Formal power series

One can characterize ${\displaystyle R[[X]]}$ abstractly as the completion of the polynomial ring ${\displaystyle R[X]}$ equipped with a particular metric. This automatically gives ${\displaystyle R[[X]]}$ the structure of a topological ring (and even of a complete metric space). But the general construction of a completion of a metric space is more involved than what is needed here, and would make formal power series seem more complicated than they are. It is possible to describe ${\displaystyle R[[X]]}$ more explicitly, and define the ring structure and topological structure separately, as follows.

As a set, ${\displaystyle R[[X]]}$ can be constructed as the set ${\displaystyle R^{\mathbb {N} }}$ of all infinite sequences of elements of ${\displaystyle R}$, indexed by the natural numbers (taken to include 0). Designating a sequence whose term at index ${\displaystyle n}$ is ${\displaystyle a_{n}}$ by ${\displaystyle (a_{n})}$, one defines addition of two such sequences by

${\displaystyle (a_{n})_{n\in \mathbb {N} }+(b_{n})_{n\in \mathbb {N} }=\left(a_{n}+b_{n}\right)_{n\in \mathbb {N} }}$

and multiplication by

${\displaystyle (a_{n})_{n\in \mathbb {N} }\times (b_{n})_{n\in \mathbb {N} }=\left(\sum _{k=0}^{n}a_{k}b_{n-k}\right)_{\!n\in \mathbb {N} }.}$

This type of product is called the Cauchy product of the two sequences of coefficients, and is a sort of discrete convolution. With these operations, ${\displaystyle R^{\mathbb {N} }}$ becomes a commutative ring with zero element ${\displaystyle (0,0,0,\ldots )}$ and multiplicative identity ${\displaystyle (1,0,0,\ldots )}$.

The product is in fact the same one used to define the product of polynomials in one indeterminate, which suggests using a similar notation. One embeds ${\displaystyle R}$ into ${\displaystyle R[[X]]}$ by sending any (constant) ${\displaystyle a\in R}$ to the sequence ${\displaystyle (a,0,0,\ldots )}$ and designates the sequence ${\displaystyle (0,1,0,0,\ldots )}$ by ${\displaystyle X}$; then using the above definitions every sequence with only finitely many nonzero terms can be expressed in terms of these special elements as

${\displaystyle (a_{0},a_{1},a_{2},\ldots ,a_{n},0,0,\ldots )=a_{0}+a_{1}X+\cdots +a_{n}X^{n}=\sum _{i=0}^{n}a_{i}X^{i};}$

these are precisely the polynomials in ${\displaystyle X}$. Given this, it is quite natural and convenient to designate a general sequence ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ by the formal expression ${\displaystyle \textstyle \sum _{i\in \mathbb {N} }a_{i}X^{i}}$, even though the latter is not an expression formed by the operations of addition and multiplication defined above (from which only finite sums can be constructed). This notational convention allows reformulation of the above definitions as

${\displaystyle \left(\sum _{i\in \mathbb {N} }a_{i}X^{i}\right)+\left(\sum _{i\in \mathbb {N} }b_{i}X^{i}\right)=\sum _{i\in \mathbb {N} }(a_{i}+b_{i})X^{i}}$

and

${\displaystyle \left(\sum _{i\in \mathbb {N} }a_{i}X^{i}\right)\times \left(\sum _{i\in \mathbb {N} }b_{i}X^{i}\right)=\sum _{n\in \mathbb {N} }\left(\sum _{k=0}^{n}a_{k}b_{n-k}\right)X^{n}.}$

which is quite convenient, but one must be aware of the distinction between formal summation (a mere convention) and actual addition.

Having stipulated conventionally that

${\displaystyle (a_{0},a_{1},a_{2},a_{3},\ldots )=\sum _{i=0}^{\infty }a_{i}X^{i},}$

(1)

one would like to interpret the right hand side as a well-defined infinite summation. To that end, a notion of convergence in ${\displaystyle R^{\mathbb {N} }}$ is defined and a topology on ${\displaystyle R^{\mathbb {N} }}$ is constructed. There are several equivalent ways to define the desired topology.

