§3. Infinite products
407
a series of analytic functions which converges (unconditionally) normally on
every disc Izl :::; r < R, i.e. on all compact subsets of Izl < R. Now we will
show in Chap. VII that if such a series converges normally on all compacta
then (i) its sum is analytic, (ii) all the derived series converge normally on all
compacta, (iii) the derivatives of the sum are the sums of the derived series.
The function v is therefore analytic on Izi < R, so (another general theorem)
is the sum of its Maclaurin series
on all the disc Izi < R, and
(19.15)
jEJ
jEJ
with an absolutely convergent series. This result can be written
b(p) =
O which is again the "obvious" result, on condition that one sums over the Pk
before summing over the ik and n, failing which one will not obtain a sum
converging unconditionally: on replacing the Ui(P) by their moduli one can
certainly obtain a divergent series, as above. One is here at the limit of
the capability of unconditional convergence, but since one never meets this
general case in situations where one has to calculate the coefficients of the
power series u(z), this is not the place to linger, other than to put the reader
on guard against the risks of excessive confidence.
The simplest case is that where un(z) = anz, treated (without considerations of convergence) by Euler, immediately providing him the expansions
(19.16)
with
(19.17)
00
00
II(1 +anz) = 1 + LAn zn
An =
L ail·· .ai,,·
O The power series obtained converges for any z provided that 2: Ian I < +00.
20 - Strange identities
Euler, as we have said, studied infinite products of arithmetic interest. For
example he proved the formula