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: , 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.

References in periodicals archive ?
The VSA channel power function makes it simple to measure signal power automatically.
For the Trécé wing trap, using the truncated dataset, the relationship between the saturating and nonsaturating traps is a power function described by the equation
As rockfill creep under low confining pressure laboratory tests cannot reflect the behavior of 200 m high CFRDs, Cheng and Ding [8] proposed a power function rockfill creep curve model under high confining pressure.
The voltage in both light towers allows operators to utilize the receptacle power function to simultaneously power tools or heaters while providing optimal illumination for oil and gas exploration, construction sites, emergency and disaster relief, as well as drilling applications.
Figure 5 also indicated that the UCS values computed from the linear and exponential functions lie above the 1:1 line in majority of the cases; whereas all the UCS values calculated from power function lie below the 1:1 line.
Two-dimensional curve is interpolated via different functions as probability distribution functions: polynomial, sinus, cosine, tangent, cotangent, logarithm, exponent, arcsin, arccos, arctan, arcctg or power function, also inverse functions.
The smart meter's third power function is its ability to safely connect and disconnect the remote customer's power during a shortage of power sources.
Their most frequently occurring appearance from the waterline to the depth of closure can be adequately described using a power function
16) have established recurrence relations for marginal and joint moment generating functions of lgos from power function and generalized exponential distributions.
It shows how a power function can explain user experience changes in relation to the QoS parameter fluctuations.
Second, although Julious and Owen (2006) studied the prescribed problem of determining the sample size under the notion of expected power, their analytical exposition for the overall or expected power function is complicated and does not conform to the adjusted sample variance approach.
Curves of soil water retention and hydraulic conductivity were obtained by power function, Van Genuchten-Maulem and Durner-Maulem models.

Site: Follow: Share:
Open / Close