152
II - Convergence: Discrete variables
theory of analytic functions of a complex variable, which would revolutionise
all of classical analysis in the XIX th century.
We defined the "derived series" of a power series f above by an algebraic
algorithm. The terminology is justified by reason of the fact that f' (z) is also
the limit, in the sense of nO 4, of the ratio
f(z + hi - f(z) = J'(z) + h [f"(z)/2! + f"'(z)hI3! + ... ]
when h E C tends to O. The expression between the brackets [ ] is indeed
a convergent power series; its sum is thus, in modulus, majorised by a fixed
number M for Ihl sufficiently small (see the end of nO 14). The preceding
relation thus shows that
(19.7)
I f(z + hi - f(z) - J'(Z)I ::; Mlhl or = O(h)
for Ihl sufficiently small, so < r once Ihl < r' = riM; this is precisely the
definition of a limit for functions defined on a subset of COne can prove this result directly without the double series which we
employed above. To do this one starts from the relation (4.9)
i(z + h)[n] - z[n] - hz[n-l]i ::; Ihl[2] (Izl + Ihl)[n-2].
On replacing f(z) by L c,."z[n] in (7), f(z + h) by the analogous expression
in z + h, and f' (z) by its definition (4), it is clear, since h[2] I h = h 12, that
the left hand side of (7) is majorised by
if one has proved directly that the series f' and f" have the same radius of
convergence R as f, then the series we have just obtained will converge for
Izl + Ihl < R and one obtains the estimate (7) again.
This argument shows in passing that, on a neighbourhood of a given
point a, a function f cannot be represented by two different power series in
z - a. If indeed f(a + h) = L cn(a)Mn] for Ihl sufficiently small, one has first
f(a) = eo(a); then the function f'(z) = lim[f(z + h) - f(z)]/h is necessarily given by the derived series cn(a)Mn-l] as the preceding argument shows,
whence Cl (a) = f' (a). Iterating, one finds successively that C2 (a) = f" (a),
c3(a) = f"'(a), etc., which determines the coefficients of the series: these are
the values at a of the functions f, f', f", ... defined as the limits of quotients and no longer as the sums of the series obtained from the derivation
algorithm, even though the results are the same.
This apparently banal property of analytic functions, the existence of
derivatives in the sense of nO 4, has far-reaching consequences, as Cauchy
II - Convergence: Discrete variables
theory of analytic functions of a complex variable, which would revolutionise
all of classical analysis in the XIX th century.
We defined the "derived series" of a power series f above by an algebraic
algorithm. The terminology is justified by reason of the fact that f' (z) is also
the limit, in the sense of nO 4, of the ratio
f(z + hi - f(z) = J'(z) + h [f"(z)/2! + f"'(z)hI3! + ... ]
when h E C tends to O. The expression between the brackets [ ] is indeed
a convergent power series; its sum is thus, in modulus, majorised by a fixed
number M for Ihl sufficiently small (see the end of nO 14). The preceding
relation thus shows that
(19.7)
I f(z + hi - f(z) - J'(Z)I ::; Mlhl or = O(h)
for Ihl sufficiently small, so < r once Ihl < r' = riM; this is precisely the
definition of a limit for functions defined on a subset of COne can prove this result directly without the double series which we
employed above. To do this one starts from the relation (4.9)
i(z + h)[n] - z[n] - hz[n-l]i ::; Ihl[2] (Izl + Ihl)[n-2].
On replacing f(z) by L c,."z[n] in (7), f(z + h) by the analogous expression
in z + h, and f' (z) by its definition (4), it is clear, since h[2] I h = h 12, that
the left hand side of (7) is majorised by
if one has proved directly that the series f' and f" have the same radius of
convergence R as f, then the series we have just obtained will converge for
Izl + Ihl < R and one obtains the estimate (7) again.
This argument shows in passing that, on a neighbourhood of a given
point a, a function f cannot be represented by two different power series in
z - a. If indeed f(a + h) = L cn(a)Mn] for Ihl sufficiently small, one has first
f(a) = eo(a); then the function f'(z) = lim[f(z + h) - f(z)]/h is necessarily given by the derived series cn(a)Mn-l] as the preceding argument shows,
whence Cl (a) = f' (a). Iterating, one finds successively that C2 (a) = f" (a),
c3(a) = f"'(a), etc., which determines the coefficients of the series: these are
the values at a of the functions f, f', f", ... defined as the limits of quotients and no longer as the sums of the series obtained from the derivation
algorithm, even though the results are the same.
This apparently banal property of analytic functions, the existence of
derivatives in the sense of nO 4, has far-reaching consequences, as Cauchy