cos2x + cos3x - ... ,
"an absolutely false result, since the series is divergent'. This is obvious for
certain values of x: for x = 7r /2 for example, one finds the "formula"
1/2 = 1 - 1 + 1 - 1 + ... ,
which we have already met. The same is true more generally for x commensurable to 7r since then among the multiples of x there occur infinitely many
integer multiples of 7r for which the corresponding terms in (3) are equal to
1 or -1. The case where x is not commensurable to 7r is even worse, the
values of cos nx then being distributed at random 8 between -1 and 1. If he
had had to, Abel could have rendered the situation even more ridiculous by
differentiating the relation (2) thrice, and setting x = 0; then one finds that
0= 1 - 4 + 9 - 16 + ... This is nevertheless what Fourier did: but instead
of starting from the relation (2.4) that he sought to establish, he started
from a series with a priori undetermined coefficients an, and, differentiating
ad libitum for x = 0, obtained linear equations in infinitely many unknowns
(I) between the an, so that he never had to write the extravagant relations
which, to verify the correctness of his calculations, he would have obtained
in giving these an the explicit values 1, 1/3, etc. that he finally found after acrobatic calculations; this must be the absolute record in mathematical
prestidigitation. His results were no less correct for all that.
In 1822, a young German, Gustav Peter Lejeune Dirichlet (1805-1859) or
Dirichlet for short, already impassioned by mathematics, arrived like Abel
at what was then the City of Light of mathematics and of physics 9 . A year
later, he had the opportunity of finding very comfortable employment, as
tutor in the household of General FaylO, a companion in arms of Napoleon.
Here he met Parisian "society" and notably Fourier whose works impressed
him as much for their staggering results as by the imaginativeness of their
proofs. He then tried to erect Fourier's theory on a solid base, particularly for
a neighbourhood of a point where the sum is discontinuous, which dropped
him right into Cauchy's "theorem" and Fourier's square wave series. His
principal result (1829), now become classical, is to be found in Chap. VII; it
completely justifies the relation (2) and Fourier's formulae.
8 See in Hairer and Wanner, Analysis by Its History, p. 41, a curious graphical
representation of the values of the function n f-+ sin n in logarithmic scale for n.
9 See Amy Dahan-Dalmedico, Mathematisations: Augustin-Louis Cauchy et l'ecole
franr;aise (Paris, Blanchard, 1992) and I. Grattan-Guinness, Convolutions in
French Mathematics, 1800-1840 (Birkhauser Verlag, 3 Vol., 1990).
10 who died in 1825. Dirichlet returned to Germany in 1826, obtained a post which
did not interest him in Breslau, arrived in Berlin in 1828 where he taught banal
mathematics at the military academy for a quarter of a century, obtained at the
same time a post, then a chair, at the University of Berlin, which, tired by the
13 hours weekly for ill paid courses, he resigned in 1855 to succeed Gauss at
Gottingen. DSB.
213
(6.3)
1/2 = cos x - cos2x + cos3x - ... ,
"an absolutely false result, since the series is divergent'. This is obvious for
certain values of x: for x = 7r /2 for example, one finds the "formula"
1/2 = 1 - 1 + 1 - 1 + ... ,
which we have already met. The same is true more generally for x commensurable to 7r since then among the multiples of x there occur infinitely many
integer multiples of 7r for which the corresponding terms in (3) are equal to
1 or -1. The case where x is not commensurable to 7r is even worse, the
values of cos nx then being distributed at random 8 between -1 and 1. If he
had had to, Abel could have rendered the situation even more ridiculous by
differentiating the relation (2) thrice, and setting x = 0; then one finds that
0= 1 - 4 + 9 - 16 + ... This is nevertheless what Fourier did: but instead
of starting from the relation (2.4) that he sought to establish, he started
from a series with a priori undetermined coefficients an, and, differentiating
ad libitum for x = 0, obtained linear equations in infinitely many unknowns
(I) between the an, so that he never had to write the extravagant relations
which, to verify the correctness of his calculations, he would have obtained
in giving these an the explicit values 1, 1/3, etc. that he finally found after acrobatic calculations; this must be the absolute record in mathematical
prestidigitation. His results were no less correct for all that.
In 1822, a young German, Gustav Peter Lejeune Dirichlet (1805-1859) or
Dirichlet for short, already impassioned by mathematics, arrived like Abel
at what was then the City of Light of mathematics and of physics 9 . A year
later, he had the opportunity of finding very comfortable employment, as
tutor in the household of General FaylO, a companion in arms of Napoleon.
Here he met Parisian "society" and notably Fourier whose works impressed
him as much for their staggering results as by the imaginativeness of their
proofs. He then tried to erect Fourier's theory on a solid base, particularly for
a neighbourhood of a point where the sum is discontinuous, which dropped
him right into Cauchy's "theorem" and Fourier's square wave series. His
principal result (1829), now become classical, is to be found in Chap. VII; it
completely justifies the relation (2) and Fourier's formulae.
8 See in Hairer and Wanner, Analysis by Its History, p. 41, a curious graphical
representation of the values of the function n f-+ sin n in logarithmic scale for n.
9 See Amy Dahan-Dalmedico, Mathematisations: Augustin-Louis Cauchy et l'ecole
franr;aise (Paris, Blanchard, 1992) and I. Grattan-Guinness, Convolutions in
French Mathematics, 1800-1840 (Birkhauser Verlag, 3 Vol., 1990).
10 who died in 1825. Dirichlet returned to Germany in 1826, obtained a post which
did not interest him in Breslau, arrived in Berlin in 1828 where he taught banal
mathematics at the military academy for a quarter of a century, obtained at the
same time a post, then a chair, at the University of Berlin, which, tired by the
13 hours weekly for ill paid courses, he resigned in 1855 to succeed Gauss at
Gottingen. DSB.