• We may give ${\displaystyle R^{\mathbb {N} }}$ the product topology, where each copy of ${\displaystyle R}$ is given the discrete topology.
• We may give ${\displaystyle R^{\mathbb {N} }}$ the I-adic topology, where ${\displaystyle I=(X)}$ is the ideal generated by ${\displaystyle X}$, which consists of all sequences whose first term ${\displaystyle a_{0}}$ is zero.
• The desired topology could also be derived from the following metric. The distance between distinct sequences ${\displaystyle (a_{n}),(b_{n})\in R^{\mathbb {N} },}$ is defined to be
  $${\displaystyle d((a_{n}),(b_{n}))=2^{-k},}$$

  
  where ${\displaystyle k}$ is the smallest natural number such that ${\displaystyle a_{k}\neq b_{k}}$; the distance between two equal sequences is of course zero.

Informally, two sequences ${\displaystyle (a_{n})}$ and ${\displaystyle (b_{n})}$ become closer and closer if and only if more and more of their terms agree exactly. Formally, the sequence of partial sums of some infinite summation converges if for every fixed power of ${\displaystyle X}$ the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient. This is clearly the case for the right hand side of (1), regardless of the values ${\displaystyle a_{n}}$, since inclusion of the term for ${\displaystyle i=n}$ gives the last (and in fact only) change to the coefficient of ${\displaystyle X^{n}}$. It is also obvious that the limit of the sequence of partial sums is equal to the left hand side.

This topological structure, together with the ring operations described above, form a topological ring. This is called the ring of formal power series over ${\displaystyle R}$ and is denoted by ${\displaystyle R[[X]]}$. The topology has the useful property that an infinite summation converges if and only if the sequence of its terms converges to 0, which just means that any fixed power of ${\displaystyle X}$ occurs in only finitely many terms.

The topological structure allows much more flexible usage of infinite summations. For instance the rule for multiplication can be restated simply as

${\displaystyle \left(\sum _{i\in \mathbb {N} }a_{i}X^{i}\right)\times \left(\sum _{i\in \mathbb {N} }b_{i}X^{i}\right)=\sum _{i,j\in \mathbb {N} }a_{i}b_{j}X^{i+j},}$

since only finitely many terms on the right affect any fixed ${\displaystyle X^{n}}$. Infinite products are also defined by the topological structure; it can be seen that an infinite product converges if and only if the sequence of its factors converges to 1 (in which case the product is nonzero) or infinitely many factors have no constant term (in which case the product is zero).

The above topology is the finest topology for which

${\displaystyle \sum _{i=0}^{\infty }a_{i}X^{i}}$

always converges as a summation to the formal power series designated by the same expression, and it often suffices to give a meaning to infinite sums and products, or other kinds of limits that one wishes to use to designate particular formal power series. It can however happen occasionally that one wishes to use a coarser topology, so that certain expressions become convergent that would otherwise diverge. This applies in particular when the base ring ${\displaystyle R}$ already comes with a topology other than the discrete one, for instance if it is also a ring of formal power series.

In the ring of formal power series ${\displaystyle \mathbb {Z} [[X]][[Y]]}$, the topology of above construction only relates to the indeterminate ${\displaystyle Y}$, since the topology that was put on ${\displaystyle \mathbb {Z} [[X]]}$ has been replaced by the discrete topology when defining the topology of the whole ring. So

${\displaystyle \sum _{i=0}^{\infty }XY^{i}}$

converges (and its sum can be written as ${\displaystyle {\tfrac {X}{1-Y}}}$); however

${\displaystyle \sum _{i=0}^{\infty }X^{i}Y}$

would be considered to be divergent, since every term affects the coefficient of ${\displaystyle Y}$. This asymmetry disappears if the power series ring in ${\displaystyle Y}$ is given the product topology where each copy of ${\displaystyle \mathbb {Z} [[X]]}$ is given its topology as a ring of formal power series rather than the discrete topology. With this topology, a sequence of elements of ${\displaystyle \mathbb {Z} [[X]][[Y]]}$ converges if the coefficient of each power of ${\displaystyle Y}$ converges to a formal power series in ${\displaystyle X}$, a weaker condition than stabilizing entirely. For instance, with this topology, in the second example given above, the coefficient of ${\displaystyle Y}$converges to ${\displaystyle {\tfrac {1}{1-X}}}$, so the whole summation converges to ${\displaystyle {\tfrac {Y}{1-X}}}$.

