Page 124 - Textos de Matemática Vol. 34

114 JAIME CARVALHO E SILVA
In the paper “On the remainder of Taylor’s formula” (Teixeira J. X. 13-17, 1890), Bruno de Cabedo studies the remainder of the Taylor formula indicated in the paper by G. Peano titled “Une nouvelle forme du reste dans la formule de Taylor” (Mathesis IX. 182-183, 1889):
(x − a)n f(n−1)(x1) − f(n−1)(a) (n) R= n! x1−a −f (a).
Bruno de Cabedo gives a new proof of this form of the remainder and studies also the case of complex variables.
In the paper “On the coefficients of the Fourier Series” (Teixeira J. XII. 81- 84, 1894), Bruno de Cabedo proves that if we consider the Fourier Series for the functions
(2.1) and
(2.2) where
(2.3)
(2.4)
(2.5)
(2.6)
then we should have
(2.7)
φ(x) = 1 a0 + 1 (am cosmx+bm sinmx) 2π π
ψ(x) = 1 a′ + 1 (a′ cosmx+b′ sinmx) 2π0πm m
am = X φ(x) cos mxdx, x0
bm = X φ(x) sin mxdx, x0
a′ = X ψ(x) cos mxdx, m x0
b′ = X ψ(x) sin mxdx, m x0
X1′1′′
φ(x)ψ(x)dx = 2π a0a0 + π (amam + bmbm).
x0
The documents known today show that in the second half of the 19th century
the Professors at the University began having some international contacts; we know that R. Sousa Pinto, Luciano Pereira da Silva and Henrique de Figueiredo visited some foreign universities in that century. Only after 1908 can we find an official university policy of sending professors in study visits abroad.
3. The textbooks
Immediately after 1772 the university began editing Portuguese translations of books by Bezout (three books), Marie and Bossut. Between 1838 and 1871 were edited three revised and very enlarged translations of the “Cours complet de Math´ematiques Pures” by L. B. Francoeur (whose first edition dates back to 1819).