Editing Formal power series (section)

==Generalizations==

=== Formal Laurent series ===
The '''formal Laurent series''' over a ring <math>R</math> are defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree.  That is, they are the series that can be written as

:<math>f = \sum_{n = N}^\infty a_n X^n</math>

for some integer <math>N</math>, so that there are only finitely many negative <math>n</math> with <math>a_n \neq 0</math>.  (This is different from the classical [[Laurent series]] of [[complex analysis]].)  For a non-zero formal Laurent series, the minimal integer <math>n</math> such that <math> a_n\neq 0</math> is called the ''order'' of <math>f</math> and is denoted <math>\operatorname{ord}(f).</math> (The order ord(0) of the zero series is <math>+\infty</math>.)

For instance, <math>X^{-3} + \frac 1 2 X^{-2} + \frac 1 3 X^{-1} + \frac 1 4 + \frac 1 5 X + \frac 1 6 X^2 + \frac 1 7 X^3 + \frac 1 8 X^4 + \dots </math> is a formal Laurent series of order –3.

Multiplication of such series can be defined. Indeed, similarly to the definition for formal power series, the coefficient of <math>X^k</math> of two series with respective sequences of coefficients <math>\{a_n\}</math> and <math>\{b_n\}</math> is
<math display="block">\sum_{i\in\Z}a_ib_{k-i}.</math>
This sum has only finitely many nonzero terms because of the assumed vanishing of coefficients at sufficiently negative indices.

The formal Laurent series form the '''ring of formal Laurent series''' over <math>R</math>, denoted by <math>R((X))</math>.{{efn|For each nonzero formal Laurent series, the order is an integer (that is, the degrees of the terms are bounded below).  But the ring <math>R((X))</math> contains series of all orders.}}  It is equal to the [[localization of a ring|localization]] of the ring <math>R[[X]]</math> of formal power series with respect to the set of positive powers of <math>X</math>.  If <math>R=K</math> is a [[field (mathematics)|field]], then <math>K((X))</math> is in fact a field, which may alternatively be obtained as the [[field of fractions]] of the [[integral domain]] <math>K[[X]]</math>.

As with <math>R[[X]]</math>, the ring <math>R((X))</math> of formal Laurent series may be endowed with the structure of a topological ring by introducing the metric
<math display="block">d(f,g)=2^{-\operatorname{ord}(f-g)}.</math>
(In particular, <math>\operatorname{ord}(0) = +\infty</math> implies that <math>d(f,f)=2^{-\operatorname{ord}(0)} = 0</math>.)

One may define formal differentiation for formal Laurent series in the natural (term-by-term) way. Precisely, the formal derivative of the formal Laurent series <math>f</math> above is
<math display="block">f' = Df = \sum_{n\in\Z} na_n X^{n-1},</math>
which is again a formal Laurent series. If <math>f</math> is a non-constant formal Laurent series and with coefficients in a field of characteristic 0, then one has
<math display="block">\operatorname{ord}(f')= \operatorname{ord}(f)-1.</math>
However, in general this is not the case since the factor <math>n</math> for the lowest order term could be equal to 0 in <math>R</math>.

====Formal residue====
Assume that <math>K</math> is a field of [[characteristic (algebra)|characteristic]] 0. Then the map

:<math>D\colon K((X))\to K((X))</math>

defined above is a <math>K</math>-[[derivation (abstract algebra)|derivation]] that satisfies

:<math>\ker D=K</math>
:<math>\operatorname{im} D= \left \{f\in K((X)) : [X^{-1}]f=0 \right \}.</math>

The latter shows that the coefficient of <math>X^{-1}</math> in <math>f</math> is of particular interest; it is called ''formal residue of <math>f</math>'' and denoted <math>\operatorname{Res}(f)</math>. The map

:<math>\operatorname{Res} : K((X))\to K</math>

is <math>K</math>-linear, and by the above observation one has an [[exact sequence]]

:<math>0 \to K \to K((X)) \overset{D}{\longrightarrow} K((X)) \;\overset{\operatorname{Res}}{\longrightarrow}\; K \to 0.</math>