This way of defining the topology is in fact the standard one for repeated constructions of rings of formal power series, and gives the same topology as one would get by taking formal power series in all indeterminates at once. In the above example that would mean constructing ${\displaystyle \mathbb {Z} [[X,Y]]}$ and here a sequence converges if and only if the coefficient of every monomial ${\displaystyle X^{i}Y^{j}}$ stabilizes. This topology, which is also the ${\displaystyle I}$-adic topology, where ${\displaystyle I=(X,Y)}$ is the ideal generated by ${\displaystyle X}$ and ${\displaystyle Y}$, still enjoys the property that a summation converges if and only if its terms tend to 0.

The same principle could be used to make other divergent limits converge. For instance in ${\displaystyle \mathbb {R} [[X]]}$ the limit

${\displaystyle \lim _{n\to \infty }\left(1+{\frac {X}{n}}\right)^{\!n}}$

does not exist, so in particular it does not converge to

${\displaystyle \exp(X)=\sum _{n\in \mathbb {N} }{\frac {X^{n}}{n!}}.}$

This is because for ${\displaystyle i\geq 2}$ the coefficient ${\displaystyle {\tbinom {n}{i}}/n^{i}}$ of ${\displaystyle X^{i}}$ does not stabilize as ${\displaystyle n\to \infty }$. It does however converge in the usual topology of ${\displaystyle \mathbb {R} }$, and in fact to the coefficient ${\displaystyle {\tfrac {1}{i!}}}$ of ${\displaystyle \exp(X)}$. Therefore, if one would give ${\displaystyle \mathbb {R} [[X]]}$ the product topology of ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$ where the topology of ${\displaystyle \mathbb {R} }$ is the usual topology rather than the discrete one, then the above limit would converge to ${\displaystyle \exp(X)}$. This more permissive approach is not however the standard when considering formal power series, as it would lead to convergence considerations that are as subtle as they are in analysis, while the philosophy of formal power series is on the contrary to make convergence questions as trivial as they can possibly be. With this topology it would not be the case that a summation converges if and only if its terms tend to 0.

The ring ${\displaystyle R[[X]]}$ may be characterized by the following universal property. If ${\displaystyle S}$ is a commutative associative algebra over ${\displaystyle R}$, if ${\displaystyle I}$ is an ideal of ${\displaystyle S}$ such that the ${\displaystyle I}$-adic topology on ${\displaystyle S}$ is complete, and if ${\displaystyle x}$ is an element of ${\displaystyle I}$, then there is a unique ${\displaystyle \Phi :R[[X]]\to S}$ with the following properties:

• ${\displaystyle \Phi }$ is an ${\displaystyle R}$-algebra homomorphism

• ${\displaystyle \Phi }$ is continuous

• ${\displaystyle \Phi (X)=x}$.

For any natural number n we have

$${\displaystyle \left(\sum _{k=0}^{\infty }a_{k}X^{k}\right)^{\!n}=\,\sum _{m=0}^{\infty }c_{m}X^{m},}$$

where

$${\displaystyle {\begin{aligned}c_{0}&=a_{0}^{n},\\c_{m}&={\frac {1}{ma_{0}}}\sum _{k=1}^{m}(kn-m+k)a_{k}c_{m-k},\ \ \ m\geq 1.\end{aligned}}}$$

(This formula can only be used if m and a₀ are invertible in the ring of coefficients.)

In the case of formal power series with complex coefficients, the complex powers are well defined at least for series f with constant term equal to 1. In this case, ${\displaystyle f^{\alpha }}$ can be defined either by composition with the binomial series (1+x)^α, or by composition with the exponential and the logarithmic series, ${\displaystyle f^{\alpha }=\exp(\alpha \log(f)),}$ or as the solution of the differential equation ${\displaystyle f(f^{\alpha })'=\alpha f^{\alpha }f'}$ with constant term 1, the three definitions being equivalent. The rules of calculus ${\displaystyle (f^{\alpha })^{\beta }=f^{\alpha \beta }}$ and ${\displaystyle f^{\alpha }g^{\alpha }=(fg)^{\alpha }}$ easily follow.

