Difference between revisions of "Power function"
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c) The power function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420033.png" /> is an odd natural number, is defined for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420034.png" />, and is negative when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420035.png" />. However, it is sometimes convenient to restrict in this case the power function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420037.png" />. The same statements apply for the power function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420038.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420039.png" /> is an irreducible fraction. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420041.png" />. | c) The power function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420033.png" /> is an odd natural number, is defined for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420034.png" />, and is negative when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420035.png" />. However, it is sometimes convenient to restrict in this case the power function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420037.png" />. The same statements apply for the power function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420038.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420039.png" /> is an irreducible fraction. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420041.png" />. | ||
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The properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420042.png" /> are usually considered when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420044.png" /> is real, although many of them also hold when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420045.png" /> and, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420046.png" /> is a natural number. | The properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420042.png" /> are usually considered when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420044.png" /> is real, although many of them also hold when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420045.png" /> and, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074200/p07420046.png" /> is a natural number. |
Revision as of 11:56, 13 December 2014
A function with
![]() |
where is a constant number. If
is an integer, the power function is a particular case of a rational function. When
and
have complex values, the power function is not single valued if
is not an integer.
For fixed real and
, the number
is a power, and the properties of
therefore follow from the properties of the power.
When , the power function
is defined and positive for any real
. When
, the power function
is defined in the following cases.
a) When , the power function
is defined to equal 0 if
, and is not defined if
. The power function
is defined to equal 1 for all
; in particular,
.
b) If is a natural number, then the power function
is defined for all
, and the power function
is defined for all
. Here
and
if
.
c) The power function , where
is an odd natural number, is defined for all real
, and is negative when
. However, it is sometimes convenient to restrict in this case the power function
to
. The same statements apply for the power function
, when
is an irreducible fraction. Here
and
.
![](/images/math/f/3/1/f31fb5ea1fafe7f63d1c42ead4d0e6e5.png)
The properties of are usually considered when
and
is real, although many of them also hold when
and, for example,
is a natural number.
Functions of the form , where
is a constant coefficient and
, express a direct proportionality (their graphs are straight lines passing through the origin of the coordinates (Fig.a)), while when
, they express an inverse proportionality (their graphs are equilateral hyperbolas with their centre at the origin of the coordinates and having the coordinate axes as their asymptotes (Fig.b)). Many laws of physics can be mathematically expressed by using functions of the form
(Fig.c).
When , the power function
is continuous, monotone (increasing when
, decreasing when
), infinitely differentiable, and, in a neighbourhood of every positive
, can be expanded into a Taylor series. Moreover,
![]() |
![]() |
![]() |
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when , where
are the binomial coefficients.
In the complex domain, the power function is defined for all
by the formula
![]() | (*) |
where . If
is an integer, then
is single valued:
![]() |
If is rational (
, where
and
are relatively prime), then the power function
takes
different values:
![]() |
where are the
-th roots of unity:
and
. If
is irrational, then
has an infinite number of values: the factor
takes different values for different
. For non-real complex values of
, the power function
is defined by the same formula (*).
Comments
Also regarding formula (*), the symbol is an abbreviation for the value
of the exponential function exp at the complex number
. This function is defined by the series
![]() |
which converges (absolutely) at each complex . Note that
if
.
Taking and
in (*) one obtains the principal value. An interesting example is obtained if
:
![]() |
References
[a1] | K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 318ff |
[a2] | K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) |
[a3] | J. Marsden, "Basic complex analysis" , Freeman (1973) |
Power function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Power_function&oldid=35618