As analytically is called in mathematics a function that is locally given by a convergent power series. Because of the differences between real and complex analysis is called for clarification often explicitly of real - analytic or complex analytic functions. In the complexes, the properties are analytic and holomorphically equivalent. If a function in the whole complex plane defined and analytically, they are called completely.
It is or. It should be an open subset. A function is called analytic at the point when there is a power series
Are converging to a neighborhood of to. If at any point of analytically so called analytic.
- An analytic function is infinitely differentiable. The converse is not true, see examples below.
- The local power series representation of an analytic function is its Taylor series. It is therefore
- Sums, differences, products, quotients (if the denominator has no zeros ) and concatenations of analytic functions are analytic.
- Is continuous and has the set of zeros of an analytic function in a cluster point, then the zero function. Accordingly, where two functions that coincide on a set that has a cluster point in, for example, on an open subset, so they are identical.
Examples of analytic functions
Many common features of real analysis such as polynomials, exponential and logarithmic functions, trigonometric functions and rational expressions in these functions are analytic. The set of all on an open set real - analytic functions is denoted by.
A known analytical function is the exponential function
The trigonometric functions sine, cosine, tangent, cotangent and their Inverse Trigonometric Functions are analytically. However, the example shows the arctangent
That in a very analytical function can have a series expansion with a finite radius of convergence.
Many special functions, such as the Euler gamma function, the Euler beta function or the Riemann ζ - function are also analytical.
Examples of non-analytic functions
The following examples of non-analytic functions are among the smooth functions: They are on their domain of infinitely differentiable but at individual points, there is no power series expansion. The following function
Is for everyone, even at the point 0, infinitely differentiable. From all follows the Taylor series of,
Which, except in the point, not with matches. Thus, at the point 0 is not analytic.
Is infinitely differentiable. All derivatives of the two sub-functions in the zero point are 0, that fit together.
There is an important class of non-analytic functions, the functions with compact support. The support of a function is the closure of the set of points where a function does not vanish:
If the support is compact, so it is called a function with compact support (or by a test function ). These functions play an important role in the theory of partial differential equations. For functions that are defined on all of, this condition is equivalent to the fact that there is a number such that the following holds for all with. A function with compact support is accordingly consistent with the zero function for big match. If the function now also analytically, it would already match at all with the zero function by the above properties of analytic functions. In other words, the only analytic function with compact support is the zero function.
Is an infinitely differentiable function with compact support.
In the previous examples it can be proved that the Taylor series at any point has a positive radius of convergence, but not always converges to the function. There are also non-analytical functions in which the Taylor series is zero radius of convergence, such as
Is on all infinitely differentiable, but its Taylor series is
And thus only for convergent.
General, one can show that any formal power series occurs as a Taylor series of a smooth function.
In the theory of functions is shown that a function of a complex variable that is complex differentiable in an open circular disk, in the same open area as often is complex differentiable, and that the power series around the center of the circular disc
For every point converges to. This is an important aspect, easier to handle under the functions in the complex plane are as functions of a real variable. In fact, one used in the theory of functions, the attributes analytically holomorphic and regular interchangeably. From the original definitions of these terms, their equivalence is not immediately apparent; it was proved later. Complex analytical functions that take on real values , are constant. A consequence of the Cauchy- Riemann equations is that the real part of an analytic function determines the imaginary part up to a constant, and vice versa.
It applies the following important relationship between real - analytic functions and complex analytic functions:
Every real analytic function can be extended to a complex - analytic, ie holomorphic function on a neighborhood of.
Conversely, each holomorphic function to real - analytic function, if they are considered first, and then to restrict only the real component (or only the imaginary part ). This is the reason why many properties of real - analytic functions are most easily proved with the help of complex function theory.
Even with functions that depend on several variables, one can as follows a Taylor series expansion at the point define:
Here, use was made of the multi- index notation, the sum extends over all multi- indices of length. In analogy to the above-discussed case of a variable, a function is analytically if the Taylor series expansion point for each of the domain having a positive radius of convergence and representing the function in the region of convergence, that is,
For all from an environment of true. In the case of complex variables one also speaks at several variables of holomorphic functions. Such functions are handled within the function theory in several complex variables.