The series

${\displaystyle A=\sum _{n=0}^{\infty }a_{n}X^{n}\in R[[X]]}$

is invertible in ${\displaystyle R[[X]]}$ if and only if its constant coefficient ${\displaystyle a_{0}}$ is invertible in ${\displaystyle R}$. This condition is necessary, for the following reason: if we suppose that ${\displaystyle A}$ has an inverse ${\displaystyle B=b_{0}+b_{1}x+\cdots }$ then the constant term ${\displaystyle a_{0}b_{0}}$ of ${\displaystyle A\cdot B}$ is the constant term of the identity series, i.e. it is 1. This condition is also sufficient; we may compute the coefficients of the inverse series ${\displaystyle B}$ via the explicit recursive formula

${\displaystyle {\begin{aligned}b_{0}&={\frac {1}{a_{0}}},\\b_{n}&=-{\frac {1}{a_{0}}}\sum _{i=1}^{n}a_{i}b_{n-i},\ \ \ n\geq 1.\end{aligned}}}$

An important special case is that the geometric series formula is valid in ${\displaystyle R[[X]]}$:

${\displaystyle (1-X)^{-1}=\sum _{n=0}^{\infty }X^{n}.}$

If ${\displaystyle R=K}$ is a field, then a series is invertible if and only if the constant term is non-zero, i.e. if and only if the series is not divisible by ${\displaystyle X}$. This means that ${\displaystyle K[[X]]}$ is a discrete valuation ring with uniformizing parameter ${\displaystyle X}$.

The computation of a quotient ${\displaystyle f/g=h}$

${\displaystyle {\frac {\sum _{n=0}^{\infty }b_{n}X^{n}}{\sum _{n=0}^{\infty }a_{n}X^{n}}}=\sum _{n=0}^{\infty }c_{n}X^{n},}$

assuming the denominator is invertible (that is, ${\displaystyle a_{0}}$ is invertible in the ring of scalars), can be performed as a product ${\displaystyle f}$ and the inverse of ${\displaystyle g}$, or directly equating the coefficients in ${\displaystyle f=gh}$:

${\displaystyle c_{n}={\frac {1}{a_{0}}}\left(b_{n}-\sum _{k=1}^{n}a_{k}c_{n-k}\right).}$

The coefficient extraction operator applied to a formal power series

${\displaystyle f(X)=\sum _{n=0}^{\infty }a_{n}X^{n}}$

in X is written

${\displaystyle \left[X^{m}\right]f(X)}$

and extracts the coefficient of X^m, so that

${\displaystyle \left[X^{m}\right]f(X)=\left[X^{m}\right]\sum _{n=0}^{\infty }a_{n}X^{n}=a_{m}.}$

Given formal power series

${\displaystyle f(X)=\sum _{n=1}^{\infty }a_{n}X^{n}=a_{1}X+a_{2}X^{2}+\cdots }$

${\displaystyle g(X)=\sum _{n=0}^{\infty }b_{n}X^{n}=b_{0}+b_{1}X+b_{2}X^{2}+\cdots ,}$

one may form the composition

${\displaystyle g(f(X))=\sum _{n=0}^{\infty }b_{n}(f(X))^{n}=\sum _{n=0}^{\infty }c_{n}X^{n},}$

where the coefficients c_n are determined by "expanding out" the powers of f(X):

${\displaystyle c_{n}:=\sum _{k\in \mathbb {N} ,|j|=n}b_{k}a_{j_{1}}a_{j_{2}}\cdots a_{j_{k}}.}$

Here the sum is extended over all (k, j) with ${\displaystyle k\in \mathbb {N} }$ and ${\displaystyle j\in \mathbb {N} _{+}^{k}}$ with ${\displaystyle |j|:=j_{1}+\cdots +j_{k}=n.}$

A more explicit description of these coefficients is provided by Faà di Bruno's formula, at least in the case where the coefficient ring is a field of characteristic 0.

Composition is only valid when ${\displaystyle f(X)}$ has no constant term, so that each ${\displaystyle c_{n}}$ depends on only a finite number of coefficients of ${\displaystyle f(X)}$ and ${\displaystyle g(X)}$. In other words, the series for ${\displaystyle g(f(X))}$ converges in the topology of ${\displaystyle R[[X]]}$.

