Power Function

the function f(x) = x^a, where a is a fixed number. Usually only real values of x^a 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 x^a 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, x^a 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 (x^a)’ = ax^a-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 = cx^a, 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 = cx^a (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 z^a is defined for all z ≠ 0 by the formula

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

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

z^a = ǀzǀ^a exp ia arg z

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

(z^a)_k = ǀ z ǀ^a∊_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 z^a has infinitely many values: the factor exp 2kπia takes on distinct values for distinct k. When a is complex, z^a 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 (z^a)₀ of a power function is the function’s value when k = 0 if – π < arg z ≤ π (or 0 ≤ arg z < 2π). Thus,

(z^a)₀ = ǀz^aǀ exp ia argz

For example, (i)₀ = exp –π/2.