power function

A real power functionDlmfDlmfPlanetmathf:+  has the form


where a is a given real number.


The power function  xxa  is differentiableMathworldPlanetmathPlanetmath with the derivativexaxa-1  and strictly increasingPlanetmathPlanetmath if  a>0  and strictly decreasing if  a<0  (and 1 if  a=0).

The power functions comprise the natural power functionsxxn  with  n=0, 1, 2,,  the root functionsxxn=x1n  with  n=1, 2, 3,  and other fraction power functionsxxa  with a any fractional number.

Note.  The power xa may of course be meaningful also for other than positive values of x, if a is an integer.  On the other hand, e.g. (-1)2 has no real values — see the general power.

Title power function
Canonical name PowerFunction
Date of creation 2013-03-22 14:46:32
Last modified on 2013-03-22 14:46:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Definition
Classification msc 26A99
Related topic PropertiesOfTheExponential
Related topic FractionPower
Related topic CubeOfANumber
Related topic Polytrope
Related topic PowerTowerSequence
Related topic LaplaceTransformOfLogarithm
Defines natural power function
Defines root function
Defines fraction power function