Formal power series

The formal power series in mathematics are a generalization of the polynomials of the polynomial rings . As with the latter, the ring-theoretical properties are in the foreground, while the power series of analysis focuses on the analytical, the ( limit value ) properties .

It is common that the coefficient of a ring to be taken, which can be very arbitrary here, it whereas in the analysis only a complete ring is usually the body of the real or the complex numbers . Another difference is that the “variable” is an indeterminate one , which is often noted with capital letters (or ) and to which a “value” is not assigned in the formal power series . The power series of analysis that are analytical in the zero point can also be understood as formal power series, since like these they can be differentiated as often as desired and are subject to coefficient comparison . ${\displaystyle R}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle X}$ ${\displaystyle T}$

Because of the many common properties and conceptualizations, the formal Laurent series are also dealt with in this article. The definitions and properties are slightly more complex in the formal Laurent series, but very often contain the formal power series as a special case.

There is support for arithmetic with formal power and Laurent series in many computer algebra systems.

Definitions

Formal power series

For a commutative ring with one element (the starting ring ) denotes the ring of the formal power series over in the indeterminate . It is isomorphic to the ring of infinite sequences ${\displaystyle R}$ ${\displaystyle R[[X]]}$ ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle R^{\mathbb {N} _{0}}}$

${\displaystyle (a_{0},a_{1},\dotsc )}$

with so that ${\displaystyle a_{n}\in R}$

${\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\dotsb =\sum _{n=0}^{\infty }a_{n}X^{n}}$

is the corresponding formal power series and the sequence corresponds to the indefinite . ${\displaystyle (0,1,0,0,\dotsc )}$ ${\displaystyle X}$

The ring in is through the figure ${\displaystyle R}$ ${\displaystyle R[[X]]}$

${\displaystyle R\ni a\mapsto (a,0,0,\dotsc )}$

embedded.

The terms of the sequence are called coefficients . Compare also polynomial ring . ${\displaystyle a_{n}}$

Formal Laurent range

The quotient ring of is the localization of after the ideal . It is called the Ring of Formal Laurent Series . He is a body when there is a body. ${\displaystyle R(\!(X)\!)}$ ${\displaystyle R[[X]]}$ ${\displaystyle R[[X]]}$ ${\displaystyle (X)}$ ${\displaystyle R}$

A formal Laurent series can have a finite number of terms with a negative index, so it has the form ${\displaystyle A(X)\in R(\!(X)\!)}$

${\displaystyle A(X)=\sum _{n=m}^{\infty }a_{n}X^{n}}$with . ${\displaystyle m\in \mathbb {Z} ,a_{n}\in R}$

These series can be embedded in the set of infinite sequences and also called ${\displaystyle R^{\mathbb {Z} }}$

${\displaystyle {\begin{array}{ll}A(X)&=\quad \sum _{n\in \mathbb {Z} }a_{n}X^{n}\\&=\quad \left(a_{n}\right)_{n\in \mathbb {Z} }\\&=\quad (\dotsc ,a_{-1},a_{0},a_{1},\dotsc )\end{array}}}$

are written under the rule that almost all coefficients with a negative index vanish. The following corresponds to the indefinite : ${\displaystyle X}$

${\displaystyle X=(\dotsc ,\,0,\,0,}$	${\displaystyle 0,}$	${\displaystyle 1,\,0,\,0,\dotsc )\qquad \in R^{\mathbb {Z} }}$
	${\displaystyle \uparrow }$	${\displaystyle \uparrow }$
index	0	1

order

The function

${\displaystyle \operatorname {ord} _{X}\colon }$	${\displaystyle R(\!(X)\!)\to }$	${\displaystyle \mathbb {Z} \cup \{+\infty \}}$
	${\displaystyle A(X)=\sum _{n\in \mathbb {Z} }a_{n}X^{n}\;\mapsto {\begin{cases}\\\\\end{cases}}}$	${\displaystyle +\infty }$ ,	if   (the zero row ) ${\displaystyle A(X)=0}$
		${\displaystyle \min \left\{n\in \mathbb {Z} \mid a_{n}\neq 0\right\}}$,	if   ${\displaystyle A(X)\neq 0}$

assigns a formal Laurent series in the indefinite its order in the indefinite . The minimum exists for because there are only finitely many indices with . ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \min \left\{n\in \mathbb {Z} \mid a_{n}\neq 0\right\}}$ ${\displaystyle A\neq 0}$ ${\displaystyle n<0}$ ${\displaystyle a_{n}\neq 0}$

The following apply to the usual requirements for comparison and addition: ${\displaystyle \pm \infty }$

And applies to everyone . ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle -\infty <n<+\infty }$ ${\displaystyle +\infty \pm n=+\infty }$

