Analytic Function - Examples

Examples

Most special functions are analytic (at least in some range of the complex plane). Typical examples of analytic functions are:

• Any polynomial (real or complex) is an analytic function. This is because if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion must immediately vanish to 0, and so this series will be trivially convergent. Furthermore, every polynomial is its own Maclaurin series.

• The exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x₀ (as in the definition) but for all values of x (real or complex).

• The trigonometric functions, logarithm, and the power functions are analytic on any open set of their domain.

Typical examples of functions that are not analytic are:

• The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0. Piecewise defined functions (functions given by different formulas in different regions) are typically not analytic where the pieces meet.

• The complex conjugate function z → z* is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic, and it is real analytic as a function from R² to R².

• See here for another example of a non-analytic smooth function.