'''Some rules of calculus'''. As a quite direct consequence of the above definition, and of the rules of formal derivation, one has, for any <math>f, g\in K((X))</math>
<ol style="list-style-type: lower-roman;"><li> <math>\operatorname{Res}(f')=0;</math></li>
<li> <math>\operatorname{Res}(fg')=-\operatorname{Res}(f'g);</math></li>
<li> <math>\operatorname{Res}(f'/f)=\operatorname{ord}(f),\qquad \forall f\neq 0;</math></li>
<li> <math>\operatorname{Res}\left(( g\circ f) f'\right) = \operatorname{ord}(f)\operatorname{Res}(g),</math> if <math>\operatorname{ord}(f)>0;</math></li>
<li> <math>[X^n]f(X)=\operatorname{Res}\left(X^{-n-1}f(X)\right).</math></li></ol>

Property (i) is part of the exact sequence above. Property (ii) follows from (i) as applied to <math>(fg)'=f'g+fg'</math>. Property (iii): any <math>f</math> can be written in the form <math>f=X^mg</math>, with <math>m=\operatorname{ord}(f)</math> and <math>\operatorname{ord}(g)=0</math>: then <math>f'/f = mX^{-1}+g'/g.</math> <math>\operatorname{ord}(g)=0</math> implies <math>g</math> is invertible in <math>K[[X]]\subset \operatorname{im}(D) = \ker(\operatorname{Res}),</math> whence <math>\operatorname{Res}(f'/f)=m.</math> Property (iv): Since <math>\operatorname{im}(D) = \ker(\operatorname{Res}),</math> we can write <math>g=g_{-1}X^{-1}+G',</math> with <math>G \in K((X))</math>. Consequently, <math>(g\circ f)f'= g_{-1}f^{-1}f'+(G'\circ f)f' = g_{-1}f'/f + (G \circ f)'</math> and (iv) follows from (i) and (iii). Property (v) is clear from the definition.

===The Lagrange inversion formula===
{{main|Lagrange inversion theorem}}
As mentioned above, any formal series <math>f \in K[[X]]</math> with ''f''<sub>0</sub> = 0 and ''f''<sub>1</sub> ≠ 0 has a composition inverse <math>g \in K[[X]].</math> The following relation between the coefficients of ''g<sup>n</sup>'' and ''f''<sup>−''k''</sup> holds ("{{Visible anchor|Lagrange inversion formula}}"):

:<math>k[X^k] g^n=n[X^{-n}]f^{-k}.</math>

In particular, for ''n''&nbsp;=&nbsp;1 and all ''k''&nbsp;≥&nbsp;1,

:<math>[X^k] g=\frac{1}{k} \operatorname{Res}\left( f^{-k}\right).</math>

Since the proof of the Lagrange inversion formula is a very short computation, it is worth reporting one proof here.{{efn|A number of different proofs exist, using techniques including Cauchy's coefficient formula for holomorphic functions, tree-counting arguments, or induction.<ref>{{cite book | last1=Stanley | first1=Richard | title=Enumerative combinatorics. Volume 1. | series =Cambridge Stud. Adv. Math. | volume=49 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-1-107-60262-5 | mr=2868112 }}</ref>{{pages?|date=March 2025}}<ref>{{Citation |last1=Gessel |first1=Ira|date=2016 |title=Lagrange inversion |journal=Journal of Combinatorial Theory, Series A |volume=144 |language=en |pages=212–249 |doi=10.1016/j.jcta.2016.06.018 |arxiv=1609.05988|mr=3534068}}</ref><ref>{{Citation |last1=Surya|first1=Erlang |last2=Warnke |first2=Lutz |date=2023 |title=Lagrange Inversion Formula by Induction |journal=The American Mathematical Monthly |volume=130 |issue=10 |language=en |pages=944–948 |doi=10.1080/00029890.2023.2251344 |arxiv=2305.17576|mr=4669236}}</ref>}} Noting <math>\operatorname{ord}(f) =1 </math>, we can apply the rules of calculus above, crucially Rule (iv) substituting <math>X \rightsquigarrow f(X)</math>, to get:

