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power series

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Definition

A power series in a variable XX and with coefficients in a ring RR is a series of the form

n=0 a nX n \sum_{n = 0}^\infty a_n X^n

where a na_n is in RR for each n0n\ge 0. Given that there are no additional convergence conditions, a power series is also termed emphatically as a formal power series. If RR is commutative, then the collection of formal power series in a variable XX with coefficients in RR forms a commutative ring denoted by R[[X]]R [ [ X ] ].

More generally, a power series in kk commuting variables X 1,,X kX_1,\ldots, X_k with coefficients in a ring RR has the form n 1=0,n 2=0,,n k=0 a n 1n kX 1 n 1X 2 n 2X k n k\sum_{n_1=0,n_2=0,\ldots, n_k = 0}^\infty a_{n_1\ldots n_k} X_1^{n_1} X_2^{n_2}\cdots X_k^{n_k}. If RR is commutative, then the collection of formal power series in kk commuting variables X 1,,X kX_1,\ldots, X_k form a formal power series ring denoted by R[[X 1,,X k]]R [ [ X_1,\ldots, X_k ] ].

More generally, we can consider noncommutative (associative unital) ring RR and words in noncommutative variables X 1,,X kX_1,\ldots, X_k of the form

w=X i 1X i m w = X_{i_1}\cdots X_{i_m}

(where mm has nothing to do with kk) and with coefficient a wRa_w \in R (here ww is a word of any length, not a multiindex in the previous sense). Thus the power sum is of the form wa wX w\sum_w a_w X_w and they form a formal power series ring in variables X 1,,X kX_1,\ldots, X_k denoted by RX 1,,X kR\langle \langle X_1,\ldots, X_k \rangle\rangle. Furthermore, RR can be even a noncommutative semiring in which case the words belong to the free monoid on the set S={X 1,,X k}S = \{ X_1,\ldots, X_k\}, the partial sums are then belong to a monoid semiring RSR\langle S\rangle. The formal power series then also form a semiring, by the multiplication rule

ra rX rb sX s= w u,v;w=uva ub vX w \sum_{r} a_r X_r \cdot \sum b_s X_s = \sum_w \sum_{u,v; w = u v} a_u b_v X_w

Of course, this implies that in a specialization, bb-s commute with variables X i kX_{i_k}; what is usually generalized to take some endomorphisms into an account (like at noncommutative polynomial level of partial sums where we get skew-polynomial rings, i.e. iterated Ore extensions).

Examples

Polynomials

For a natural number kk, a power series n=0 a nX n\sum_{n=0}^\infty a_n X^n such that a n=0a_n = 0 for all n>kn \gt k is a polynomial of degree at most kk.

Taylor series

MacLaurin series

For fC ()f \in C^\infty(\mathbb{R}) a smooth function on the real line, and for f (n)C ()f^{(n)} \in C^\infty(\mathbb{R}) denoting its nnth derivative its MacLaurin series (its Taylor series at 00) is the power series

n=0 1n!f (n)(0)x n. \sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(0) x^n \,.

If this power series converges to ff, then we say that ff is analytic.

Laurent series

Puiseux series

Properties

  • An element a=a 0+a 1x+a 2x 2+a = a_0 + a_1 x + a_2 x^2 + \ldots in R[[x]]R[ [x] ] is (multiplicatively) invertible iff a 0a_0 is invertible.

This follows easily from the observation that we can invert 1+xb1 + x b for any power series bb by forming 1xb+x 2b 21 - x b + x^2 b^2 - \ldots and collecting only finitely many terms in each degree. As a simple corollary,

  • If RR is a local ring, then the power series ring R[[X]]R[ [X] ] is also a local ring.

Functional substitution and inversion

Proposition

R[[x 1,,x n]]R[ [x_1, \ldots, x_n] ] equipped with the ideal (x 1,,x n)(x_1, \ldots, x_n) is the free adic RR-algebra on nn generators, in the sense that it is the value of the left adjoint PowPow to the forgetful functor

Ideal:AdicRAlgSet:(A,I)IIdeal: AdicRAlg \to Set: (A, I) \mapsto I

as applied to the set {x 1,,x n}\{x_1, \ldots, x_n\}.

Proof

The idea is that for each adic RR-algebra (S,I)(S, I) and element (s 1,s n)I n(s_1, \ldots s_n) \in I^n, there is a unique adic algebra map R[[x 1,,x n]]SR[ [x_1, \ldots, x_n] ] \to S that sends x ix_i to s is_i; this adic algebra map sends a power series a k 1,,k nx 1 k 1x n k n\sum a_{k_1, \ldots, k_n} x_1^{k_1} x_n^{k_n} to the sequence of truncations

( k 1++k n<ka k 1,,k ns 1 k 1s n k nmodI k) k\left(\sum_{k_1 + \ldots + k_n \lt k} a_{k_1, \ldots, k_n} s_1^{k_1} \ldots s_n^{k_n} \mod I^k\right)_k

belonging to lim kS/I kS\underset{\longleftarrow}{\lim}_k S/I^k \cong S.

