Advanced Engineering Mathematics, 7th Edition

17.2 Powers and Roots

INTRODUCTION

Recall from calculus that a point (x, y) in rectangular coordinates can also be expressed in terms of polar coordinates (r, θ). We shall see in this section that the ability to express a complex number z in terms of r and θ greatly facilitates finding powers and roots of z.

Polar Form

Rectangular coordinates (x, y) and polar coordinates (r, θ) are related by the equations x = r cos θ and y = r sin θ (see Section 14.1). Thus a nonzero complex number z = x + iy can be written as z = (r cos θ) + i(r sin θ) or

(1)

We say that (1) is the polar form of the complex number z. We see from FIGURE 17.2.1 that the polar ...