This means that the formal Laurent rows can be used as rows

${\displaystyle {\begin{array}{lll}R(\!(X)\!)&=\quad {\bigl \{}A(X)=\sum _{n\in \mathbb {Z} }a_{n}X^{n}&{\big |}\;\operatorname {ord} _{X}(A)>-\!\infty {\bigr \}}\\&=\quad {\bigl \{}\left(a_{n}\right)_{n\in \mathbb {Z} }\in R^{\mathbb {Z} }&{\big |}\;\exists \,u\in \mathbb {Z} :a_{n}\neq 0\implies n\geq u{\bigr \}}\end{array}}}$

with order bounded downwards and the formal power series

${\displaystyle {\begin{array}{lll}R[[X]]&=\quad {\bigl \{}A(X)\in R(\!(X)\!)&{\big |}\;\operatorname {ord} _{X}(A)\geq 0{\bigr \}}\\&=\quad {\bigl \{}(\dotsc ,a_{-1},a_{0},a_{1},\dotsc )\in R^{\mathbb {Z} }&{\big |}\;a_{n}\neq 0\implies n\geq 0{\bigr \}}\end{array}}}$

characterize as such with non-negative order.

For the sake of simplicity, we generally assume that a coefficient of a formal power or Laurent series , if it is accessed with an index , returns the value 0. ${\displaystyle a_{n}}$ ${\displaystyle A(X)}$ ${\displaystyle n<\operatorname {ord} _{X}(A)}$

Addition and multiplication

Be with

${\displaystyle B(X)=\sum _{n\in \mathbb {Z} }b_{n}X^{n}}$

given a second formal power or Laurent series, then add them

${\displaystyle A(X)+B(X)=\sum _{n\in \mathbb {Z} }(a_{n}+b_{n})X^{n}}$

component by component . The sum of two formal power series results in a formal power series.

The multiplication

${\displaystyle A(X)\;B(X)=\sum _{n=\operatorname {ord} _{X}(A)+\operatorname {ord} _{X}(B)}^{\infty }{\Bigl (}\sum _{i=\operatorname {ord} _{X}(A)}^{n}a_{i}\,b_{n-i}{\Bigr )}X^{n}}$

is a fold . Again the product of two formal power series results in a formal power series.

properties

• The laws of commutative rings apply to the ring operations addition and multiplication .
• The formal power or Laurent series in which all coefficients are 0 is called the zero series . It is the neutral element 0 of addition in both rings, and . ${\displaystyle R[[X]]}$ ${\displaystyle R(\!(X)\!)}$
• A scalar is multiplied as in the usual scalar multiplication . So 1 is the one series . ${\displaystyle a\in R}$
• Coefficient comparison: Two formal power or Laurent series and are exactly the same if they are in all coefficients ${\displaystyle \textstyle A(X)=\sum _{n\in \mathbb {Z} }a_{n}X^{n}}$ ${\displaystyle \textstyle B(X)=\sum _{n\in \mathbb {Z} }b_{n}X^{n}}$

${\displaystyle \forall n\in \mathbb {Z} :a_{n}=b_{n}}$

to match.

• The units of are precisely those formal power series whose absolute term (constant term) is a unit in (see also the § Multiplicative Inverse ). ${\displaystyle R[[X]]}$ ${\displaystyle a_{0}}$ ${\displaystyle R}$
• If a Noetherian ring , an integrity ring or a local ring , this also applies to . ${\displaystyle R}$ ${\displaystyle R[[X]]}$
• The polynomial ring can be embedded in homomorphic (and injective ) as a ring of sequences with only finitely many non-vanishing coefficients. If there is a body, then the rational function body can be embedded in homomorphic (and injective). The embeddings apply ${\displaystyle R[X]}$ ${\displaystyle R[[X]]}$
  ${\displaystyle K}$ ${\displaystyle K(X)}$ ${\displaystyle K(\!(X)\!)}$
  ${\displaystyle \rightarrow }$

${\displaystyle {\begin{array}{ccc}K[X]&\rightarrow &K[[X]]\\\downarrow &&\downarrow \\K(X)&\rightarrow &K(\!(X)\!)\end{array}}}$

with the quotient fields in the bottom line.

• If there is a body , there is a complete discrete evaluation ring with the uniformizing element . It is the completion of the polynomial ring with respect to the ideal . Its remainder class field is , its quotient field, the body of the formal Laurent series . ${\displaystyle K}$ ${\displaystyle K[[X]]}$ ${\displaystyle X}$ ${\displaystyle K[X]}$ ${\displaystyle (X)}$ ${\displaystyle K}$ ${\displaystyle K(\!(X)\!)}$
• Conversely, according to the structural theorems of Irving S. Cohen, every complete discrete evaluation ring with the same characteristic is isomorphic to the ring of the formal power series over its remainder class field.

