148
II - Convergence: Discrete variables
§3. First concepts of analytic functions
19 - The Taylor series
First recall that given an open set 44 Gee and a scalar function J defined
on G one says that J is analytic in G if, for all a E G, there is a series
expansion
J(z) = L In(a)(z - a)n
valid for Iz - al sufficiently small, with coefficients In(a) which necessarily
depend on a and on J. The first example of an analytic function is provided
by the following theorem:
Theorem 14. The sum oj a power series J(z) = L anz n oj radius oj convergence R> 0 is analytic in the disc D : Izl < R.
We need to show that, for all a ED, the sum J(z) can be represented by
a power series in z - a = h on a neighbourhood of a and even, as we shall
see, for Ihl < R - lal, i.e. in the largest disc with centre a contained in D.
Let us first calculate formally, using, to simplify the calculations, the
divided powers z[n] = zn In! as in nO 1. Now every power series can be written
in the form
(19.1)
J(z) = L Cnz[n) = Co + clz/l! + c2 z2 /2! + ...
with scalar coefficients Cn. Using the binomial formula of nO 1, one has, if lal
and la + hi are < R,
L cna[n- p ) hlp]
(p,n)EI
where leN x N is the set of pairs (p, n) such that 0 S; p S; n. Theorem 13
shows that, iJ this sum converges unconditionally, then one can group its
terms arbitrarily. It is natural to consider the partition (Ip) of I, where,
for each pEN, Ip denotes the set of pairs (n,p), n ;::: p: the partial sum
will be the product of hlp) by the power series Ln~p cna[n- p ) = Jp(a) and
J(z) = J(a + h) will thus, for given a, be a power series in h = z - a.
To prove unconditional convergence of the sum that we have just obtained
we use another partition (In) of I, that obtained by grouping together all
the pairs (n, p) for which n (and no longer p) has a given value. To be able
to apply Theorem 13 we need to verify that if one replaces the general term
44 Recall the definition: for all a E G there is an open disc B(a, r) such that
B(a,r) c G.
II - Convergence: Discrete variables
§3. First concepts of analytic functions
19 - The Taylor series
First recall that given an open set 44 Gee and a scalar function J defined
on G one says that J is analytic in G if, for all a E G, there is a series
expansion
J(z) = L In(a)(z - a)n
valid for Iz - al sufficiently small, with coefficients In(a) which necessarily
depend on a and on J. The first example of an analytic function is provided
by the following theorem:
Theorem 14. The sum oj a power series J(z) = L anz n oj radius oj convergence R> 0 is analytic in the disc D : Izl < R.
We need to show that, for all a ED, the sum J(z) can be represented by
a power series in z - a = h on a neighbourhood of a and even, as we shall
see, for Ihl < R - lal, i.e. in the largest disc with centre a contained in D.
Let us first calculate formally, using, to simplify the calculations, the
divided powers z[n] = zn In! as in nO 1. Now every power series can be written
in the form
(19.1)
J(z) = L Cnz[n) = Co + clz/l! + c2 z2 /2! + ...
with scalar coefficients Cn. Using the binomial formula of nO 1, one has, if lal
and la + hi are < R,
L cna[n- p ) hlp]
(p,n)EI
where leN x N is the set of pairs (p, n) such that 0 S; p S; n. Theorem 13
shows that, iJ this sum converges unconditionally, then one can group its
terms arbitrarily. It is natural to consider the partition (Ip) of I, where,
for each pEN, Ip denotes the set of pairs (n,p), n ;::: p: the partial sum
will be the product of hlp) by the power series Ln~p cna[n- p ) = Jp(a) and
J(z) = J(a + h) will thus, for given a, be a power series in h = z - a.
To prove unconditional convergence of the sum that we have just obtained
we use another partition (In) of I, that obtained by grouping together all
the pairs (n, p) for which n (and no longer p) has a given value. To be able
to apply Theorem 13 we need to verify that if one replaces the general term
44 Recall the definition: for all a E G there is an open disc B(a, r) such that
B(a,r) c G.