Assume that the ring ${\displaystyle R}$ has characteristic 0 and the nonzero integers are invertible in ${\displaystyle R}$. If we denote by ${\displaystyle \exp(X)}$ the formal power series

${\displaystyle \exp(X)=1+X+{\frac {X^{2}}{2!}}+{\frac {X^{3}}{3!}}+{\frac {X^{4}}{4!}}+\cdots ,}$

then the expression

${\displaystyle \exp(\exp(X)-1)=1+X+X^{2}+{\frac {5X^{3}}{6}}+{\frac {5X^{4}}{8}}+\cdots }$

makes perfect sense as a formal power series. However, the statement

${\displaystyle \exp(\exp(X))\ {\stackrel {?}{=}}\ e\exp(\exp(X)-1)\ =\ e+eX+eX^{2}+{\frac {5eX^{3}}{6}}+\cdots }$

is not a valid application of the composition operation for formal power series. Rather, it is confusing the notions of convergence in ${\displaystyle R[[X]]}$ and convergence in ${\displaystyle R}$; indeed, the ring ${\displaystyle R}$ may not even contain any number ${\displaystyle e}$ with the appropriate properties.

Whenever a formal series

${\displaystyle f(X)=\sum _{k}f_{k}X^{k}\in R[[X]]}$

has f₀ = 0 and f₁ being an invertible element of R, there exists a series

${\displaystyle g(X)=\sum _{k}g_{k}X^{k}}$

that is the composition inverse of ${\displaystyle f}$, meaning that composing ${\displaystyle f}$ with ${\displaystyle g}$ gives the series representing the identity function ${\displaystyle x=0+1x+0x^{2}+0x^{3}+\cdots }$. The coefficients of ${\displaystyle g}$ may be found recursively by using the above formula for the coefficients of a composition, equating them with those of the composition identity X (that is 1 at degree 1 and 0 at every degree greater than 1). In the case when the coefficient ring is a field of characteristic 0, the Lagrange inversion formula (discussed below) provides a powerful tool to compute the coefficients of g, as well as the coefficients of the (multiplicative) powers of g.

Given a formal power series

${\displaystyle f=\sum _{n\geq 0}a_{n}X^{n}\in R[[X]],}$

we define its formal derivative, denoted Df or f ′, by

${\displaystyle Df=f'=\sum _{n\geq 1}a_{n}nX^{n-1}.}$

The symbol D is called the formal differentiation operator. This definition simply mimics term-by-term differentiation of a polynomial.

This operation is R-linear:

${\displaystyle D(af+bg)=a\cdot Df+b\cdot Dg}$

for any a, b in R and any f, g in ${\displaystyle R[[X]].}$ Additionally, the formal derivative has many of the properties of the usual derivative of calculus. For example, the product rule is valid:

${\displaystyle D(fg)\ =\ f\cdot (Dg)+(Df)\cdot g,}$

and the chain rule works as well:

${\displaystyle D(f\circ g)=(Df\circ g)\cdot Dg,}$

whenever the appropriate compositions of series are defined (see above under composition of series).

Thus, in these respects formal power series behave like Taylor series. Indeed, for the f defined above, we find that

${\displaystyle (D^{k}f)(0)=k!a_{k},}$

where D^k denotes the kth formal derivative (that is, the result of formally differentiating k times).

If ${\displaystyle R}$ is a ring with characteristic zero and the nonzero integers are invertible in ${\displaystyle R}$, then given a formal power series

${\displaystyle f=\sum _{n\geq 0}a_{n}X^{n}\in R[[X]],}$

we define its formal antiderivative or formal indefinite integral by

${\displaystyle D^{-1}f=\int f\ dX=C+\sum _{n\geq 0}a_{n}{\frac {X^{n+1}}{n+1}}.}$

for any constant ${\displaystyle C\in R}$.

This operation is R-linear:

${\displaystyle D^{-1}(af+bg)=a\cdot D^{-1}f+b\cdot D^{-1}g}$

for any a, b in R and any f, g in ${\displaystyle R[[X]].}$ Additionally, the formal antiderivative has many of the properties of the usual antiderivative of calculus. For example, the formal antiderivative is the right inverse of the formal derivative:

${\displaystyle D(D^{-1}(f))=f}$

for any ${\displaystyle f\in R[[X]]}$.