Operations and other properties

Coefficient extraction

The operator for extracting the coefficient of degree from the power or Laurent series in is written as ${\displaystyle m\in \mathbb {Z} }$ ${\displaystyle \textstyle A(X)=\sum _{n\in \mathbb {Z} }a_{n}X^{n}}$ ${\displaystyle X}$

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

It is a projection of the formal row to the right onto the -th component in . So is ${\displaystyle m}$ ${\displaystyle R^{\mathbb {Z} }}$

${\displaystyle \left[X^{m}\right]A(X)=\left[X^{m}\right]\sum _{n\in \mathbb {Z} }a_{n}X^{n}=a_{m}}$

and

${\displaystyle A(X)=\sum _{n\in \mathbb {Z} }X^{n}\left[Y^{n}\right]A(Y)}$ .

For formal power series is for by definition . ${\displaystyle A(X)}$ ${\displaystyle m<0}$ ${\displaystyle \left[X^{m}\right]A(X)=0}$

Guide coefficient

The order has a certain analogy to the degree function in polynomial rings. This is the name of the coefficient ${\displaystyle \operatorname {ord} }$

${\displaystyle l(A):={\begin{cases}\\\\\end{cases}}}$	${\displaystyle \textstyle a_{\operatorname {ord} (A)}=\left[X^{\operatorname {ord} (A)}\right]A(X)}$ ,	if   ${\displaystyle A(X)\neq 0}$
	${\displaystyle 0}$ ,	if   ${\displaystyle A(X)=0}$

also leading coefficient .

It applies to everyone ${\displaystyle A,B\in R(\!(X)\!)}$

• ${\displaystyle \operatorname {ord} (A\cdot B)\geq \operatorname {ord} (A)+\operatorname {ord} (B)}$

(Does not contain zero divisors - more precisely: if the leading coefficients are not zero divisors - equality applies.) ${\displaystyle R}$

• ${\displaystyle \operatorname {ord} (A+B)\geq \min\{\operatorname {ord} (A),\operatorname {ord} (B)\}}$.

The function

${\displaystyle |A|:=2^{-\!\operatorname {ord} (A)}}$

meets all requirements of a non-Archimedean pseudo-amount .

Is a body, then a (discrete) rating (a logarithmic written non-Archimedean amount, Eng. Valuation ) to the ring as the (mentioned above) corresponding valuation ring . One recognizes the -adic topology, where the ideal generated by is the multiple of . It is the associated maximal ideal and the remainder class field. ${\displaystyle K}$ ${\displaystyle \operatorname {ord} }$ ${\displaystyle K[[X]]}$ ${\displaystyle I}$ ${\displaystyle I:=(X)}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle K}$

Exponentiation

For is ${\displaystyle n\in \mathbb {N} _{0}}$

${\displaystyle {\bigl (}A(X){\bigr )}^{n}=\left(\sum _{k=0}^{\infty }a_{k}X^{k}\right)^{n}=:\sum _{m=0}^{\infty }c_{m}X^{m}}$

With

${\displaystyle c_{0}=a_{0}^{n}}$

and recursive

${\displaystyle c_{m}\,m\,a_{0}=\sum _{k=1}^{m}(kn-m+k)\,a_{k}\,c_{m-k}}$     for , ${\displaystyle m\in \mathbb {N} }$

so for example

${\displaystyle c_{1}={\binom {n}{1}}a_{0}^{n-1}a_{1}}$,

${\displaystyle c_{2}={\binom {n}{1}}a_{0}^{n-1}a_{2}+{\binom {n}{2}}a_{0}^{n-2}a_{1}^{2}}$,

${\displaystyle c_{3}={\binom {n}{1}}a_{0}^{n-1}a_{3}+{\binom {n}{n\!-\!2,1,1}}a_{0}^{n-2}a_{1}a_{2}+{\binom {n}{3}}a_{0}^{n-3}a_{1}^{3}}$, ....

They are polynomials in the with integer (multinomial) coefficients , even if the recursion formula can only be easily solved if and are invertible in the ring . (For the case see also § Composition .) ${\displaystyle c_{m}}$ ${\displaystyle a_{k}}$ ${\displaystyle c_{m}}$ ${\displaystyle m}$ ${\displaystyle a_{0}}$ ${\displaystyle R}$ ${\displaystyle a_{0}=0}$

Multiplicative inverse

The formal power series has a multiplicative inverse if and only if the absolute term ${\displaystyle \textstyle A(X)=\sum _{n=0}^{\infty }a_{n}X^{n}\in R[[X]]}$ ${\displaystyle \textstyle B(X):=\sum _{n=0}^{\infty }b_{n}X^{n}\in R[[X]]}$

${\displaystyle a_{0}=\left[X^{0}\right]A(X)}$

is invertible in the ring . Then is too ${\displaystyle R}$

${\displaystyle \operatorname {ord} _{X}(A)=0}$

and recursive

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

If a field, then a formal power series is invertible if and only if the absolute term is not 0, that is, if it is not divisible by . ${\displaystyle K}$ ${\displaystyle K[[X]]}$ ${\displaystyle X}$