:<math>
\begin{align}
k[X^k] g^n & 
\ \stackrel{\mathrm{(v)}}=\ 
k\operatorname{Res}\left( g^n X^{-k-1}  \right)
\ \stackrel{\mathrm{(iv)}}=\ 
k\operatorname{Res}\left(X^n f^{-k-1}f'\right)
\ \stackrel{\mathrm{chain}}=\ 
-\operatorname{Res}\left(X^n (f^{-k})'\right) \\
& 
\ \stackrel{\mathrm{(ii)}}=\ 
\operatorname{Res}\left(\left(X^n\right)' f^{-k}\right)
\ \stackrel{\mathrm{chain}}=\ 
n\operatorname{Res}\left(X^{n-1}f^{-k}\right)
\ \stackrel{\mathrm{(v)}}=\ 
n[X^{-n}]f^{-k}.
\end{align}
</math>

'''Generalizations.''' One may observe that the above computation can be repeated plainly in more general settings than ''K''((''X'')): a generalization of the Lagrange inversion formula is already available working in the <math>\Complex((X))</math>-modules <math>X^{\alpha}\Complex((X)),</math> where α is a complex exponent. As a consequence, if ''f'' and ''g'' are as above, with <math>f_1=g_1=1</math>, we can relate the complex powers of ''f'' / ''X'' and ''g'' / ''X'': precisely, if α and β are non-zero complex numbers with negative integer sum, <math>m=-\alpha-\beta\in\N,</math> then

:<math>\frac{1}{\alpha}[X^m]\left( \frac{f}{X} \right)^\alpha=-\frac{1}{\beta}[X^m]\left( \frac{g}{X} \right)^\beta.</math>

For instance, this way one finds the power series for [[Lambert W function#Integer and complex powers|complex powers of the Lambert function]].

=== Power series in several variables ===
Formal power series in any number of indeterminates (even infinitely many) can be defined. If ''I'' is an index set and ''X<sub>I</sub>'' is the set of indeterminates ''X<sub>i</sub>'' for ''i''∈''I'', then a [[monomial]] ''X''<sup>''α''</sup> is any finite product of elements of ''X<sub>I</sub>'' (repetitions allowed); a formal power series in ''X<sub>I</sub>'' with coefficients in a ring ''R'' is determined by any mapping from the set of monomials ''X''<sup>''α''</sup> to a corresponding coefficient ''c''<sub>''α''</sub>, and is denoted <math display="inline">\sum_\alpha c_\alpha X^\alpha</math>. The set of all such formal power series is denoted <math>R[[X_I]],</math> and it is given a ring structure by defining

:<math>\left(\sum_\alpha c_\alpha X^\alpha\right)+\left(\sum_\alpha d_\alpha X^\alpha \right)= \sum_\alpha (c_\alpha+d_\alpha) X^\alpha</math>

and

:<math>\left(\sum_\alpha c_\alpha X^\alpha\right)\times\left(\sum_\beta d_\beta X^\beta\right)=\sum_{\alpha,\beta} c_\alpha d_\beta X^{\alpha+\beta}</math>

==== Topology ====
The topology on <math>R[[X_I]]</math> is such that a sequence of its elements converges only if for each monomial ''X''<sup>α</sup> the corresponding coefficient stabilizes. If ''I'' is finite, then this the ''J''-adic topology, where ''J'' is the ideal of <math>R[[X_I]]</math> generated by all the indeterminates in ''X<sub>I</sub>''. This does not hold if ''I'' is infinite. For example, if <math>I=\N,</math> then the sequence <math>(f_n)_{n\in \N}</math> with <math>f_n = X_n + X_{n+1} + X_{n+2} + \cdots </math> does not converge with respect to any ''J''-adic topology on ''R'', but clearly for each monomial the corresponding coefficient stabilizes.

As remarked above, the topology on a repeated formal power series ring like <math>R[[X]][[Y]]</math> is usually chosen in such a way that it becomes isomorphic as a [[topological ring]] to <math>R[[X,Y]].</math>

====Operations====
All of the operations defined for series in one variable may be extended to the several variables case.

* A series is invertible if and only if its constant term is invertible in ''R''.
* The composition ''f''(''g''(''X'')) of two series ''f'' and ''g'' is defined if ''f'' is a series in a single indeterminate, and the constant term of ''g'' is zero. For a series ''f'' in several indeterminates a form of "composition" can similarly be defined, with as many separate series in the place of ''g'' as there are indeterminates.

