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" />. | ||
| − | < | + | <center><asy> |
| + | import graph; | ||
| + | picture whole; | ||
| − | + | real sc=0.8; | |
| − | + | draw ( graph( new real(real x) {return x;}, -2, 2), red+1.2, "$y=x$" ); | |
| + | draw ( graph( new real(real x) {return 2x;}, -1, 1), blue+1.2, "$y=2x$" ); | ||
| + | draw ( graph( new real(real x) {return x/2;}, -2, 2), green+1.2, "$y=x/2$" ); | ||
| − | + | xaxis(-2.1,2.1, LeftTicks(Label(fontsize(8pt)),Step=1,step=0.2,Size=2,size=1,NoZero)); | |
| + | yaxis(-2,2, RightTicks(Label(fontsize(8pt)),Step=0.5,step=0.1,Size=2,size=1,NoZero)); | ||
| + | labelx("$x$",(2.3,0.25)); | ||
| + | labely("$y$",(0.15,2.3)); | ||
| − | + | add(scale(0.72sc,1.2sc)*legend(),(0.5,-0.75)); | |
| − | + | real mrg=1.3; | |
| + | draw( scale(mrg)*box((-2,-2),(2,2)), white ); | ||
| + | |||
| + | add (whole,shift(-sc*230,0)*currentpicture.fit(sc*mrg*6.5cm)); | ||
| + | erase(); | ||
| + | |||
| + | |||
| + | draw ( graph( new real(real x) {return 1/x;}, -4, -0.25), red+1.2, "$y=1/x$" ); | ||
| + | draw ( graph( new real(real x) {return 1/x;}, 0.25, 4), red+1.2 ); | ||
| + | draw ( graph( new real(real x) {return 2/x;}, -4, -0.5), blue+1.2, "$y=2/x$" ); | ||
| + | draw ( graph( new real(real x) {return 2/x;}, 0.5, 4), blue+1.2 ); | ||
| + | draw ( graph( new real(real x) {return 1/(2x);}, -4, -0.125), green+1.2, "$y=1/(2x)$" ); | ||
| + | draw ( graph( new real(real x) {return 1/(2x);}, 0.125, 4), green+1.2 ); | ||
| + | |||
| + | xaxis(-4.2,4.2, LeftTicks(Label(fontsize(8pt)),Step=2,step=0.5,Size=2,size=1,NoZero)); | ||
| + | yaxis(-4,4, RightTicks(Label(fontsize(8pt)),Step=1,step=0.2,Size=2,size=1,NoZero)); | ||
| + | labelx("$x$",(4.6,0.5)); | ||
| + | labely("$y$",(0.3,4.6)); | ||
| + | |||
| + | add(scale(0.75sc,0.75sc)*legend(),(0.95,-1.2)); | ||
| + | |||
| + | real mrg=1.3; | ||
| + | draw( scale(mrg)*box((-4,-4),(4,4)), white ); | ||
| + | |||
| + | add (whole,shift(0,0)*currentpicture.fit(sc*mrg*6.5cm,mrg*6.5cm,false)); | ||
| + | erase(); | ||
| + | |||
| + | |||
| + | draw ( graph( new real(real x) {return x^3;}, -4^(1/3), 4^(1/3)), red+1.2, "$y=x^3$" ); | ||
| + | draw ( graph( new real(real x) {return x^2;}, -2, 2), blue+1.2, "$y=x^2$" ); | ||
| + | draw ( graph( new real(real x) {return sqrt(x);}, 0, 4), green+1.2, "$y=x^{1/2}$" ); | ||
| + | draw ( graph( new real(real x) {return -sqrt(x);}, 0, 4), green+1.2 ); | ||
| + | |||
| + | xaxis(-4.2,4.2, LeftTicks(Label(fontsize(8pt)),Step=2,step=0.5,Size=2,size=1,NoZero)); | ||
| + | yaxis(-4,4, RightTicks(Label(fontsize(8pt)),Step=1,step=0.2,Size=2,size=1,NoZero)); | ||
| + | labelx("$x$",(4.6,0.5)); | ||
| + | labely("$y$",(0.3,4.6)); | ||
| + | |||
| + | add(scale(0.5sc,0.75sc)*legend(),(0.6,-2.5)); | ||
| + | |||
| + | real mrg=1.3; | ||
| + | draw( scale(mrg)*box((-4,-4),(4,4)), white ); | ||
| + | |||
| + | add (whole,shift(sc*230,0)*currentpicture.fit(sc*mrg*6.5cm,mrg*6.5cm,false)); | ||
| + | erase(); | ||
| + | |||
| + | shipout(whole); | ||
| + | </asy></center> | ||
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 07:55, 2 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 
.
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,
 |
 |
 |
 |
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=11351