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Taylor series

Contents

Idea

The Taylor series of a smooth function ff (at a given point cc) is a formal power series (in xcx - c) whose partial sum?s are the Taylor polynomials of ff (at cc). As the Taylor polynomials are approximations of ff by polynomials (up to a given degree), so the Taylor series is an approximation of ff by an analytic function (or at least an asymptotic expansion that attempts to be this).

See also Taylor's theorem for error estimates in the convergence of Taylor series.

Defintion

Let fC ()f \in C^\infty(\mathbb{R}) a smooth function with nnth derivative f (n)C ()f^{(n)} \in C^\infty(\mathbb{R}) and let cc be a real number.

Definition

The Taylor series of ff at cc is the formal power series

n=0 1n!f (n)(c)(xc) n. \sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(c) (x-c)^n \,.
Definition

For fC ()f \in C^\infty(\mathbb{R}) a smooth function with nnth derivative f (n)C ()f^{(n)} \in C^\infty(\mathbb{R}), its Maclaurin series is its Taylor series at zero:

n=0 1n!f (n)(0)x n. \sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(0) x^n \,.
Remark

Similar definitions apply to functions on any Cartesian space or smooth manifold.

Properties

Recall that a (partial) function ff is analytic at cc (in the interior of domf\dom f) if there exists a power series PP at cc and a neighbourhood UU of cc such that, for all xUdomfx \in U \cap \dom f, P(x)P(x) converges to f(x)f(x).

Proposition

If ff is analytic at cc, then the only power series witnessing this is the Taylor series of ff at cc (so in particular, the Taylor series exists; analytic functions are smooth).

In contrast, a smooth function need not be analytic; the classic counterexample is a bump function. In fact, the Taylor series of ff at cc might not converge to ff anywhere except at cc, either because the Taylor series has vanishing radius of convergence or because it converges to something else (an analytic function with the same jet as ff but a different germ). However, we can say this:

Proposition

The Taylor series of a smooth function ff is always an asymptotic expansion of ff.

Theorem

(Borel’s theorem)

The map

C ( k+l)C ( k)[[X 1,X l]] C^\infty(\mathbb{R}^{k+l}) \to C^\infty(\mathbb{R}^k) [ [ X_1, \cdots X_l] ]

obtained by forming Taylor series in ll variables is surjective.

In particular, every power series in one real variable is the Taylor series of some smooth function on the real line (even if it has vanishing radius of convergence and so is not the Taylor series of any analytic function).

Proof

The proof is reproduced for instance in MSIA, I, 1.3

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Revised on March 7, 2014 08:58:23 by Toby Bartels (64.89.53.137)