In the case of the formal derivative, there are now separate [[partial derivative]] operators, which differentiate with respect to each of the indeterminates. They all commute with each other.

==== Universal property ====
In the several variables case, the universal property characterizing <math>R[[X_1, \ldots, X_r]]</math> becomes the following. If ''S'' is a commutative associative algebra over ''R'', if ''I'' is an ideal of ''S'' such that the ''I''-adic topology on ''S'' is complete, and if ''x''<sub>1</sub>, ..., ''x<sub>r</sub>'' are elements of ''I'', then there is a ''unique'' map <math>\Phi: R[[X_1, \ldots, X_r]] \to S</math> with the following properties:

* Φ is an ''R''-algebra homomorphism
* Φ is continuous
* Φ(''X''<sub>''i''</sub>) = ''x''<sub>''i''</sub> for ''i'' = 1, ..., ''r''.

===Non-commuting variables===
The several variable case can be further generalised by taking ''non-commuting variables'' ''X<sub>i</sub>'' for ''i'' ∈ ''I'', where ''I'' is an index set and then a [[monomial]] ''X''<sup>α</sup> is any [[word (mathematics)|word]] in the ''X<sub>I</sub>''; a formal power series in ''X<sub>I</sub>'' with coefficients in a ring ''R'' is determined by any mapping from the set of monomials ''X''<sup>α</sup> to a corresponding coefficient ''c''<sub>α</sub>, and is denoted <math>\textstyle\sum_\alpha c_\alpha X^\alpha </math>. The set of all such formal power series is denoted ''R''«''X<sub>I</sub>''», and it is given a ring structure by defining addition pointwise

:<math>\left(\sum_\alpha c_\alpha X^\alpha\right)+\left(\sum_\alpha d_\alpha X^\alpha\right)=\sum_\alpha(c_\alpha+d_\alpha)X^\alpha</math>

and multiplication by

:<math>\left(\sum_\alpha c_\alpha X^\alpha\right)\times\left(\sum_\alpha d_\alpha X^\alpha\right)=\sum_{\alpha,\beta} c_\alpha d_\beta X^{\alpha} \cdot X^{\beta}</math>

where · denotes concatenation of words. These formal power series over ''R'' form the '''Magnus ring''' over ''R''.<ref>{{cite book | first=Helmut | last=Koch | title=Algebraic Number Theory | publisher=[[Springer-Verlag]] | year=1997 | isbn=978-3-540-63003-6 | zbl=0819.11044 | series=Encycl. Math. Sci. | volume=62 | edition=2nd printing of 1st | page=167 }}</ref><ref>{{cite book | title=The Mathematical Theory of Knots and Braids: An Introduction | volume=82 | series=North-Holland Mathematics Studies | first=Siegfried | last=Moran | publisher=Elsevier | year=1983 | isbn=978-0-444-86714-8 | page=211 | zbl=0528.57001 }}</ref>

=== On a semiring ===
{{expand section|sum, product, examples|date=August 2014}}

Given an [[Alphabet (formal languages)|alphabet]] <math>\Sigma</math> and a [[semiring]] <math>S</math>. The formal power series over <math>S</math> supported on the language <math>\Sigma^*</math> is denoted by <math>S\langle\langle \Sigma^*\rangle\rangle</math>. It consists of all mappings <math>r:\Sigma^*\to S</math>, where <math>\Sigma^*</math> is the [[free monoid]] generated by the non-empty set <math>\Sigma</math>.

The elements of <math>S\langle\langle \Sigma^*\rangle\rangle</math> can be written as formal sums

:<math>r = \sum_{w \in \Sigma^*} (r,w)w.</math>

where <math>(r,w)</math> denotes the value of <math>r</math> at the word <math>w\in\Sigma^*</math>. The elements <math>(r,w)\in S</math> are called the coefficients of <math>r</math>.

For <math>r\in S\langle\langle \Sigma^*\rangle\rangle</math> the support of <math>r</math> is the set

:<math>\operatorname{supp}(r)=\{w\in\Sigma^*|\ (r,w)\neq 0\}</math>

A series where every coefficient is either <math>0</math> or <math>1</math> is called the characteristic series of its support.

