§3. First concepts of analytic functions
175
operations of algebra are enough to get down to work. After all, this is algebra for computers, apart from the fact that machines do not know where
to go if one does not tell them where to start; this is one of the numerous
differences between what they call artificial intelligence and what one used
to call intelligence in short and what must now be called natural intelligence
to avoid confusion, assuming that there is a risk of confusion ...
We have just spoken of "formal series"; what is a formal series with
coefficients in an arbitrary field K? It is an expression of the form E anX n
with coefficients an E K depending on an index nEZ, but zero for n negative
and sufficiently large; the so-called "series" contains only a finite number of
negative powers of the so-called "variable" X, which it must contain if one
wants to be able to write something like
1/X 3 (1 - X) = X- 3 + X- 2 + ....
Such a "series" has no a priori meaning: it is possible to give one to certain
infinite sums in lR or C, but quite impossible in a field where one has no
concept of a limit. One extricates oneself by considering - this was Hamilton's
idea for defining the complex numbers - that a formal series is simply a family
(an) of elements of the field K, and defining the sum and the product of two
such series by the natural formulae:
(an).(bn ) = (en) where en = 2: apbq ;
p+q=n
again a convolution product ... The fact that the an and bn are zero for
n < 0 large ensures that the definition of en involves only a finite sum, so
has a meaning in an arbitrary field. With these definitions of the sum and
of the product, the formal series form a new commutative field, i.e. satisfy
the axioms (I) of nO 1: an elementary exercise in algebra, asking only a little
patience. If the field K appears too abstract to you, abstract yourself from K
and calculate without trying to understand the concrete significance of the
letters: they have none, so here again we have mechanics for computers.
The notation E anXn can be justified - or rather explained - as follows.
First, one identifies each a E K with the formal series all of whose coefficients
are zero except ao = a. The letter X denotes the formal series all of whose
coefficients are zero except al = 1. On applying the rules of calculus above,
one finds that, for all nEZ, the series xn has all its coefficients zero except
an = 1, so that the product anXn is the series all of whose the terms are zero
except the nth, which equals an. The addition law then shows that a formal
series having no more than a finite number of nonzero coefficients is just the
sum of the series anXn for those n such that an =f. OJ one generally calls this
175
operations of algebra are enough to get down to work. After all, this is algebra for computers, apart from the fact that machines do not know where
to go if one does not tell them where to start; this is one of the numerous
differences between what they call artificial intelligence and what one used
to call intelligence in short and what must now be called natural intelligence
to avoid confusion, assuming that there is a risk of confusion ...
We have just spoken of "formal series"; what is a formal series with
coefficients in an arbitrary field K? It is an expression of the form E anX n
with coefficients an E K depending on an index nEZ, but zero for n negative
and sufficiently large; the so-called "series" contains only a finite number of
negative powers of the so-called "variable" X, which it must contain if one
wants to be able to write something like
1/X 3 (1 - X) = X- 3 + X- 2 + ....
Such a "series" has no a priori meaning: it is possible to give one to certain
infinite sums in lR or C, but quite impossible in a field where one has no
concept of a limit. One extricates oneself by considering - this was Hamilton's
idea for defining the complex numbers - that a formal series is simply a family
(an) of elements of the field K, and defining the sum and the product of two
such series by the natural formulae:
(an).(bn ) = (en) where en = 2: apbq ;
p+q=n
again a convolution product ... The fact that the an and bn are zero for
n < 0 large ensures that the definition of en involves only a finite sum, so
has a meaning in an arbitrary field. With these definitions of the sum and
of the product, the formal series form a new commutative field, i.e. satisfy
the axioms (I) of nO 1: an elementary exercise in algebra, asking only a little
patience. If the field K appears too abstract to you, abstract yourself from K
and calculate without trying to understand the concrete significance of the
letters: they have none, so here again we have mechanics for computers.
The notation E anXn can be justified - or rather explained - as follows.
First, one identifies each a E K with the formal series all of whose coefficients
are zero except ao = a. The letter X denotes the formal series all of whose
coefficients are zero except al = 1. On applying the rules of calculus above,
one finds that, for all nEZ, the series xn has all its coefficients zero except
an = 1, so that the product anXn is the series all of whose the terms are zero
except the nth, which equals an. The addition law then shows that a formal
series having no more than a finite number of nonzero coefficients is just the
sum of the series anXn for those n such that an =f. OJ one generally calls this