Editing Formal power series (section)

==Interpreting formal power series as functions==
{{Complex analysis sidebar}}
In [[mathematical analysis]], every convergent [[power series]] defines a [[function (mathematics)|function]] with values in the [[real number|real]] or [[complex number|complex]] numbers. Formal power series over certain special rings can also be interpreted as functions, but one has to be careful with the [[function domain|domain]] and [[codomain]]. Let

:<math>f = \sum a_n X^n \in R[[X]],</math>

and suppose <math>S</math> is a commutative associative algebra over <math>R</math>, <math>I</math> is an ideal in <math>S</math> such that the [[I-adic topology]] on <math>S</math> is complete, and <math>x</math> is an element of <math>I</math>. Define:

:<math>f(x) = \sum_{n\ge 0} a_n x^n.</math>

This series is guaranteed to converge in <math>S</math> given the above assumptions on <math>x</math>. Furthermore, we have

:<math> (f+g)(x) = f(x) + g(x)</math>

and

:<math> (fg)(x) = f(x) g(x).</math>

Unlike in the case of bona fide functions, these formulas are not definitions but have to be proved.

Since the topology on <math>R[[X]]</math> is the <math>(X)</math>-adic topology and <math>R[[X]]</math> is complete, we can in particular apply power series to other power series, provided that the arguments don't have [[constant coefficients]] (so that they belong to the ideal <math>(X)</math>): <math>f(0)</math>, <math>f(X^2-X)</math> and <math>f((1-X)^{-1}-1)</math> are all well defined for any formal power series <math>f \in R[[X]].</math>

With this formalism, we can give an explicit formula for the multiplicative inverse of a power series <math>f</math> whose constant coefficient <math>a=f(0)</math> is invertible in <math>R</math>:

:<math>f^{-1} = \sum_{n \ge 0} a^{-n-1} (a-f)^n.</math>

If the formal power series <math>g</math> with <math>g(0)=0</math> is given implicitly by the equation

:<math>f(g) =X</math>

where <math>f</math> is a known power series with <math>f(0)=0</math>, then the coefficients of <math>g</math> can be explicitly computed using the [[#The Lagrange inversion formula|Lagrange inversion formula]].