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

204 Chapter 10. Toward rigorization of analysis
“[. . . ] and thus it shows that
� �
i
= (1 + x)
a
i a ,
which demonstrates that our theorem is true when for the exponent n any fraction
i
a is taken, from this its truth is evident for all positive numbers taken in place of
the exponent n.” 36
BOLZANO was slightly more specific on binomial expansion for irrational expo-
nents. As a consequence of the meaning of (1 + x) i for i an irrational number, BOLZANO
claimed, (1 + x) i could be approached as closely as desired by (1 + x) m n for m, n inte-
gers. Inserting m n for i everywhere in the series and letting m n
approach i, the sum
would approach (1 + x) i as closely as desired. Thus, BOLZANO alluded to his concept
of continuity applied to the exponentiation and to power series in order to obtain the
binomial theorem for all real exponents. 37
10.4 New types of series
The series discussed thus far have all been power series but in the early nineteenth
century, this situation changed. Series which were not power series had emerged
in various contexts in the eighteenth century but became very important in the first
decades of the nineteenth century, mainly through investigations in the theory of heat
conducted by J. B. J. FOURIER (1768–1830). 38
FOURIER’S term-wise integration. From the first decade of the nineteenth century,
FOURIER had begun representing physical phenomena — mainly heat conduction —
by trigonometric series. In 1822, his investigations were published as a monograph. 39
One of FOURIER’S central tricks was the term-wise integration of an infinite series em-
ployed to obtain the Fourier coefficients in the following way. Assuming that a function
φ (x) could be expanded as
φ (x) =
36 “[. . . ] atque hinc in genere manifestum fore
∞
∑ ai sin ix, (10.2)
i=1
� �
i
= (1 + x)
a
i a ,
ita ut iam demonstratum sit theorema nostrum verum esse, si pro exponente n fractio quaecunque
i
a accipiatur, unde veritas iam est evicta pro omnibus numeris positivis loco exponentis n accipiendis.”
(L. Euler, 1775, 215–216).
37 (Bolzano, 1816, §46).
38 FOURIER and his works leading to Fourier series have been widely studied, see e.g. (Bottazzini,
1986; I. Grattan-Guinness, 1972).
39 (Fourier, 1822).