§3. First concepts of analytic functions
167
as Euler called it with characteristic aplomb - no one before or after him has
ever met this kind of object in algebra -, has for its roots the points where
the function sin(x)/x vanishes, i.e. mC", n 1= O. [The function sinz happily has
no complex nonreal roots ... J Since the first term ao is equal to 1, the general
formula (11) shows "obviously" that the "polynomial of infinite degree" (12)
is identical to the product of all the expressions of the form 1 - x/mr; on
replacing x by 7rX and grouping the terms for nand -n you obtain (9) with
the greatest of ease 54 .
Providence was manifestly on Euler's side - Providence, and a formidable
intuition for correct formulae - since his argument could just as well prove
that any other everywhere convergent power series has an expansion as an
infinite product featuring the points where it vanishes; this is flagrantly not
possible, particularly, but not only, when the series never vanishes: the exponential is an example of an "algebraic equation of infinite degree" having no
root. The theory of analytic functions here again provides general theorems
(Weierstrass, Chap. VII, nO 20), but much less simple than Euler's ideas; he
later gave, in his Introductio in Analysin Infinitorum of 1748, a proof which,
without being perfectly correct, is at least salvageable (Chap. IV, nO 18) since
it is based on a correct (and very ingenious) idea.
22 - Multiplication of series. Composition of analytic functions.
Formal series
Let (Ui), i E I and (Vj), j E J be two absolutely summable families and let
us consider the family of products UiVj, indexed by the Cartesian product
I x J. As we have seen in nO 18, example 2 the product family converges
unconditionally and its sum is the product of those of the two given series:
(22.1)
As an application, consider two series Ln>o Un and Ln>o V n, absolutely
convergent in the sense of n° 14, so convergIng unconditionally as we have
seen in nO 15. Writing U and V for their sums we then have
(22.2)
where the right hand side is the sum of a double series which also converges
unconditionally. Let us group the terms according to the value of p + q = n
(associativity! see also the much more general example 3 of n° 14). We obtain
54 Moritz Cantor, Vorlesungen ... , Vol. III, pp. 658-659. One can judge Euler's
influence by the fact that, more than seventy years later, Fourier reproduced his
proof in his works on trigonometric series without voicing the slightest doubt as
to its validity. It is true that, with his grossly divergent series, he was ill-placed
to argue with Euler ...
167
as Euler called it with characteristic aplomb - no one before or after him has
ever met this kind of object in algebra -, has for its roots the points where
the function sin(x)/x vanishes, i.e. mC", n 1= O. [The function sinz happily has
no complex nonreal roots ... J Since the first term ao is equal to 1, the general
formula (11) shows "obviously" that the "polynomial of infinite degree" (12)
is identical to the product of all the expressions of the form 1 - x/mr; on
replacing x by 7rX and grouping the terms for nand -n you obtain (9) with
the greatest of ease 54 .
Providence was manifestly on Euler's side - Providence, and a formidable
intuition for correct formulae - since his argument could just as well prove
that any other everywhere convergent power series has an expansion as an
infinite product featuring the points where it vanishes; this is flagrantly not
possible, particularly, but not only, when the series never vanishes: the exponential is an example of an "algebraic equation of infinite degree" having no
root. The theory of analytic functions here again provides general theorems
(Weierstrass, Chap. VII, nO 20), but much less simple than Euler's ideas; he
later gave, in his Introductio in Analysin Infinitorum of 1748, a proof which,
without being perfectly correct, is at least salvageable (Chap. IV, nO 18) since
it is based on a correct (and very ingenious) idea.
22 - Multiplication of series. Composition of analytic functions.
Formal series
Let (Ui), i E I and (Vj), j E J be two absolutely summable families and let
us consider the family of products UiVj, indexed by the Cartesian product
I x J. As we have seen in nO 18, example 2 the product family converges
unconditionally and its sum is the product of those of the two given series:
(22.1)
As an application, consider two series Ln>o Un and Ln>o V n, absolutely
convergent in the sense of n° 14, so convergIng unconditionally as we have
seen in nO 15. Writing U and V for their sums we then have
(22.2)
where the right hand side is the sum of a double series which also converges
unconditionally. Let us group the terms according to the value of p + q = n
(associativity! see also the much more general example 3 of n° 14). We obtain
54 Moritz Cantor, Vorlesungen ... , Vol. III, pp. 658-659. One can judge Euler's
influence by the fact that, more than seventy years later, Fourier reproduced his
proof in his works on trigonometric series without voicing the slightest doubt as
to its validity. It is true that, with his grossly divergent series, he was ill-placed
to argue with Euler ...