Power Function

power function

[′pau̇·ər ‚fəŋk·shən]
(mathematics)
A function whose value is the product of a constant and a power of the independent variable.
(statistics)
The function that indicates the probability of rejecting the null hypothesis for all possible values of the population parameter for a given critical region.

Power Function

 

the function f(x) = xa, where a is a fixed number. Usually only real values of xa are considered for real values of the base x and exponent a. The function has real values for all x > 0. If a is a rational number with an odd denominator, the function also has real values for all x < 0. If, however, a is a rational number with an even denominator or if a is irrational, then xa has no real values for any x < 0. When x = 0, the power function is equal to 0 for all a > 0 and is undefined for a < 0; 00 has no definite meaning.

Figure 1

The power function is single-valued in the domain of real numbers except when a is a rational number that can be represented by an irreducible fraction with an even denominator. When a is such a rational number, the function is double-valued and assumes values equal in absolute value but opposite in sign for the same value of the argument x > 0. Only the nonnegative value of the function is generally considered in this case. For x > 0, xa is increasing if a > 0 and decreasing if a < 0.

The power function is continuous and differentiable at all points of its domain of definition except at the point x = 0 when 0 < a < 1 (continuity is preserved in this case, but the derivative becomes infinite). The derivative is given by the equation (xa)’ = axa-1. Furthermore,

when a ≠ – 1, and

These two equations hold in any interval in the domain of definition of the integrand.

Functions of the form y = cxa, where c is a constant, play an important role in pure and applied mathematics. When a = 1, such functions express a direct proportion, and their graphs are lines that pass through the origin (see Figure 1). When a = – 1, the functions express an inverse proportion; their graphs are equilateral hyperbolas whose center is at the origin and whose asymptotes are the coordinate axes (see Figure 2).

Figure 2

Many laws of physics are expressed mathematically by functions of the form y = cxa (see Figure 3). For example, y = ex2 expresses the law of uniformly accelerated or decelerated motion. Here, y is the distance traveled, x is the time, and 2c is the acceleration; the initial distance and speed are both 0.

Figure 3

In the complex domain the power function za is defined for all z ≠ 0 by the formula

(*) za = exp a Ln z = exp a(lnǀzǀ + i argz + 2kπi)

where k = 0, ±1, ±2, .... If a is an integer, za is single-valued:

za = ǀzǀa exp ia arg z

Suppose a is rational—that is, a = p/q, where p and q are relatively prime. Then za takes on q distinct values:

(za)k = ǀ z ǀak exp ia arg z

Here, ∊k = exp 2kπi/q are the q th roots of 1: Power Function, and k = 0, 1,. . . , q – 1. If a is irrational, then za has infinitely many values: the factor exp 2kπia takes on distinct values for distinct k. When a is complex, za is defined by the same formula (*). For example,

zi = exp i(In ǀzǀ + i arg z + 2kπi)

= exp (i In ǀzǀ – argz – 2kπ)

In particular, ii = exp (–π/2 – 2kπ), where k = 0, ±1, ±2, ...

The principal value (za)0 of a power function is the function’s value when k = 0 if – π < arg z ≤ π (or 0 ≤ arg z < 2π). Thus,

(za)0 = ǀzaǀ exp ia argz

For example, (i)0 = exp –π/2.

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