Real analytic function

2020 Mathematics Subject Classification: Primary: 14Pxx [MSN][ZBL]

Let $I\subset \mathbb R$ be an open set. A function $f:I \to \mathbb R$ is called analytic if for any $x_0\in I$ there is a neighborhood $J$ of $x_0$ and a power series $\sum a_n (x-x_0)^n$ such that \begin{equation}\label{e:power_series} f(x) = \sum_n a_n (x-x_0)^n \qquad \forall x\in J \end{equation} (here and in the rest of the entry we follow two conventions: $0^0$ is set to be $1$ and when we write an identity as \eqref{e:power_series} we implicitely assume that the series on the left hand side converges). The notion can be extended to functions of several variables. Namely, if $U\subset \mathbb R^k$ is an open set, a function $f:U\to \mathbb R$ is called analytic if for each $x_0\in U$ there is a neighborhood $V$ of $x_0$ and a sequence $P_n$ of homogeneous polynomials of degree $n$ in $k$ variables such that \begin{equation}\label{e:power_series_k} f(x) = \sum_n P_n (x-x_0)\qquad \forall x\in V\, . \end{equation} Finally, a map $f: U \to \mathbb R^m$ is called analytic if each component function of $f$ is analytic.

Properties

Taylor series

An analytic function is infinitely differentiable and its power expansion coincides with the Taylor series. Namely, the coefficients $a_n$ in \eqref{e:power_series} are given by \[ a_n = \frac{f^{(n)} (x_0)}{n!}\, . \] A similar, more complicated formula, can be written for the polynomials in the expansion \eqref{e:power_series}. More precisely, using multiindex notation, we have \[ P_n (y) = \sum_{|\alpha| = n} \frac{1}{\alpha!}\frac{\partial^n f}{\partial x^\alpha} (x_0)\, y^\alpha \] where

• $\alpha$ denotes a multiindex, namely an element $(\alpha_1, \ldots, \alpha_k)\in \mathbb Z^k$ with $\alpha_i \geq 0$;
• $|\alpha|$ equals $\alpha_1 + \ldots + \alpha_k$;
• $\alpha!$ equals $\alpha_1! \alpha_2! \ldots \alpha_k!$;
• $\frac{\partial^n f}{\partial x^\alpha}$ is the partial derivative

\[ \frac{\partial^n f}{\partial x_1^{\alpha_1} \partial x_2^{\alpha_2} \ldots \partial x_k^{\alpha_k}}\, ; \]

• $y^\alpha$ denotes the polynomial $y_1^{\alpha_1} y_2^{\alpha_2} \ldots y_k^{\alpha_k}$.

Closure properties

Analytic functions are closed under the most common operations, namely: linear combinations, products and compositions of real analytic functions remain real analytic. The same holds for quotients on the set where the divisor is different from zero. The power-series expansion for the resulting function can be obtained from the power series expansions of the original functions through suitable formulas. The power series of the linear combination is the linear combination of the power series, whereas the power series of the product is the Cauchy product of the power series. More complicated formulas hold for the quotient and the composition.

Similarly, derivatives and primitives of analytic functions are analytic as well and their power series can be found differentiating (resp.integrating) the original series term by term. Namely, if \eqref{e:power_series} holds, then the following identities \[ f' (x) = \sum_n n a_n (x-x_0)^{n-1} \] \[ \int f(x)\, dx = C + \sum_n \frac{a_n}{n+1} (x-x_0)^{n+1} \] are valid on any symmetric open interval where \eqref{e:power_series} holds. Similar conclusions apply to analytic functions of several variables.

Implicit and inverse function theorems

Both the implicit and inverse function theorems (see Implicit function and Inverse function) hold in the category of analytic functions. Namely, under the usual assumption of nondegeneracy of the differential of a map $F$ at the point $x_0$, if the map $F$ is analytic then

• the local inverse (in the case of the inverse function theorem)
• or the locally implicitly defined function solving the equation $F(y, f(y))= F(x_0)$ (in the case of the inverse function theorem)

are analytic.

Analytic varieties and unique continuation

The set $Z_f$ of zeros of a (nonconstant) real analytic function is a real analytic subvariety and as such has very strong properties. In particular,

• if the domain is $1$-dimensional, then the set consists of isolated points
• if the domain is $k$-dimensional, then the set $Z_f$ can be written as the union $S_j$ (for $j$ ranging from $0$ to $k-1$) of $j$-dimensional (real analytic) submanifolds.

The sets $S_j$ are called strata and they might be empty.

As a corollary, if two real analytic function $f$ and $g$ coincide on an open set $V$, then they coincide on any connected open subset (of the intersection of their domain of definition) which has nontrivial intersection with $V$.

The properties of real analytic functions and real analytic subvarieties are deeply related to those of polynomials and real algebraic subvarieties. For instance, a deep useful property of real analytic functions is the Lojasiewicz inequality.

Comparison with complex analytic functions

Any real analytic function can be locally extended to an holomorphic (or complex analytic) function. More precisely, assume that the left hand side of \eqref{e:power_series} converges for some $x$ with $|x-x_0|=r$. Then the series converges for any complex value $x$ with $|x- (x_0 + 0i)| < r$ and defines an holomorphic function which coincides with $f$ on the interval $]x_0-r, x_0+r[$. Similarly, for any real analytic function $f:U\to \mathbb R$ and any $x_0\in U$ there is an open neighborhood $W$ in $\mathbb C^n$ of $x_0+ 0i$ and an holomorphic map $g: W \to \mathbb C$ such that $g$ coincides with $f$ on $W \cap \{z: {\rm Im}\, (z) =0\}$.

However, many of the properties of holomorphic functions do not extend to real analytic functions. For instance, if $g: \mathbb C \to \mathbb C$ is holomorphic, then the corresponding power series expansion $\sum_n a_n z^n$ of $f$ at $0$ converges to $g(z)$ for any $z\in \mathbb C$ (see Entire function). On the other hand it can be readily checked that the map $f: \mathbb R\to \mathbb R$ given by $f(x) = \frac{1}{1+x^2}$ is real analytic on the whole real line, but its power series at $0$ converges only for $|x|\leq 1$.

References

[KP]	S. Krantz, H. R. Parks, "A Primer of Real Analytic Functions", Birkhäuser (2002).
[Ru]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) MR0385023 Zbl 0346.26002