${\displaystyle R[[X]]}$ is an associative algebra over ${\displaystyle R}$ which contains the ring ${\displaystyle R[X]}$ of polynomials over ${\displaystyle R}$; the polynomials correspond to the sequences which end in zeros.

The Jacobson radical of ${\displaystyle R[[X]]}$ is the ideal generated by ${\displaystyle X}$ and the Jacobson radical of ${\displaystyle R}$; this is implied by the element invertibility criterion discussed above.

The maximal ideals of ${\displaystyle R[[X]]}$ all arise from those in ${\displaystyle R}$ in the following manner: an ideal ${\displaystyle M}$ of ${\displaystyle R[[X]]}$ is maximal if and only if ${\displaystyle M\cap R}$ is a maximal ideal of ${\displaystyle R}$ and ${\displaystyle M}$ is generated as an ideal by ${\displaystyle X}$ and ${\displaystyle M\cap R}$.

Several algebraic properties of ${\displaystyle R}$ are inherited by ${\displaystyle R[[X]]}$:

• if ${\displaystyle R}$ is a local ring, then so is ${\displaystyle R[[X]]}$ (with the set of non units the unique maximal ideal),
• if ${\displaystyle R}$ is Noetherian, then so is ${\displaystyle R[[X]]}$ (a version of the Hilbert basis theorem),
• if ${\displaystyle R}$ is an integral domain, then so is ${\displaystyle R[[X]]}$, and
• if ${\displaystyle K}$ is a field, then ${\displaystyle K[[X]]}$ is a discrete valuation ring.

The metric space ${\displaystyle (R[[X]],d)}$ is complete.

The ring ${\displaystyle R[[X]]}$ is compact if and only if R is finite. This follows from Tychonoff's theorem and the characterisation of the topology on ${\displaystyle R[[X]]}$ as a product topology.

The ring of formal power series with coefficients in a complete local ring satisfies the Weierstrass preparation theorem.

The formal Laurent series over a ring ${\displaystyle R}$ 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

${\displaystyle f=\sum _{n=N}^{\infty }a_{n}X^{n}}$

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

Multiplication of such series can be defined. Indeed, similarly to the definition for formal power series, the coefficient of ${\displaystyle X^{k}}$ of two series with respective sequences of coefficients ${\displaystyle \{a_{n}\}}$ and ${\displaystyle \{b_{n}\}}$ is

$${\displaystyle \sum _{i\in \mathbb {Z} }a_{i}b_{k-i}.}$$

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 ${\displaystyle R}$, denoted by ${\displaystyle R((X))}$.[lower-alpha 1] It is equal to the localization of the ring ${\displaystyle R[[X]]}$ of formal power series with respect to the set of positive powers of ${\displaystyle X}$. If ${\displaystyle R=K}$ is a field, then ${\displaystyle K((X))}$ is in fact a field, which may alternatively be obtained as the field of fractions of the integral domain ${\displaystyle K[[X]]}$.

As with ${\displaystyle R[[X]]}$, the ring ${\displaystyle R((X))}$ of formal Laurent series may be endowed with the structure of a topological ring by introducing the metric

$${\displaystyle d(f,g)=2^{-\operatorname {ord} (f-g)}.}$$

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 ${\displaystyle f}$ above is

$${\displaystyle f'=Df=\sum _{n\in \mathbb {Z} }na_{n}X^{n-1},}$$

which is again a formal Laurent series. If ${\displaystyle f}$ is a non-constant formal Laurent series and with coefficients in a field of characteristic 0, then one has

$${\displaystyle \operatorname {ord} (f')=\operatorname {ord} (f)-1.}$$

However, in general this is not the case since the factor ${\displaystyle n}$ for the lowest order term could be equal to 0 in ${\displaystyle R}$.