It follows that we may define a clone or cartesian operad as follows: the n thn^{th} component is the set I n=(x 1,,x n)R[[x 1,,x n]]I_n = (x_1, \ldots, x_n) \subset R[ [x_1, \ldots, x_n] ] which is the monad value IdealPow({x 1,,x n})Ideal Pow(\{x_1, \ldots, x_n\}). Letting MM denote the monad IdealPowIdeal \circ Pow, with monad multiplication μ\mu, and [n][n] the set {x 1,,x n}\{x_1, \ldots, x_n\}, the clone multiplication

I n×I k nI kI_n \times I_k^n \to I_k

is the composition of the maps

M(n)×M(k) nM(n)×hom([n],M(k))1×funcM(n)×hom(M(n),MM(k))evalMM(k)μ(k)M(k)M(n) \times M(k)^n \cong M(n) \times \hom([n], M(k)) \stackrel{1 \times func}{\to} M(n) \times \hom(M(n), M M(k)) \stackrel{eval}{\to} M M(k) \stackrel{\mu(k)}{\to} M(k)

The clone multiplication thus defined is called substitution of power series; it takes a tuple consisting of p(x 1,,x n)I n,q 1(x 1,x k)I k,q n(x 1,,x k)I k)p(x_1, \ldots, x_n) \in I_n, q_1(x_1, \ldots x_k) \in I_k, \ldots q_n(x_1, \ldots, x_k) \in I_k) to a power series denoted as

p(q 1(x 1,,x k),q n(x 1,,x k)).p(q_1(x_1, \ldots, x_k), \ldots q_n(x_1, \ldots, x_k)).

The resulting clone or operad yields, in the particular case k=n=1k = n = 1, an associative substitution operation

xR[[x]]×xR[[x]]subxR[[x]]x R[ [x] ] \times x R[ [x] ] \stackrel{sub}{\to} x R[ [x] ]

with sub(p,q)=pqsub(p, q) = p \circ q the power series p(q(x))p(q(x)).

Proposition

The group of invertible elements in the substitution monoid xR[[x]]x R[ [x] ] consists of power series of the form a 1x+a 2x 2+a_1 x + a_2 x^2 + \ldots where a 1a_1 is multiplicatively invertible in the ring RR.

In other words, we can functionally invert a power series provided that the linear coefficient a 1a_1 is invertible in RR.

Proof

Given power series a=a 1x+a 2x 2+a = a_1 x + a_2 x^2 + \ldots and b=b 1x+b 2x 2+b = b_1 x + b_2 x^2 + \ldots, we may read off coefficients of the composite aba \circ b as

(ab) k= n1a n k=k 1++k nb k 1b k 2b k n(a \circ b)_k = \sum_{n \geq 1} a_n \sum_{k = k_1 + \ldots + k_n} b_{k_1} b_{k_2} \ldots b_{k_n}

where in particular (ab) 1=a 1b 1(a \circ b)_1 = a_1 b_1. Now aa is the left functional inverse of bb, or bb is the right inverse of aa, if (ab)(x)=x(a \circ b)(x) = x, i.e., if (ab) k=1(a \circ b)_k = 1 if k=1k = 1 and 00 otherwise. The first equation says simply (ab) 1=a 1b 1=1(a \circ b)_1 = a_1 b_1 = 1 which implies a 1a_1 is invertible. Conversely, if a 1a_1 is multiplicatively invertible and b 1=a 1 1b_1 = a_1^{-1}, then the equations

n1a n k=k 1++k nb k 1b k 2b k n =1ifk=1 =0ifk1\array{ \sum_{n \geq 1} a_n \sum_{k = k_1 + \ldots + k_n} b_{k_1} b_{k_2} \ldots b_{k_n} & = 1\; if\; k = 1 \\ & = 0\; if\; k \neq 1 }

may be uniquely solved for the remaining a ia_i's given the b jb_j's, and uniquely solved for the remaining b jb_j's given the a ia_i's, by an inductive procedure: for k1k \neq 1 we have

a 1b k+a kb 1 k+termsa nb k 1b k n=0a_1 b_k + a_k b_1^k + \; terms\; a_n b_{k_1} \ldots b_{k_n} = 0

and this allows us to solve for b kb_k,

b k=a 1 1(a kb 1 k+termsa nb k 1b k n)b_k = -a_1^{-1}(a_k b_1^k + \; terms\; a_n b_{k_1} \ldots b_{k_n})

given the values a 1,,a ka_1, \ldots, a_k and earlier bb-values b k jb_{k_j} for k j<kk_j \lt k given by inductive hypothesis. Similarly we can solve for a ka_k in terms of given coefficients b 1,,b kb_1, \ldots, b_k and earlier aa-values a na_n, n<kn \lt k. Thus every power series aa has a right inverse if a 1 1a_1^{-1} exists, and bb has a left inverse if b 1 1b_1^{-1} exists, and this completes the proof.

References

A formalization in homotopy type theory and there in Coq is discussed in section 4 of

The discussion of the differentiation of a converging power series term by term is at

category: analysis, algebra

Revised on February 9, 2018 15:49:19 by Todd Trimble (69.112.103.83)