Analytic Function - Definitions

Definitions

Formally, a function ƒ is real analytic on an open set D in the real line if for any x0 in D one can write

\begin{align} f(x) & = \sum_{n=0}^\infty a_{n} \left( x-x_0 \right)^{n} \\ & = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots \end{align}

in which the coefficients a0, a1, ... are real numbers and the series is convergent to ƒ(x) for x in a neighborhood of x0.

Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point x0 in its domain

converges to ƒ(x) for x in a neighborhood of x0 (in the mean-square sense). The set of all real analytic functions on a given set D is often denoted by Cω(D).

A function ƒ defined on some subset of the real line is said to be real analytic at a point x if there is a neighborhood D of x on which ƒ is real analytic.

The definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane."

Read more about this topic:  Analytic Function

Famous quotes containing the word definitions:

    “Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
    There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
    —Edmond De Goncourt (1822–1896)

    “What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
    —G.C. (Georg Christoph)

    “The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
    —Pamela Patrick Novotny (20th century)