§3. First concepts of analytic functions
151
for Ihl < R - Izl, agreeing to put 1(0) = I. This is the famous formula for
power series named after (Brook) Taylor (1715) even though it was more
or less known to Newton by 1691, who did not exploit it further than his
exponential series, as well as to Johann Bernoulli who published it, replacing
f by its primitive, in 1694 (M. Cantor, III, pp. 229 and 383); one can verify it
very easily algebraically when 1 reduces to a polynomial in z. If, in particular,
in (6) one replaces z by 0, h by z and p by n, one finds MacLaurin's formula
(19.6') I(z) = L I(n) (O)z[n] = 1(0) + f'(O)z/l! + f'(0)z2/2! + ...
We shall see later (Chap. V, nO 18) that there is a similar result, but with
a finite number of terms and a "remainder" that one can evaluate, valid for
functions of a real variable possessing derivatives (in the usual sense) up to
a certain order, much more general than those which can be represented by
power series.
Finally, the algorithm for passing from the series 1 to the series f' can be
inverted; one thus obtains the primitive series
of I, with an arbitrary constant c, the same radius of convergence and the
relation F' = 1 which "justifies" the terminology to those who have learnt
that the primitive, in the elementary sense, of the function xn is xn+ 1 / (n + 1);
see nO 11.
As we have already said, Newton was the first to make systematic use of
power series to solve all sorts of problems. The greatest mathematician of the
following century, Euler, also used them freely. But neither Newton nor Euler
elaborated any general theory of power series, and it was Lagrange (17361813) who, in his Theorie des fonctions analytiques (1797), founded it, as he
said, to liberate analysis from infinitely small quantities, from limits, from
"fluxions" (derivatives) a la Newton, etc. and to transform "infinitesimal"
analysis into an "algebraic" analysis. He it was who introduced the notation
/'(x) and the word "derivative" for analytic functions; he proved the Taylor
and MacLaurin formulae as we have done above, though without worrying too
much about questions of convergence, nor working in
was doomed to failure, but his theory prepared the ground for Cauchy and his
46 This is false in general for functions of a real variable, no matter how differentiable
they may be, because, as Cauchy was first to remark, the function I(x) equal to
exp( -1 / x 2 ) for x 'I 0 and to 0 for x = 0 possesses at x = 0 successive derivatives
of every order, all of which are zero, as we will show later; MacLaurin's formula
would then show that I is identically zero. In fact, on R one can construct
indefinitely differentiable functions having arbitrarily given successive derivatives
at x = 0 (Chap. V, nO 29).
151
for Ihl < R - Izl, agreeing to put 1(0) = I. This is the famous formula for
power series named after (Brook) Taylor (1715) even though it was more
or less known to Newton by 1691, who did not exploit it further than his
exponential series, as well as to Johann Bernoulli who published it, replacing
f by its primitive, in 1694 (M. Cantor, III, pp. 229 and 383); one can verify it
very easily algebraically when 1 reduces to a polynomial in z. If, in particular,
in (6) one replaces z by 0, h by z and p by n, one finds MacLaurin's formula
(19.6') I(z) = L I(n) (O)z[n] = 1(0) + f'(O)z/l! + f'(0)z2/2! + ...
We shall see later (Chap. V, nO 18) that there is a similar result, but with
a finite number of terms and a "remainder" that one can evaluate, valid for
functions of a real variable possessing derivatives (in the usual sense) up to
a certain order, much more general than those which can be represented by
power series.
Finally, the algorithm for passing from the series 1 to the series f' can be
inverted; one thus obtains the primitive series
of I, with an arbitrary constant c, the same radius of convergence and the
relation F' = 1 which "justifies" the terminology to those who have learnt
that the primitive, in the elementary sense, of the function xn is xn+ 1 / (n + 1);
see nO 11.
As we have already said, Newton was the first to make systematic use of
power series to solve all sorts of problems. The greatest mathematician of the
following century, Euler, also used them freely. But neither Newton nor Euler
elaborated any general theory of power series, and it was Lagrange (17361813) who, in his Theorie des fonctions analytiques (1797), founded it, as he
said, to liberate analysis from infinitely small quantities, from limits, from
"fluxions" (derivatives) a la Newton, etc. and to transform "infinitesimal"
analysis into an "algebraic" analysis. He it was who introduced the notation
/'(x) and the word "derivative" for analytic functions; he proved the Taylor
and MacLaurin formulae as we have done above, though without worrying too
much about questions of convergence, nor working in
was doomed to failure, but his theory prepared the ground for Cauchy and his
46 This is false in general for functions of a real variable, no matter how differentiable
they may be, because, as Cauchy was first to remark, the function I(x) equal to
exp( -1 / x 2 ) for x 'I 0 and to 0 for x = 0 possesses at x = 0 successive derivatives
of every order, all of which are zero, as we will show later; MacLaurin's formula
would then show that I is identically zero. In fact, on R one can construct
indefinitely differentiable functions having arbitrarily given successive derivatives
at x = 0 (Chap. V, nO 29).