If the formal power series is the absolute term or if it is a formal Laurent series, then with invertible leading coefficients the series can be converted into via the intermediate step ${\displaystyle A(X)}$ ${\displaystyle a_{0}=0}$ ${\displaystyle l(A)=a_{\operatorname {ord} _{X}(A)}}$ ${\displaystyle A(X)}$ ${\displaystyle R(\!(X)\!)}$

${\displaystyle B(X):=X^{-\!\operatorname {ord} _{X}(A)}A(X)\in K[[X]]}$

invert multiplicative - with the result:

${\displaystyle A(X)^{-1}=X^{-\!\operatorname {ord} _{X}(A)}B(X)^{-1}\in K(\!(X)\!)}$

Is a field, then is the quotient field of . ${\displaystyle K}$ ${\displaystyle K(\!(X)\!)}$ ${\displaystyle K[[X]]}$

division

If the divisor is invertible in , then the quotient has ${\displaystyle A(X)}$ ${\displaystyle R[[X]]}$

${\displaystyle C(X)=\sum _{n=0}^{\infty }c_{n}X^{n}:=Z(X)/A(X)={\Bigl (}\sum _{n=0}^{\infty }z_{n}X^{n}{\Bigr )}\;/\;{\Bigl (}\sum _{n=0}^{\infty }a_{n}X^{n}{\Bigr )}}$

two power series and according to the calculation scheme ${\displaystyle Z(X)}$ ${\displaystyle A(X)}$

										quotient
dividend					divisor
${\displaystyle (z_{0}}$	${\displaystyle +z_{1}X}$	${\displaystyle +z_{2}X^{2}}$	${\displaystyle +\dotsb )}$	${\displaystyle /}$	${\displaystyle (a_{0}}$	${\displaystyle +a_{1}X}$	${\displaystyle +a_{2}X^{2}}$	${\displaystyle +\dotsb )}$	${\displaystyle =}$
${\displaystyle {-a_{0}{\tfrac {z_{0}}{a_{0}}}}}$	${\displaystyle {-a_{1}c_{0}X}}$	${\displaystyle {-a_{2}c_{0}X^{2}}}$	${\displaystyle -\dotsb }$	${\displaystyle {\tfrac {z_{0}}{a_{0}}}\;\,=}$						${\displaystyle c_{0}}$
	${\displaystyle (z_{1}\!-\!a_{1}c_{0})X}$	${\displaystyle +(z_{2}\!-\!a_{2}c_{0})X^{2}}$	${\displaystyle +\dotsb }$
	${\displaystyle {-a_{0}{\tfrac {z_{1}-a_{1}c_{0}}{a_{0}}}X}}$	${\displaystyle {-a_{1}c_{1}X^{2}}}$	${\displaystyle -\dotsb }$	${\displaystyle +{\tfrac {z_{1}-a_{1}c_{0}}{a_{0}}}X=}$						${\displaystyle +c_{1}X}$
		${\displaystyle (z_{2}\!-\!a_{2}c_{0}\!-\!a_{1}c_{1})X^{2}}$	${\displaystyle +\dotsb }$
		${\displaystyle {\;\;-a_{0}{\tfrac {z_{2}-a_{2}c_{0}-a_{1}c_{1}}{a_{0}}}X^{2}}}$	${\displaystyle -\dotsb }$	${\displaystyle +{\tfrac {z_{2}-a_{2}c_{0}-a_{1}c_{1}}{a_{0}}}X^{2}=}$						${\displaystyle +c_{2}X^{2}}$
			${\displaystyle +\dotsb }$
			${\displaystyle -\dotsb }$							${\displaystyle +\dotsb }$

of the polynomial division mirrored in the monomial order recursively the coefficients

${\displaystyle {\begin{aligned}c_{n}=&a_{0}^{-1}\left(z_{n}-\sum _{i=1}^{n}a_{i}c_{n-i}\right)\qquad \qquad (n\geq 0).\\\end{aligned}}}$

The intermediate step in § Multiplicative Inverse indicates how the shown calculation scheme can be expanded into a division algorithm . ${\displaystyle R(\!(X)\!)}$

Inverse of polynomials

For bodies , the body of the rational functions (polynomial quotients) of the form ${\displaystyle K}$ ${\displaystyle K(X)}$

${\displaystyle {\frac {Z(X)}{A(X)}}={\frac {z_{0}+z_{1}X+\dotsb +z_{e}X^{e}}{a_{0}+a_{1}X+\dotsb +a_{d}X^{d}}}}$

embed in the ring in a similar manner as in . An important example is ${\displaystyle K(\!(X)\!)}$ ${\displaystyle K[X]}$ ${\displaystyle K[[X]]}$

${\displaystyle K(X)\ni \quad {\frac {1}{1-X}}=\sum _{n=0}^{\infty }X^{n}\quad \in K(\!(X)\!)}$.

