168
II - Convergence: Discrete variables
the algebraically obvious formula (apart from the fact that we are not doing
algebra 55 •.. )
00
00
00
(22.3) L Un· L Vn = L Wn where Wn = UOVn + UIVn-l + ... + UnVo·
o
0
0
The principal application of this formula is
Theorem 16. Let J(z) = L anz n and g(z) = L bnz n be two power series
which converge absolutely Jor Izl < R. Then
(22.4)
J(z)g(z) = L Cnz n with Cn = anbo + ... + aobn
and the series L Cnz n converges absolutely Jor Izl < R.
In other words, the rule Jor multiplication oj polynomials applies to power
series, of course under the restriction that one is at a point z where the two
series are absolutely convergent, e.g. in the interior of the smaller of the discs
of convergence of the two given series. It applies also more generally to the
Laurent series
where this time one sums over all n E Z; being the sum of a power series in
z and of a power series in liz, such a series converges, in general, only in an
annulus (possibly empty: the case of the series L zn where one sums over Z)
defined by inequalities 0 :::; R' < Izl < R"; there it converges absolutely as do
power series in their disc of convergence, and for the same reason: comparison
with geometric series. We shall return to this point in Chap. VII. So, if one
has two Laurent series converging in the same nonempty annulus, one can
form the unordered product and then group the terms:
where the series is extended over all pairs (p, q) such that p + q = n: one
recovers the convolution product over Z of n° 18, though the series Lan
and L bn need not be absolutely convergent even if the series defining their
convolution product is, as we have just seen. These series arise in the study
of analytic functions on the neighbourhood of an isolated singular point;
example: the function exp(z + liz), whose Laurent series expansion uses the
following result (the fact that this is presented as a "Corollary" should not
lead one to forget how fundamental it is):
55 This is so little algebraic that, if one applies the rule to non-absolutely convergent
series, one may obtain a divergent series, as Cauchy showed by choosing Un =
Vn = (_l)n+l In. Then Iwnl > 2nl(n-1), an expression which tends to 2 and not
to O. Paul Dugac, Sur les /ondements de l'Analyse de Cauchy d Baire (doctoral
thesis, Universite Pierre et Marie Curie, 1978) p. 19.
II - Convergence: Discrete variables
the algebraically obvious formula (apart from the fact that we are not doing
algebra 55 •.. )
00
00
00
(22.3) L Un· L Vn = L Wn where Wn = UOVn + UIVn-l + ... + UnVo·
o
0
0
The principal application of this formula is
Theorem 16. Let J(z) = L anz n and g(z) = L bnz n be two power series
which converge absolutely Jor Izl < R. Then
(22.4)
J(z)g(z) = L Cnz n with Cn = anbo + ... + aobn
and the series L Cnz n converges absolutely Jor Izl < R.
In other words, the rule Jor multiplication oj polynomials applies to power
series, of course under the restriction that one is at a point z where the two
series are absolutely convergent, e.g. in the interior of the smaller of the discs
of convergence of the two given series. It applies also more generally to the
Laurent series
where this time one sums over all n E Z; being the sum of a power series in
z and of a power series in liz, such a series converges, in general, only in an
annulus (possibly empty: the case of the series L zn where one sums over Z)
defined by inequalities 0 :::; R' < Izl < R"; there it converges absolutely as do
power series in their disc of convergence, and for the same reason: comparison
with geometric series. We shall return to this point in Chap. VII. So, if one
has two Laurent series converging in the same nonempty annulus, one can
form the unordered product and then group the terms:
where the series is extended over all pairs (p, q) such that p + q = n: one
recovers the convolution product over Z of n° 18, though the series Lan
and L bn need not be absolutely convergent even if the series defining their
convolution product is, as we have just seen. These series arise in the study
of analytic functions on the neighbourhood of an isolated singular point;
example: the function exp(z + liz), whose Laurent series expansion uses the
following result (the fact that this is presented as a "Corollary" should not
lead one to forget how fundamental it is):
55 This is so little algebraic that, if one applies the rule to non-absolutely convergent
series, one may obtain a divergent series, as Cauchy showed by choosing Un =
Vn = (_l)n+l In. Then Iwnl > 2nl(n-1), an expression which tends to 2 and not
to O. Paul Dugac, Sur les /ondements de l'Analyse de Cauchy d Baire (doctoral
thesis, Universite Pierre et Marie Curie, 1978) p. 19.