Lecture
In this section we introduce the notion of a formal power series
(the word «power» will often be omitted), a convenient tool
for studying singularities of functions, maps, differential equations, and many other «continuous» objects.
DEFINITION 5.1. A formal power series in the variables x1,...,xp over an algebraic field K is a series, i.e. an infinite sum
aₓ₋...₋ₓ⋅x1ⁿ¹⋅...⋅xpⁿ₉, (5.1)
∑n1,...,np≥0
where aₓ₋...₋ₓ are elements of the field K, and n1,....,np are nonnegative integers.
Note that an algebraic field by itself does not contain the notion of convergence of its elements, so that without additional assumptions the question of the convergence of such a series simply does not arise, and
the series (5.1) is regarded as a formal algebraic expression.
For our purposes the situation of interest is the one where K = R, the field
of real numbers, considered with the standard convergence
determined by the standard Euclidean structure (inner product). In this case the word «formal»
means that the series (5.1)
may diverge at every point (except the origin, at which a power series always converges).
One can define the addition and multiplication of formal series by
the same rule as for convergent series (and this is in fact simpler,
50
since one need not worry about convergence). As a result one obtains
a commutative ring, and indeed an algebra, of formal power series
over the field R. It is denoted by the symbol
R[[x1,...,xp]].
The double square brackets serve to distinguish it from the algebra of polynomials. Moreover, one can also substitute formal series into one another and
divide them by one another. The latter operation is not always
defined; more precisely, its result is not always a formal power series (just as the ratio of two integers is not always,
an integer). Formal series can also be formally
differentiated and integrated.
Finally, on the set R[[x1,...,xp]] one can introduce a topology, which defines a notion of convergence on it. We shall not go into this question, since it is not needed for what follows. The interested reader is referred to the book [29], in which formal series
are also called generating functions (of the sequences
of their coefficients).
Formal series are related to smooth functions in a completely natural way. If one takes an arbitrary smooth
function F(x1,...,xp) and forms its Taylor series at the point 0, one obtains, in general, precisely a formal series. As is known, it may either
diverge, or converge to some analytic function which, in general, does not coincide with the original function F (for example, for
the function (5.3) constructed below).
It is natural to ask the converse question as well. Given an arbitrary formal series, does there exist a smooth function whose Taylor
series it is? The answer to this question is given by the Borel–Whitney lemma (Lemma 5.1), which we shall prove.
Let us introduce one more kind of formal series: formal
power series in one of the variables, while the remaining
variables play the role of parameters on which the coefficients of the series — which are already «genuine» functions — depend. To emphasize this
circumstance, such series are often called semiformal, but we
shall also call them formal, since this causes no confusion.
'Émile Borel (1871-1956) — a French mathematician. Hassler Whitney (1907-
1989) — an American mathematician who made a great contribution to differential
topology and singularity theory.
51
Namely, let (x, y1,...,ym) be a set of variables with one «distinguished» variable x. Denote y = (y1,..., ym). Consider a formal power series in the variable x:
∑ an(y)⋅xⁿ, (5.2)
n≥0
in which the coefficients an(y) are smooth functions. Addition and multiplication of formal series (5.2), as well as other operations on them,
are defined analogously to the above.
For simplicity of the statements of the results given below we shall assume that the functions an(y) : Rⁿ → R are smooth on the whole space.
Comments