More generally:
Is

${\displaystyle A(X)=\sum _{n=0}^{d}a_{n}X^{n}}$

a polynomial different from 0, then the (leading) coefficient can be inverted in and with ${\displaystyle k:=\operatorname {ord} _{X}(A)\in \mathbb {N} _{0}}$ ${\displaystyle a_{k}\neq 0}$ ${\displaystyle K}$

${\displaystyle C(X):=X^{-k}\,A(X)=:\sum _{n=0}^{d-k}a_{n+k}X^{n+k}\in K[[X]]}$

${\displaystyle \operatorname {ord} _{X}(C)=0}$. Is therefore multiplicative inverted in the multiplicative inverse . The multiplicative inverse of is then ${\displaystyle C(X)}$ ${\displaystyle K[[X]]}$ ${\displaystyle D(X):=C(X)^{-1}\in K[[X]]}$ ${\displaystyle A(X)}$

${\displaystyle {\begin{array}{rll}B(X)=&\sum _{n=-k}^{\infty }b_{n}X^{n}\\:=&X^{-k}\,D(X)&\qquad {\bigl (}\in K(\!(X)\!)\,{\bigr )}\\=&X^{-k}\,C(X)^{-1}\\=&X^{-k}\,{\bigl (}X^{-k}\,A(X){\bigr )}^{-1}\\=&A(X)^{-1}\end{array}}}$

with the coefficients

${\displaystyle {\begin{array}{lll}b_{-k}&=a_{k}^{-1}\\b_{n}&=-a_{k}^{-1}\sum _{i=k+1}^{\min(d,2k+n)}a_{i}b_{k+n-i}&\quad (n\geq -k+1).\end{array}}}$

example

Is , then is and for . They are therefore the Fibonacci sequence (shifted by 1 position) and its generating function . Thus, a Polynomquotient is on its coefficient sequence not as easy as rational to recognize as a rational number in its periodic g -adischen development . ${\displaystyle A(X)=1-X-X^{2}}$ ${\displaystyle b_{0}=b_{1}=1}$ ${\displaystyle b_{n}=b_{n-1}+b_{n-2}}$ ${\displaystyle n\geq 2}$ ${\displaystyle b_{n}}$ ${\displaystyle X/A(X)=X\,B(X)}$
${\displaystyle B=1/A}$ ${\displaystyle \left(b_{n}\right)}$

${\displaystyle K(\!(X)\!)}$is the completion of the body with respect to the metric described in § Convergence . ${\displaystyle K(X)}$

convergence

A formal power series

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

is under the metric

${\displaystyle \operatorname {d} (A,B):=|A-B|=2^{-\!\operatorname {ord} (A-B)}}$.

Limit of the sequence of polynomials with ${\displaystyle {\bigl (}A_{k}(X){\bigr )}_{k\in \mathbb {N} }}$

${\displaystyle A_{k}(X):=\sum _{n=0}^{k}a_{n}X^{n}}$ .

The relevant convergence criterion is a Cauchy criterion for sequences , and is the completion of the polynomial ring with respect to this metric. ${\displaystyle R[[X]]}$ ${\displaystyle R[X]}$

This metric creates the Krull topology in the rings and . ${\displaystyle R[[X]]}$ ${\displaystyle R(\!(X)\!)}$

Two series of formal Laurent series and have exactly then the same limit, if for every one there, so for all ${\displaystyle {\bigl (}A_{k}(X){\bigr )}_{k\in \mathbb {N} }\in R(\!(X)\!)}$ ${\displaystyle {\bigl (}B_{l}(X){\bigr )}_{l\in \mathbb {N} }\in R(\!(X)\!)}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle N\in \mathbb {Z} }$ ${\displaystyle k,l>N}$

${\displaystyle |A_{k}(X)-B_{l}(X)|<\varepsilon }$

is, which means nothing else than that, for sufficiently large indices, the differences in terms of the two sequences are divisible by arbitrarily high powers of - in short: that the two limit values ​​have the same coefficients . ${\displaystyle X}$

For the convergence of power series and Laurent series for “inserted values” of (interpreted as a variable) in real / complex metrics see Laurent series # Convergence of Laurent series . ${\displaystyle X}$

Concatenation (composition)

A formal power series without an absolute term can be converted into a formal power or Laurent series with the result ${\displaystyle \textstyle P(X):=\sum _{i=1}^{\infty }p_{i}X^{i}=p_{1}X+p_{2}X^{2}+\cdots }$ ${\displaystyle \textstyle A(X):=\sum _{n\in \mathbb {Z} }a_{j}X^{j}}$