The subset of <math>S\langle\langle \Sigma^*\rangle\rangle</math> consisting of all series with a finite support is denoted by <math>S\langle \Sigma^*\rangle</math> and called polynomials.

For <math>r_1, r_2\in S\langle\langle \Sigma^*\rangle\rangle</math> and <math>s\in S</math>, the sum <math>r_1+r_2</math> is defined by 
:<math>(r_1+r_2,w)=(r_1,w)+(r_2,w)</math>
The (Cauchy) product <math>r_1\cdot r_2</math> is defined by
:<math>(r_1\cdot r_2,w) = \sum_{w_1w_2=w}(r_1,w_1)(r_2,w_2)</math>
The Hadamard product <math>r_1\odot r_2</math> is defined by
:<math>(r_1\odot r_2,w)=(r_1,w)(r_2,w)</math>
And the products by a scalar <math>sr_1</math> and <math>r_1s</math> by
:<math>(sr_1,w)=s(r_1,w)</math> and <math>(r_1s,w)=(r_1,w)s</math>, respectively.

With these operations <math>(S\langle\langle \Sigma^*\rangle\rangle,+,\cdot,0,\varepsilon)</math> and <math>(S\langle \Sigma^*\rangle, +,\cdot,0,\varepsilon)</math> are semirings, where <math>\varepsilon</math> is the empty word in <math>\Sigma^*</math>.

These formal power series are used to model the behavior of [[weighted automata]], in [[theoretical computer science]], when the coefficients <math>(r,w)</math> of the series are taken to be the weight of a path with label <math>w</math> in the automata.<!-- the correct symbols for the double angled braces are ⟪ and ⟫; but they work poorly in many browsers. Wikipedia's TeX doesn't support \llangle and \rrangle. Also no support for Greek italics in wiki TeX it seems --><ref>Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. ''Handbook of Weighted Automata'', 3–28. {{doi|10.1007/978-3-642-01492-5_1}}, p. 12</ref>

===Replacing the index set by an ordered abelian group===
{{Main|Hahn series}}

Suppose <math>G</math> is an ordered [[abelian group]], meaning an abelian group with a total ordering <math><</math> respecting the group's addition, so that <math>a<b</math> if and only if <math>a+c<b+c</math> for all <math>c</math>. Let '''I''' be a [[well-order]]ed subset of <math>G</math>, meaning '''I''' contains no infinite descending chain. Consider the set consisting of

:<math>\sum_{i \in I} a_i X^i </math>

for all such '''I''', with <math>a_i</math> in a commutative ring <math>R</math>, where we assume that for any index set, if all of the <math>a_i</math> are zero then the sum is zero. Then <math>R((G))</math> is the ring of formal power series on <math>G</math>; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same. Sometimes the notation <math>[[R^G]]</math> is used to denote <math>R((G))</math>.<ref>{{cite journal | first1=Khodr | last1=Shamseddine | first2=Martin | last2=Berz | url= http://www.physics.umanitoba.ca/~khodr/Publications/RS-Overview-offprints.pdf | title=Analysis on the Levi-Civita Field: A Brief Overview | journal= Contemporary Mathematics | volume=508 | pages=215–237 | date=2010| doi=10.1090/conm/508/10002 | isbn=9780821847404 }}</ref>

Various properties of <math>R</math> transfer to <math>R((G))</math>. If <math>R</math> is a field, then so is <math>R((G))</math>. If <math>R</math> is an [[ordered field]], we can order <math>R((G))</math> by setting any element to have the same sign as its leading coefficient, defined as the least element of the index set '''I''' associated to a non-zero coefficient. Finally if <math>G</math> is a [[divisible group]] and <math>R</math> is a [[real closed field]], then <math>R((G))</math> is a real closed field, and if <math>R</math> is [[algebraically closed]], then so is <math>R((G))</math>.

This theory is due to [[Hans Hahn (mathematician)|Hans Hahn]], who also showed that one obtains subfields when the number of (non-zero) terms is bounded by some fixed infinite cardinality.