Power function

Fig.1.  versus  for various 

Power function (Степенная функция) is one of the primary elementary functions;

(1) 

where  is parameter; often it is assumed that  is real.

Notations

In C++,  can be expressed as . In some realizations, there may be some problems while one of arguments are integer and another is complex. In some programming languages, including Mathematica, Maple, Matlab aln Latex (in the "math.mode"), the same can be written as z^a.

There is description of the Power function on Wolfram MathWorld.[1]

Often, the function is considered with 2 parameters, including the leading coefficient into definition of function [2]. For several real values , the graphic  is shown in figure 1.

In some cases of integer  the function has specific names.

For , the power function  is cubic function.

For , the power function  is quadratic function.

For , the power function , referring to the identity function.

For , the power function  refers to constant.

For , the power function  and the power function is its own inverse function. For this case, often  is called "inverse function" (without to indicate, for which function it is "inverse"); using such a term in a context, where the inversion of function takes place, may cause confusions.

The confusion may appear also at the use of the superscript after the name of a function to indicate that after to evaluate the function, the power function should be applied to the result. Often this appears dealing with trigonometric functions, one writes, for example,  instead of . In particular, at , this writing may refer also to the ArcSin function. Such a confusion happens at the use of the Mathematica operation TeXForm: expressions  are automatically replaced to , mimicking the th iteration of function . In TORI, the upper superscript after the function indicates its iteration.

Properties of the power function

the following expressions hold in the wide ranges of values of parameters :







However, the use for negative and comlex values of parameters may cause errors. For integer value of power , the expression  can be simplified as follows







and so on.

Derivative

The derivative of a power function can be derived from the properties of exp and ln:



Such an expression is valid in wide range of parameters.

Integral

Form the expression for the derivative, the integration of the power function also can be derived:



Such an expression is also valid in wide range of parameters, and, especially, while .

Iterates of power function

Fig.2. Iterates of quadratic function,  for various 

The th Iterate of power function can be expressed again through the power function in the following way:

(2) 

with . This corresponds to the school–level rules

(3) 

(4) 

however, such a rules should be used carefully, because for come complex values of ,  and  the relations (3) and (4) are not valid. Since year 2005, the Mathematica software recognizes this trouble and the Simplify and FullSimplify operations do not use substitutions (3) and (4) automatically before to get some additional information about values of the parameters involved.

The interesting property of iteration of the Power function is that

(5) 

For this reason, the expression

(7) 

can be interpreted in a way in the left hand side of (5) as well as the expression in the right hand side of (5) in wide range of values of parameters and the argument. (Assuming the same for other functions may lead to confusions.) For $a\!=\!2$, iterates of the quadratic function are shown in Figure 2.

The power function is one of few functions, for which the iterates can be expressed in so simple way. For other function , the th iterate  can be expressed through its superfunction  and the Abel function ,

(8) 

However, for , the expression (8) is also valid; the corresponding superfunction  and the Abel function  are

(9) 

(10) 

Another often confusion is related with the order of operations, while  is interpreted as , while the more usual interpretation is . in particular, the Gaussian exponential  is never interpreted as . In order to avoid confusions, the use of parenthesis is strongly recommended; also, the representation through exponential and logarithm is appropriate.

Use of the power function

Initially, the power function could be used for short writing of expressions with polynomials with numbers. However, after discovery of exponential and logarithm, by default,  is interpreted as ; the power function can be applied to objects of any sets, where operations  and  are already defined. Even if some of these operations are not yet defined,  for natural number  can be considered as just iteration of function "multiplication to " applied to unity  times.

In such a way the power function is interpreted in quantum mechanics; for example,  refers to iterative application of operator  to the state .