Editing Formal power series (section)

=== 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.