${\displaystyle {\begin{array}{lll}C(X)&:=&(A\circ P)(X):=A(P(X)\!)\\&=&\sum _{j\in \mathbb {Z} }a_{j}{\bigl (}P(X){\bigr )}^{j}\\&=&\sum _{j\in \mathbb {Z} }a_{j}\left(\sum _{i=1}^{\infty }p_{i}X^{i}\right)^{j}\\&=:&\sum _{n\in \mathbb {Z} }c_{n}X^{n}\end{array}}}$

insert ( concatenate with it ).
For the usability of the power series it is important that it has no constant term (no absolute term), that is, that is. Because then only depends on a finite number of coefficients. ${\displaystyle P(X)}$ ${\displaystyle \operatorname {ord} _{X}(P)\geq 1}$ ${\displaystyle c_{n}}$

If there is a power series, then there is also a power series, and the formula applies to the coefficients ${\displaystyle A(X)}$ ${\displaystyle \operatorname {ord} _{X}(A)\geq 0}$ ${\displaystyle C(X)}$ ${\displaystyle c_{n}}$

${\displaystyle c_{n}=[X^{n}]\,\sum _{j=0}^{\infty }a_{j}\left(\sum _{i=1}^{\infty }p_{i}X^{i}\right)^{j}=\sum _{j\in \mathbb {N} _{0},\,\vert {\boldsymbol {i}}\vert =n}a_{j}p_{i_{1}}p_{i_{2}}\cdots p_{i_{n}}}$

with and (see multi-index # conventions of multi-index notation ). ${\displaystyle {\boldsymbol {i}}:=(i_{1},\ldots ,i_{n})}$ ${\displaystyle \vert {\boldsymbol {i}}\vert :=i_{1}+\cdots +i_{n}}$

Otherwise, if there is with , then powers with negative exponents can be formed over the multiplicative inverse . ${\displaystyle n<0}$ ${\displaystyle a_{n}\neq 0}$ ${\displaystyle P(X)^{n}}$ ${\displaystyle P(X)^{-1}}$

They are polynomials in those with integer coefficients. A more explicit representation can be found in ${\displaystyle c_{n}}$ ${\displaystyle p_{i},\dotsc ,a_{j},\dotsc }$

→ Main article : Faà di Bruno's formula

Formal differentiation

The formal derivation of the formal power or Laurent series is denoted by or (as in analysis) with : ${\displaystyle \textstyle A(X)=\sum _{n\in \mathbb {Z} }a_{n}X^{n}}$ ${\displaystyle \operatorname {D} _{X}A(X)=\operatorname {D} A(X)=\operatorname {D} A}$ ${\displaystyle A^{\prime }}$

${\displaystyle \operatorname {D} A(X)=\sum _{n\in \mathbb {Z} }na_{n}X^{n-1}}$ .

The derivation of a formal power series results in a formal power series. It is a -Derivation , and it obeys the well-known calculation rules of differential calculus including the chain rule : ${\displaystyle R}$

${\displaystyle \operatorname {D} (A\circ B)(X)=(\operatorname {D} A)\left(B(X)\right)\cdot \operatorname {D} B(X)}$ .

In relation to the derivative, formal power or Laurent series behave like (infinite) Taylor series or Laurent series. Indeed it is for ${\displaystyle k\leq m}$

${\displaystyle \left[X^{m-k}\right](\operatorname {D} ^{k}A)(X)=\prod _{j=0}^{k-1}(m-j)\,\left[X^{m}\right]A(X)=k!\,{\binom {m}{k}}\,a_{m}}$

and

${\displaystyle \left[X^{0}\right](\operatorname {D} ^{m}A)(X)=m!\,a_{m}}$ .

Thus, in a ring with a characteristic different from 0, only a finite number of formal derivatives are different from the zero series. Furthermore applies

${\displaystyle \operatorname {ord} (A^{\prime })\geq \operatorname {ord} (A)-1}$ .

For rows with     the equal sign applies. ${\displaystyle \operatorname {ord} (A)\;l(A)\;\neq 0}$

Formal residual

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

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

a -Derivation that ${\displaystyle K}$

${\displaystyle \ker \operatorname {D} =K}$

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

Fulfills. This shows that the coefficient of in is of particular interest; it is called the formal residual of and is also noted. The image ${\displaystyle X^{-1}}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle \operatorname {Res} (A)}$

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

is -linear, and you have the exact sequence ${\displaystyle K}$

${\displaystyle 0\to K\to K(\!(X)\!)\;{\xrightarrow[{\operatorname {D} }]{\;}}\,K(\!(X)\!)\;{\xrightarrow[{\operatorname {Res} }]{\;}}\,K\to 0}$.