Assume that ${\displaystyle K}$ is a field of characteristic 0. Then the map

${\displaystyle D\colon K((X))\to K((X))}$

is a ${\displaystyle K}$-derivation that satisfies

${\displaystyle \ker D=K}$

${\displaystyle \operatorname {im} D=\left\{f\in K((X)):[X^{-1}]f=0\right\}.}$

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

${\displaystyle \operatorname {Res} :K((X))\to K}$

is ${\displaystyle K}$-linear, and by the above observation one has an exact sequence

${\displaystyle 0\to K\to K((X)){\overset {D}{\longrightarrow }}K((X))\;{\overset {\operatorname {Res} }{\longrightarrow }}\;K\to 0.}$

Some rules of calculus. As a quite direct consequence of the above definition, and of the rules of formal derivation, one has, for any ${\displaystyle f,g\in K((X))}$

1. ${\displaystyle \operatorname {Res} (f')=0;}$
2. ${\displaystyle \operatorname {Res} (fg')=-\operatorname {Res} (f'g);}$
3. ${\displaystyle \operatorname {Res} (f'/f)=\operatorname {ord} (f),\qquad \forall f\neq 0;}$
4. ${\displaystyle \operatorname {Res} \left((g\circ f)f'\right)=\operatorname {ord} (f)\operatorname {Res} (g),}$ if ${\displaystyle \operatorname {ord} (f)>0;}$
5. ${\displaystyle [X^{n}]f(X)=\operatorname {Res} \left(X^{-n-1}f(X)\right).}$

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

As mentioned above, any formal series ${\displaystyle f\in K[[X]]}$ with f₀ = 0 and f₁ ≠ 0 has a composition inverse ${\displaystyle g\in K[[X]].}$ The following relation between the coefficients of gⁿ and f^−k holds ("Lagrange inversion formula"):

${\displaystyle k[X^{k}]g^{n}=n[X^{-n}]f^{-k}.}$

In particular, for n = 1 and all k ≥ 1,

${\displaystyle [X^{k}]g={\frac {1}{k}}\operatorname {Res} \left(f^{-k}\right).}$

Since the proof of the Lagrange inversion formula is a very short computation, it is worth reporting it here. Noting ${\displaystyle \operatorname {ord} (f)=1}$, we can apply the rules of calculus above, crucially Rule (iv) substituting ${\displaystyle X\rightsquigarrow f(X)}$, to get:

${\displaystyle {\begin{aligned}k[X^{k}]g^{n}&\ {\stackrel {\mathrm {(v)} }{=}}\ k\operatorname {Res} \left(g^{n}X^{-k-1}\right)\ {\stackrel {\mathrm {(iv)} }{=}}\ k\operatorname {Res} \left(X^{n}f^{-k-1}f'\right)\ {\stackrel {\mathrm {chain} }{=}}\ -\operatorname {Res} \left(X^{n}(f^{-k})'\right)\\&\ {\stackrel {\mathrm {(ii)} }{=}}\ \operatorname {Res} \left(\left(X^{n}\right)'f^{-k}\right)\ {\stackrel {\mathrm {chain} }{=}}\ n\operatorname {Res} \left(X^{n-1}f^{-k}\right)\ {\stackrel {\mathrm {(v)} }{=}}\ n[X^{-n}]f^{-k}.\end{aligned}}}$

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 ${\displaystyle \mathbb {C} ((X))}$-modules ${\displaystyle X^{\alpha }\mathbb {C} ((X)),}$ where α is a complex exponent. As a consequence, if f and g are as above, with ${\displaystyle f_{1}=g_{1}=1}$, 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, ${\displaystyle m=-\alpha -\beta \in \mathbb {N} ,}$ then

${\displaystyle {\frac {1}{\alpha }}[X^{m}]\left({\frac {f}{X}}\right)^{\alpha }=-{\frac {1}{\beta }}[X^{m}]\left({\frac {g}{X}}\right)^{\beta }.}$

For instance, this way one finds the power series for complex powers of the Lambert function.

Formal power series in any number of indeterminates (even infinitely many) can be defined. If I is an index set and X_I is the set of indeterminates X_i for i∈I, then a monomial X^α is any finite product of elements of X_I (repetitions allowed); a formal power series in X_I with coefficients in a ring R is determined by any mapping from the set of monomials X^α to a corresponding coefficient c_α, and is denoted ${\textstyle \sum _{\alpha }c_{\alpha }X^{\alpha }}$. The set of all such formal power series is denoted ${\displaystyle R[[X_{I}]],}$ and it is given a ring structure by defining

${\displaystyle \left(\sum _{\alpha }c_{\alpha }X^{\alpha }\right)+\left(\sum _{\alpha }d_{\alpha }X^{\alpha }\right)=\sum _{\alpha }(c_{\alpha }+d_{\alpha })X^{\alpha }}$