Complex Numbers
We develop the properties of the number system called the complex numbers. We also describe derivatives and integrals of some basic functions of complex numbers.
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Complex Numbers
Abstract We develop the properties of the number system called the complex numbers. We also describe derivatives and integrals of some basic functions of complex numbers.
9.1 Complex Numbers Most people first encounter complex numbers as solutions of quadratic equations x2 + bx + c = 0, that is, zeros z of the function f (x) = x2 + bx + c. Example 9.1. Take as an example the equation x2 + 1 = 0. The √ √ quadratic formula for the roots gives z = ± −1. There is no real number −1. We introduce a √ new number i = −1. Definition 9.1. A complex number z is defined as the sum of a real number x and a real multiple y of i, z = x + iy, where i denotes a square root of minus one, i2 = −1. The number x = Re (z) is called the real part of z, and the real number y = Im (z) is called its imaginary part. A complex number whose imaginary part is zero is called (naturally enough) real. A complex number whose real part is zero is called purely imaginary. The complex conjugate of z is z = x − iy. You might be wondering how solving an equation such as r2 + 1 = 0 might arise in calculus. Consider the differential equation y + y = 0. We know that sin x and cos x solve the equation. But what about y = erx ? The second derivative of y = erx is r2 times y, so y = erx solves the equation P.D. Lax and M.S. Terrell, Calculus With Applications, Undergraduate Texts in Mathematics, 347 DOI 10.1007/978-1-4614-7946-8 9, © Springer Science+Business Media New York 2014
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9 Complex Numbers
y + y = (r2 + 1)erx = 0 if r solves r2 + 1 = 0. We will see in this chapter that eix , sin x, and cos x are related, and we will see in Chap. 10 that functions of complex numbers help us solve many useful differential equations. Complex numbers also have very practical applications; they are used to analyze alternating current circuits.1
9.1a Arithmetic of Complex Numbers We now describe a natural way of doing arithmetic with complex numbers. To add complex numbers, we add their real and imaginary parts separately: (x + iy) + (u + iv) = x + u + i(y + v). Similarly, for subtraction, (x + iy) − (u + iv) = x − u + i(y − v). To multiply complex numbers, we use the distributive law: (x + iy)(u + iv) = xu + iyu + xiv + iyiv. Rewrite xi as ix and yi as iy, since we are assuming that multiplication of real numbers and i is commutative. Then, since i2 = −1, we can write the product above as (xu − yv) + i(yu + xv). Example 9.2. Multiplication includes squaring: (−i)2 = −1,
(3 − i)2 = 9 − 6i + i2 = 8 − 6i,
(5i)2 = −25.
y x + iy x = + i . To It is easy to divide a complex number by a real number r: r r r x + iy express the quotient of two complex numbers as a complex number in the u + iv form s + it, multiply the numerator and denominator by the complex conjugate2 u + iv = u − iv. We get x + iy (x + iy)(u − iv) xu + yv + i(yu − xv) xu + yv yu − xv = = = 2 +i 2 . 2 2 2 u + iv (u + iv)(u − iv) u +v u +v u + v2 Notice that the indicated division by u2 + v2 can be carried out unless both u and v are zero. In that case, the divisor u + iv is zero, so
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