A few rules from differential calculus

The following applies to all : ${\displaystyle A=\textstyle \sum _{n\in \mathbb {Z} }a_{n}X^{n},B\in K(\!(X)\!)}$

i.	${\displaystyle \operatorname {Res} (A^{\prime })=0}$.
ii.	${\displaystyle \operatorname {Res} (AB^{\prime })=-\operatorname {Res} (A^{\prime }B)}$.
iii.	${\displaystyle A\neq 0}$	${\displaystyle \implies \,\operatorname {Res} (A^{\prime }/A)=\operatorname {ord} (A)}$.
iv.	${\displaystyle \operatorname {ord} (B)>0}$	${\displaystyle \implies \,\operatorname {Res} \left(\!(A\circ B)B^{\prime }\right)=\operatorname {ord} (B)\operatorname {Res} (A)}$.
v.	${\displaystyle [X^{n}]A(X)=\operatorname {Res} \left(X^{-n-1}A(X)\right).}$

Property (i) is part of the exact sequence.
Property (ii) follows from (i) when applied to. Property (iii): each can be written as with and , from which paths are invertible in which follows. Property (iv): You can write with . Hence and (iv) follows from (i) and (iii). Property (v) follows directly from the definition. ${\displaystyle (AB)^{\prime }=A^{\prime }B+AB^{\prime }}$
${\displaystyle A\in K(\!(X)\!)}$ ${\displaystyle A=:X^{m}B}$ ${\displaystyle m:=\operatorname {ord} (A)}$ ${\displaystyle C\in K[[X]]}$ ${\displaystyle A^{\prime }/A=mX^{-1}+C^{\prime }/C.}$ ${\displaystyle \operatorname {ord} (C)=0}$ ${\displaystyle C}$ ${\displaystyle K[[X]]\subset \operatorname {im} (\operatorname {D} )=\ker(\operatorname {Res} ),}$ ${\displaystyle \operatorname {Res} (A^{\prime }/A)=m}$
${\displaystyle \operatorname {im} (\operatorname {D} )=\ker(\operatorname {Res} ),}$ ${\displaystyle A=a_{-1}X^{-1}+F^{\prime },}$ ${\displaystyle F\in K[[X]]}$ ${\displaystyle (A\circ B)B^{\prime }=a_{-1}B^{-1}B^{\prime }+(F^{\prime }\circ B)B^{\prime }=a_{-1}B^{\prime }/B+(F\circ B)^{\prime }}$

Inverse of the composition (inverse function)

If the formal power series has the coefficient and is invertible in , then the inverse of the composition , the (formal) inverse function , can be formed by . Their coefficients are integer polynomials in and den . ${\displaystyle \textstyle A(X)=\sum _{n=1}^{\infty }a_{n}X^{n}\in R[[X]]}$ ${\displaystyle \left[X^{0}\right]A(X)=0}$ ${\displaystyle \left[X^{1}\right]A(X)=a_{1}}$ ${\displaystyle R}$ ${\displaystyle \textstyle B(X):=A^{-1}(X)=\sum _{n=1}^{\infty }b_{n}X^{n}}$ ${\displaystyle A}$ ${\displaystyle b_{n}}$ ${\displaystyle a_{1}^{-1}}$ ${\displaystyle a_{n}(n\geq 2)}$

→ Main article : Lagrangian inversion formula # Explicit formula

The following statements are somewhat weaker, but easier to write down:

If a body of characteristic 0, then the formula becomes ${\displaystyle K}$

${\displaystyle b_{n}=1/n\left[X^{n-1}\right]\left({\frac {X}{A(X)}}\right)^{n}}$

${\displaystyle (\mathbf {I} )}$

acted as another version of the Lagrangian inversion formula.

The formula can be used a little more broadly:
Is anything, then is ${\displaystyle \textstyle C(X)=\sum _{n=0}^{\infty }c_{n}X^{n}\in K[[X]]}$

${\displaystyle (C\circ A^{-1})(X)=c_{0}+\sum _{n=1}^{\infty }{\frac {X^{n}}{n}}\left[Y^{n-1}\right]C^{\prime }(Y)\left({\frac {Y}{A(Y)}}\right)^{n}}$

${\displaystyle (\mathbf {II} )}$

There are various formulations of the Lagrange inversion formula (including the Lagrange-Bürmann formula ), often using higher derivatives and Bell polynomials .

example

The too

${\displaystyle A(X):=X-{\frac {aX^{2}}{1!}}+{\frac {a^{2}X^{3}}{2!}}\mp \cdots =X\exp(-aX)={\frac {X}{\exp(aX)}}}$

inverse series is

${\displaystyle B(X):=\sum _{n=1}^{\infty }{\frac {\left(na\right)^{n-1}}{n!}}X^{n}}$ ,

because it is ${\displaystyle (\mathbf {I} )}$

${\displaystyle b_{n}=1/n\left[X^{n-1}\right]{\frac {X}{A(X)}}=1/n\left[X^{n-1}\right]\exp(aX)={\frac {\left(na\right)^{n-1}}{n!}}}$ ,

whence the claim.

