§3. First concepts of analytic functions
149
in the preceding sum by its absolute value then (i) for fixed n, the sum over
p converges (obvious: it is a finite sum), (ii) the sum over n of the sums
over p converges. But the sum over p is obviously equal to Icnl (Ial + Ihl)[n]
- it suffices to retrace the calculation -, so it all reduces to proving the
convergence of the series
where u = lal + Ihl. By definition of the radius of convergence R this is the
case for lui < R, i.e. for Ihl < R -Ial·
Since the sum considered converges unconditionally, one can calculate it
according to the partition (Ip), in other words permute the order of summations relative to nand p, qed.
Writing z for what we have just denoted by a, we see that, subject to the
conditions Izl < Rand Ihl < R - Izl, we have a relation of the form
(19.2) J(z + h) = L Jp(z)h[p] = Jo{z) + JI(z)h/1! + h(z)h2/2! + ...
p~o
with coefficients Jo{z) = J(z) and, for p > 0,
(19.3)
Jp(z) = L Cnz[n- p ] = cp + cp+lz/1! + Cp+2 z2 / 2! + ....
n~p
Replacing n by n + p, this relation can be written as
(19.3')
Jp(z) = L Cn+pz[n] ,
n~O
which exhibits how these series are formed starting from the given series
L Cnz[n]: one shifts the coefficients Cn to the left.
These formulae can be interpreted in a much more striking way. We
showed in nO 4, equation (4.8), that, in C, the function z[n] possesses a derivative (in the complex sense) equal to z[n-l], so has a pth derivative equal to
z[n-p]. The series (3) may thus be obtained from the initial power series by replacing each of its monomials by its pth derivative; in other words, one passes
from J to J P by applying the traditional rule of calculus for the derivatives
of a polynomial to the power series J{z). Another dangerous bend: do not
generalise this to arbitrary series of differentiable functions on lR ...
To clarify the situation, let us agree to call the series defined by the
formula
(19.4)
J'{z)
if J{z)
L Cn+lZ[n] = Cl + C2Z + C3 Z[2] + .. .
L cnz[n] = Co + CIZ + C2 Z[2] + .. .