Real analytic functions are locally given by a convergent power series (i.e. it has a power series on a particular neighborhood). More specifically, they can be expressed by a power series with non-empty radius of convergence—an interval of positive radius centered at α :
Real analytic functions are infinitely differentiable up to the nth derivative (e.g. first derivative, second derivative, third derivative,…).
Real Analytic Function Definition with Taylor Series
The power series expansion of an analytic function coincides with the Taylor series. This gives us another way to define a real analytic function, as one agrees with its Taylor series in a neighborhood of every point. In other words, a Taylor series will converge to the series at a certain point.
As an example, a function is real analytic at zero if there is some R > 0 so that:
In order to be classified as real analytic, a function doesn’t have to agree with its Taylor series everywhere, just when R < 0.
More generally, these functions can be described as analytic at an arbitrary point a, in which case the expression would be differentiated at a:
This tells that as long as |x – a| is less than R (i.e. we’re near point a), the function f can be written as a power series.
Properties of a Real Analytic Function
Real analytic functions are a very small class of functions within the set of smooth (infinitely differentiable) functions. In order to be classified as “real analytic”, a function must be all of the following:
- A real-numbered function,
- A continuous function,
- A differentiable function,
- A smooth function,
- In agreement with its Taylor series in a neighborhood of every point.
 Stefanski, R. (2004). Factorization of Polynomials and Real Analytic Functions. Retrieved July 8, 2021 from: https://radekstefanski.weebly.com/uploads/1/3/6/4/13643663/stefanski2004-factorization1.pdf
Stephanie Glen. "Real Analytic Function: Simple Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/real-analytic-function-definition/
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