Determining if a function is real anaytic at the point $a$?

Question

Is there a method, other then using refer to the Taylor series to determine if a real function is analytic at the point $a$. If so please, if possible, could you give a source.

Answer 1

The most convenient method I know is to show that $f$ extends to a holomorphic function in a neighborhood of $a$ in the complex plane. The property of being holomorphic can be verified just by considering the first order derivatives (the Cauchy-Riemann equations). It implies, by a theorem of complex analysis (see any textbook), that the function is represented by a power series.

Example

Consider the function $f(x)=1/(x^2+1)$ near the point $x=1$. Its natural extension to the complex plane is $$F(x+iy) = \frac{1}{(x+iy)^2+1}$$ which is holomorphic as long as $x+iy\notin \{-i,i\}$. In particular, $F$ is holomorphic in the open disk of radius $\sqrt{2}$ centered at $1$. It follows that $F$ is represented by a power series centered at $1$ with radius of convergence $\sqrt{2}$. Restricting attention to the real line again, conclude that $f$ is represented by a power series, $$ \frac{1}{x^2+1} = \sum_{n=0}^\infty c_n(x-1)^n,\qquad |x-1|<\sqrt{2} $$