Universal property

The ring can be characterized by the following universal property: Be a commutative associative algebra over the commutative and unitary ring . If now is an ideal of such that the -adic topology is complete , and then there is a unique one with the following properties: ${\displaystyle R[[X]]}$
${\displaystyle B}$ ${\displaystyle R}$ ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle I}$ ${\displaystyle B}$ ${\displaystyle x\in I,}$ ${\displaystyle \Phi \colon R[[X]]\to B}$

• ${\displaystyle \Phi (X)=x}$
• ${\displaystyle \Phi }$is a homomorphism of -algebras ${\displaystyle R}$
• ${\displaystyle \Phi }$is steady .

In several indeterminates

If a commutative ring with 1, then and are commutative rings with 1 and therefore also recursive ${\displaystyle R}$ ${\displaystyle R_{1}:=R[[X_{1}]]}$ ${\displaystyle {}_{1}\!R:=R(\!(X_{1})\!)}$

${\displaystyle R_{m}:=R_{m-1}[[X_{m}]]=:R[[X_{1},\dotsc ,X_{m}]]}$

and

${\displaystyle {}_{m}\!R:={}_{m-1}\!R(\!(X_{m})\!)=:R(\!(X_{1},\dotsc ,X_{m})\!)}$.

The order of the m. a. W .: the rings of all permutations are isomorphic, and each intermediate ring can be regarded as a starting ring. ${\displaystyle X_{1},\dotsc ,X_{m}}$

Generally one understands any sum

${\displaystyle A(X_{1},\dotsc ,X_{m})=\sum _{n_{1},\dotsc ,n_{m}\in \mathbb {Z} }a_{n_{1},\dotsc ,n_{m}}X_{1}^{n_{1}}\dotsm X_{m}^{n_{m}}}$

of monomials the form with integral exponents as a formal series in multiple indeterminate, as a power series when all the coefficients with a negative index component disappear, or as a Laurent series, if there is a lower bound to exist. ${\displaystyle a_{n_{1},\dotsc ,n_{m}}X_{1}^{n_{1}}\dotsm X_{m}^{n_{m}}}$ ${\displaystyle n_{1},\dotsc ,n_{m}}$ ${\displaystyle n_{i}}$ ${\displaystyle u\in \mathbb {Z} }$ ${\displaystyle a_{n_{1},\dotsc ,n_{m}}\neq 0\implies \forall i:n_{i}\geq u}$

A monomial order makes it possible to arrange the monomials accordingly and thereby generalize terms such as guide coefficient .

The size is called the total degree of a monomial . If the (non-vanishing) monomials of a formal power or Laurent series all have the same total degree, then it is a homogeneous series; a formal power series is then a homogeneous polynomial . ${\displaystyle n_{1}+\dotsb +n_{m}}$ ${\displaystyle X_{1}^{n_{1}}\dotsm X_{m}^{n_{m}}}$

For the coefficient extraction operator

${\displaystyle \left[X_{m}^{n_{m}}\right]A(X_{1},\dotsc ,X_{m})}$

From the power or Laurent series , all monomials in which the indeterminate has the degree must be combined as a power or Laurent series in the other indeterminates . ${\displaystyle A}$ ${\displaystyle X_{m}}$ ${\displaystyle n_{m}}$ ${\displaystyle X_{1},\dotsc ,X_{m-1}}$

With the above successive formation of , the topology of the output ring, here :, is lost: the topology of the subspace in is, according to the construction, the discrete . However, if this is not desired, you can equip the result with the product of the topologies of and . The same applies to rings from formal Laurent series. ${\displaystyle R_{2}:=R_{1}[[X_{2}]]={\bigl (}R[[X_{1}]]{\bigr )}[[X_{2}]]}$ ${\displaystyle R_{1}:=R[[X_{1}]]}$ ${\displaystyle R_{1}}$ ${\displaystyle R_{2}=R_{1}[[X_{2}]]}$ ${\displaystyle R[[X_{1},X_{2}]]}$ ${\displaystyle R[[X_{1}]]}$ ${\displaystyle R[[X_{2}]]}$ ${\displaystyle R(\!(X_{1},X_{2})\!)}$

See also

• Local body
• Generating function
• Monoid ring

literature

• LV Kuz'min: Formal power series . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Josef Hofbauer : Lagrange inversion. Retrieved May 6, 2018 . 
• Alan D. Sokal: A Ridiculously Simple and Explicit Implicit Function Theorem. January 31, 2009, accessed May 6, 2018 . 
• Eric W. Weisstein : Lagrangian inversion formula . In: MathWorld (English).

References and comments

1. ↑ which is an additive group under the addition to be defined, but for which the following definition of the multiplication does not work
2. ↑ A. Sokal
3. ↑ J. Hofbauer