Everything about Analytic Function totally explained
In
mathematics, an
analytic function is a
function that's locally given by a convergent
power series. Analytic functions can be thought of as a bridge between
polynomials and general functions. There exist both
real analytic functions and
complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that don't hold generally for real analytic functions. A function is analytic if it's equal to its
Taylor series in some
neighborhood.
Definitions
Formally, a function
f is
real analytic on an
open set D in the
real line if for any
x0 in
D one can write
»
Also, if a complex analytic function is defined in an open
ball around a point
x0, its power series expansion at
x0 is convergent in the whole ball. This isn't true in general for real analytic functions. (Note that an open ball in the complex plane would be a
disk, while on the real line it would be an
interval.)
Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane. However, not every real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function
f (
x) defined in the paragraph above is a counterexample, as it isn't defined for
x =
±i.
Analytic functions of several variables
One can define analytic functions in several variables by means of power series in those variables (see
power series). Analytic functions of several variables have some of the same properties as analytic functions of one variable. However, especially for complex analytic functions, new and interesting phenomena show up when working in 2 or more dimensions. For instance, zero sets of complex analytic functions in more than one variables are never discrete.
Further Information
Get more info on 'Analytic Function'.
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