power function[′pau̇·ər ‚fəŋk·shən]
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.
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).
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.
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 ǀa∊k 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.