Formally, a function $${\displaystyle f}$$ is real analytic on an open set $${\displaystyle D}$$ in the real line if for any $${\displaystyle x_{0}\in D}$$ one can write
in which the … See more

Typical examples of analytic functions are
• The following elementary functions:
• Most special functions (at least in some range of the complex plane): See more

• The sums, products, and compositions of analytic functions are analytic.
• The reciprocal of an analytic function that is nowhere zero is … See more

Real and complex analytic functions have important differences (one could notice that even from their different relationship with … See more

The following conditions are equivalent:
1. $${\displaystyle f}$$ is real analytic on an open set $${\displaystyle D}$$.
2. There is a complex analytic … See more

As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or $${\displaystyle {\mathcal {C}}^{\infty }}$$). (Note that this differentiability is in … See more

One can define analytic functions in several variables by means of power series in those variables (see power series). Analytic functions … See more