**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 [1]:

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.

## References

[1] 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

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