406
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
n° 18, Theorem 13). The sum over PI, ... ,Pn is equal to the product of the
Wi(Z) for i = il, ... , in (multiplication of series). So it remains to prove
(19.11)
Wi 1 (z) ... Win (Z) < +00;
which is what Theorem 16 affirms if we know that
(19.12)
p,n
for Izi < R (associativity: sum first over P, whence wn(z), then over n). This
condition is realised if, for all n, the coefficents an(p) are real and of the same
sign (Le. all > 0 or all < 0, the sign maybe depending on n), or, another important case, if the power series un(z) each contain only one nonzero term,
since wn(z) = lun(lzl)l in both these cases. This is also the case for a product
such as
II 1/(1 + qnz) = L(1- qn z + q2n z2 - ... )
with Iql < 1, since here wn(z) = I: Iqnzl P = Iqnzl/(l -Iqnzl) and the series
Wn converges. The three cases, curiously, will be found again in Euler, as we
shall see in the following n O •
In a case of this kind, one can calculate formally with the sum (9) without
risk of falling into divergent series; one can perform arbitrary groupings of
terms and, in particular, group all the monomials of the same degree in z,
whence a power series whose coefficients are calculated "as in Algebra", but
with the help of absolutely convergent series.
In the general case we know only that
L lun(z)1 = L IL an (p)zP I < +00,
n
n
P
which is not enough to ensure (12). Another way of proceeding consists of
replacing (8) by
(19.13)
Vj(z) = uitCz) ... Ui n (z) = L bj(p)zP
P
where
(19.14) bj(p) =
Pl+···+Pn=P
Then, by Theorem 16,
II[l + un(z)] = 1 + L L Vj(z) = 1 + L Vj(z) = v(z),
n2::1
jEJn
jEJ