Taylor series

2020 Mathematics Subject Classification: Primary: 26A09 Secondary: 30B10 [MSN][ZBL]

Also known as Maclaurin series. The series was published by B. Taylor in 1715, whereas a series reducible to it by a simple transformation was published by Johann I. Bernoulli in 1694.

One real variable

Let $U$ be an open set of $\mathbb R$ and consider a function $f: U \to \mathbb R$. If $f$ is infinitely differentiable at $x_0$, its Taylor series at $x_0$ is the power series given by \begin{equation}\label{e:Taylor_series} \sum_{n=0}^\infty \frac{f^{(n)} (x_0)}{n!} (x-x_0)^n\, , \end{equation} where we use the convention that $0^0=1$.

The partial sums \[ P_k(x) := \sum_{n=0}^k \frac{f^{(n)} (x_0)}{n!} (x-x_0)^n \] of a Taylor series are called Taylor polynomial of degree $k$ and the "remainder" $f(x)- P_k (x)$ can be estimated in several ways, see Taylor formula.

Analyticity

The property of being infinitely differentiable does not guarantee the convergence of the Taylor series to the function $f$: a well-known example is given by the function \[ f (x) = \left\{\begin{array}{ll} e^{-1/x^2} \quad &\mbox{if } x\neq 0\\ 0 \quad &\mbox{otherwise.} \end{array} \right. \] Indeed the function $f$ defined above is infinitely differentiable everywhere, its Taylor series at $0$ vanishes identically, but $f(x)>0$ for any $x\neq 0$.

If the Taylor series of a function $f$ at $x_0$ converges to the values of $f$ in a neighborhood of $x_0$, then $f$ is real analytic (in a neighborhood of $x_0$). The Taylor series is also unique in the following sense: if for some given function $f$ defined in a neighborhood of $x_0$ there is a power series $\sum a_n (x-x_0)^n$ which converges to the values of $f$, then such series coincides necessarily with the Taylor series.

With the aid of the formulas for the difference $f (x) - P_n (x)$ (see Taylor formula) one can establish several criterions for the analyticity of $f$. A popular one is the existence of positive constants $C$, $R$ and $\delta$ such that \[ |f^{(n)} (x)| \leq C n! R^n \qquad \forall x\in ]x_0-\delta, x_0+\delta[\quad \forall n\in \mathbb N\, . \]

One complex variable

If $U$ is a subset of the complex plane and $f:U\to \mathbb C$ an holomorphic function (i.e. complex differentiable at every point of $U$), then the Taylor series at $x_0\in U$ is given by the same formula, where $f^{(n)} (x_0)$ denotes the complex $n$-th derivative. The existence of all derivatives is guaranteed by the holomorphy of $f$, which also implies the convergence of the power series to $f$ in a neighborhood of $x_0$ (in sharp contrast with the real differentiability!), see Analytic function.

Several variables

The Taylor series can be generalized to functions of several variables. More precisely, if $U\subset \mathbb R^n$ and $f:U\to \mathbb R$ is infinitely differentiable at $\alpha\in U$, the Taylor series of $f$ at $\alpha$ is given by \begin{equation}\label{e:power_series_nd} \sum_{k_1, \ldots, k_n =0}^\infty \frac{1}{k_1!\cdots k_n!} \frac{\partial^{k_1+\cdots + k_n} f}{\partial x_1^{k_1} \cdots \partial x_n^{k_n}} (\alpha)\, (x_1-\alpha_1)^{k_1} \ldots (x_n - \alpha_n)^{k_n}\, \end{equation} (see also Multi-index notation for other ways of expressing \eqref{e:power_series_nd}). The (real) analyticity of $f$ is defined by the property that such series converges to $f$ in a neighborhood of $\alpha$. An entirely analogous formula can be written for holomorphic functions of several variables (see Analytic function).

Further generalizations

The Taylor series can be generalized to the case of mappings of subsets of linear normed spaces into similar spaces.

References

[Di]	J.A. Dieudonné, "Foundations of modern analysis" , Acad. Press (1960) (Translated from French)
[IS]	V.A. Il'in, V.A. Sadovnichii, B.Kh. Sendov, "Mathematical analysis" , Moscow (1979) (In Russian)
[Ni]	S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)
[Ru]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78 MR0385023 Zbl 0346.26002
[St]	K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)