formal power series
Formal power series allow one to employ much of the analytical machinery of power series in settings which don’t have natural notions of convergence. They are also useful in order to compactly describe sequences and to find closed formulas for recursively described sequences; this is known as the method of generating functions and will be illustrated below.
We start with a commutative ring . We want to define the ring of formal power series over in the variable , denoted by ; each element of this ring can be written in a unique way as an infinite sum of the form , where the coefficients are elements of ; any choice of coefficients is allowed. is actually a topological ring so that these infinite sums are well-defined and convergent. The addition and multiplication of such sums follows the usual laws of power series.
Start with the set of all infinite sequences in . Define addition of two such sequences by
and multiplication by
This turns into a commutative ring with multiplicative identity (1,0,0,…). We identify the element of with the sequence (,0,0,…) and define . Then every element of of the form can be written as the finite sum
In order to extend this equation to infinite series, we need a metric on . We define , where is the smallest natural number such that (if there is not such , then the two sequences are equal and we define their distance to be zero). This is a metric which turns into a topological ring, and the equation
can now be rigorously proven using the notion of convergence arising from ; in fact, any rearrangement of the series converges to the same limit.
This topological ring is the ring of formal power series over and is denoted by .
The geometric series formula is valid in :
An element of is invertible in if and only if its constant coefficient is invertible in (see invertible formal power series). This implies that the Jacobson radical of is the ideal generated by and the Jacobson radical of .
Several algebraic properties of are inherited by :
The metric space is complete. The topology on is equal to the product topology on where is equipped with the discrete topology. It follows from Tychonoff’s theorem that is compact if and only if is finite. The topology on can also be seen as the -adic topology, where is the ideal generated by (whose elements are precisely the formal power series with zero constant coefficient).
If is a field, we can consider the quotient field of the integral domain ; it is denoted by and called a (formal) power series field. It is a topological field whose elements are called formal Laurent series; they can be uniquely written in the form
where is an integer which depends on the Laurent series .
Formal power series as functions
In analysis, every convergent power series defines a function with values in the real or complex numbers. Formal power series can also be interpreted as functions, but one has to be careful with the domain and codomain. If is an element of , if is a commutative associative algebra over , if an ideal in such that the -adic topology on is complete, and if is an element of , then we can define
This latter series is guaranteed to converge in given the above assumptions. Furthermore, we have
(unlike in the case of bona fide functions, these formulas are not definitions but have to proved).
Since the topology on is the -adic topology and is complete, we can in particular apply power series to other power series, provided that the arguments don’t have constant coefficients: , and are all well-defined for any formal power series .
Differentiating formal power series
If , we define the formal derivative of as
and the chain rule:
(in case g(0)=0).
In a sense, all formal power series are Taylor series, because if , then
(here denotes the element .
Power series in several variables
The fastest way to define the ring of formal power series over in variables starts with the ring of polynomials over . Let be the ideal in generated by , consider the -adic topology on , and form its completion. This results in a complete topological ring containing which is denoted by .
For , we write . Then every element of can be written in a unique was as a sum
where the sum extends over all . These sums converge for any choice of the coefficients and the order in which the summation is carried out does not matter.
If is the ideal in generated by (i.e. consists of those power series with zero constant coefficient), then the topology on is the -adic topology.
Since is a commutative ring, we can define its power series ring, say . This ring is naturally isomorphic to the ring just defined, but as topological rings the two are different.
If is a field, then is a unique factorization domain.
Similar to the situation described above, we can “apply” power series in several variables to other power series with zero constant coefficients. It is also possible to define partial derivatives for formal power series in a straightforward way. Partial derivatives commute, as they do for continuously differentiable functions.
Then one can easily show that
as well as
(the latter being valid in the ring ).
As an example of the method of generating functions, consider the problem of finding a closed formula for the Fibonacci numbers defined by , , and . We work in the ring and define the power series
is called the generating function for the sequence . The generating function for the sequence is while that for is . From the recurrence relation, we therefore see that the power series agrees with except for the first two coefficients. Taking these into account, we find that
(this is the crucial step; recurrence relations can almost always be translated into equations for the generating functions). Solving this equation for , we get
Using the golden ratio and , we can write the latter expression as
These two power series are known explicitly because they are geometric series; comparing coefficients, we find the explicit formula
In algebra, the ring (where is a field) is often used as the “standard, most general” complete local ring over .
The power series ring can be characterized by the following universal property: if is a commutative associative algebra over , if is an ideal in such that the -adic topology on is complete, and if are given, then there exists a unique with the following properties:
is an -algebra homomorphism
is continuous (http://planetmath.org/Continuous)
|Title||formal power series|
|Date of creation||2013-03-22 12:49:30|
|Last modified on||2013-03-22 12:49:30|
|Last modified by||AxelBoldt (56)|
|Defines||formal power series|
|Defines||formal Laurent series|
|Defines||power series field|