§3. Infinite products
401
Theorem 15. Let L un(p) be a series depending on an integer n. Suppose
that (i) limn un(p) = u(p) exists for all p, (ii) there exists an absolutely
convergent series dominating all the series considered. Then
00
00
J!"'~ IT [1 + un(p)] = IT [1 + u(p)]
p=l
p=l
A variant of this result, with the same proof, is obtained on replacing the
"discrete" variable n by a continuous variable:
Theorem 15 bis. Let X be a set and L fp(x) a series of scalar functions defined on X converging normally on X. Then the infinite product
TI[I + fp(x)] converges absolutely for all x E X and its partial products converge uniformly on X.
In the case where X c C, we deduce that if the fp are continuous, then
so is the product. If, when x tends to a point a adherent to X (or to infinity),
the fp(x) tend to limits up, then p(x) = TI[I + fp(x)] tends to the product
of 1 + up, etc.
In general, the series L fp will converge normally only on all compact
subsets of X, in which case, of course, the same holds for the partial products.
To deduce that, if the fp are continuous on X, so similarly is the product,
one needs to know that, for all a E X and all sufficiently small r > 0, the set
of x E X such that d(a, x) ::; r is compact, in other words that X is locally
compact.
In these statements, one should pay more attention than we have done to
the possibility that some factors of the products considered might be zero.
In the situation of Theorem 15 bis, we have Ifp(x)1 < ! for all x once p > N;
one should therefore suppress the first terms N of the product to obtain a
meaningful statement 31 •
From the infinite product
(18.16)
one can deduce a slightly more general formula. On replacing z by z + u, the
general term becomes, after a line of calculation,
now the product of the factors 1 - u 2 jn 2 rr2 is equal to sin uju. One deduces
that
(18.17) sin(z + u) (1 j ) n°O (
z) ( z)
.
=
+z u
1 + - - -
1 + - - - ;
SIn u
u - nrr
u + nrr
n=l
31 See the ultrarapid and ultraprecise exposition of R. Remmert, Funktionentheorie 2, pp. 2-10.
401
Theorem 15. Let L un(p) be a series depending on an integer n. Suppose
that (i) limn un(p) = u(p) exists for all p, (ii) there exists an absolutely
convergent series dominating all the series considered. Then
00
00
J!"'~ IT [1 + un(p)] = IT [1 + u(p)]
p=l
p=l
A variant of this result, with the same proof, is obtained on replacing the
"discrete" variable n by a continuous variable:
Theorem 15 bis. Let X be a set and L fp(x) a series of scalar functions defined on X converging normally on X. Then the infinite product
TI[I + fp(x)] converges absolutely for all x E X and its partial products converge uniformly on X.
In the case where X c C, we deduce that if the fp are continuous, then
so is the product. If, when x tends to a point a adherent to X (or to infinity),
the fp(x) tend to limits up, then p(x) = TI[I + fp(x)] tends to the product
of 1 + up, etc.
In general, the series L fp will converge normally only on all compact
subsets of X, in which case, of course, the same holds for the partial products.
To deduce that, if the fp are continuous on X, so similarly is the product,
one needs to know that, for all a E X and all sufficiently small r > 0, the set
of x E X such that d(a, x) ::; r is compact, in other words that X is locally
compact.
In these statements, one should pay more attention than we have done to
the possibility that some factors of the products considered might be zero.
In the situation of Theorem 15 bis, we have Ifp(x)1 < ! for all x once p > N;
one should therefore suppress the first terms N of the product to obtain a
meaningful statement 31 •
From the infinite product
(18.16)
one can deduce a slightly more general formula. On replacing z by z + u, the
general term becomes, after a line of calculation,
now the product of the factors 1 - u 2 jn 2 rr2 is equal to sin uju. One deduces
that
(18.17) sin(z + u) (1 j ) n°O (
z) ( z)
.
=
+z u
1 + - - -
1 + - - - ;
SIn u
u - nrr
u + nrr
n=l
31 See the ultrarapid and ultraprecise exposition of R. Remmert, Funktionentheorie 2, pp. 2-10.