The Identity Theorem for real analytic functions

What is the condition for two real analytic functions to be identically equal? We know that there is a nice condition (Identity Theorem) for holomorphic function to check if they are the same. What is its version for real analytic functions?

Solution 1:

As a reference, I suggest A Primer of Real Analytic Functions by Krantz and Parks.

The two versions of the Identity Theorem stated by Daniel Fisher can be unified at the expense of more complicated statement.

If $U$ is a domain, and $f,g$ are two real-analytic functions defined on $U$, and if $V\subset U$ is a nonempty open set with $f\lvert_V \equiv g\lvert_V$, then $f \equiv g$. If the domain is one-dimensional (an interval in $\mathbb{R}$), then it suffices that $f\lvert_M \equiv g\lvert_M$ for some $M\subset U$ that has an accumulation point in $U$.

Claim. If $f,g$ are real analytic and there is a point $p\in \mathbb R^n$ such that the set of all limits $$ \left\{\lim_{n\to \infty} \frac{x_n-p}{|x_n-p|} : \qquad f(x_n)=g(x_n),\ x_n\to p, \ x_n\ne p\right\} \tag{1}$$ has an interior point in the topology of the sphere $S^{n-1}$, then $f\equiv g$.

Indeed, suppose $f-g$ is not identically zero. Express its Taylor series at $p$ as the sum of homogeneous polynomials $P_d$. Let $d$ be the smallest degree for which $P_d$ is not identically zero. Then the set defined by (1) can be shown to be precisely $$ S^{n-1} \cap \{ P_d =0\} \tag{2}$$ Since the zero set of a polynomial has empty interior in $\mathbb R^n$, it follows that (2) has empty interior in $S^{n-1}$. $\quad \Box$

When $n=1$, the sphere $S^0$ is a two-point set, so any nonempty subset of it has nonempty interior. We thus recover the one-dimensional result.