Complex Analysis/Complex Numbers/Introduction

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A complex number z is a number written in the form:

z = x + i y

,

where x and y are real numbers and i is the imaginary number $i 2 = - 1$ . We call x the real part and y the imaginary part of z, and denote them by $\Re z$ and $\Im z$ , respectively. Note that for the number $z = 3 - 2 i$ , $\Im z=y=-2$ , not $- 2 i$ . Also, to distinguish between complex and purely real numbers, we will often use the letter z for the complex ones.

We say two complex numbers are equal if and only if their respective parts are equal:

Given

z = a + i b

,

w = c + i d

,

z = w

iff

a = c

and

b = d

.

Also, we define the standard operators:

z + w = (a + i b) + (c + i d) = (a + c) + i (b + d)

z - w = (a + i b) - (c + i d) = (a - c) + i (b - d)

z w = (a + i b)(c + i d) = a c + i a d + i b c + i 2 b d = (a c - b d) + i (a d + b c)

,

because by definition $i 2 = - 1$ .

Commutativity, associativity, and distributivity follow easily from the corresponding laws for real numbers. Also, note that the real numbers are contained within the complex numbers by setting y to 0. Also, the unity from the reals also functions as a unity within the complex number system.

With the introduction of the complex numbers, we can now solve any polynomial equation; i.e., the complex numbers are algebraically closed, and are in fact the closure of the reals.

We define the complex conjugate $\bar z$ of a complex number $z = x + i y$ by

$\bar z=x-iy$ . Also, we define the modulus |z| of a complex number z by

$\left |z\right |^2=z\bar z=(x+yi)(x-yi)=x^2+y^2$ .

We can represent the number z in the complex plane by the point with rectangular coordinates $(x, y)$ . Also, by converting to polar coordinates, we may write

x = r cosθ

,

y = r sinθ

and z may be written in the polar form

z = r (cosθ + i sinθ)

.

Here, r must be positive, and θ is unique modulo $2π$ . When $-\pi<\theta\le\pi$ , we call θ the principal argument and denote it by $arg z$ . Note that we cannot unambiguously define zero in polar coordinates. By the Pythagorean theorem, $r=\sqrt{x^2+y^2}$ (where the root is the positive one), and $r^2=\left |z\right |^2=x^2+y^2=z\bar z$ .

As a shorthand, we may write $\operatorname{cis} \theta = \cos \theta + i\sin \theta$ , so $z=r\operatorname{cis} \theta$ . This notation simplifies multiplication and taking powers, because

$z_1 z_2 \,\!$	$=(r_1 \operatorname{cis} \theta_1)(r_2 \operatorname{cis} \theta_2)$
	$=r_1 r_2 \left [ \left ( \cos \theta_1 + i \sin \theta_1 \right )\left ( \cos \theta_2 + i \sin \theta_2 \right ) \right ]$
	$=r_1 r_2 \left [ \left ( \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \right ) + i \left ( \sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2 \right ) \right ]$
	$=r_1 r_2 \left ( \cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2) \right )$
	$=r_1 r_2 \operatorname{cis} (\theta_1 + \theta_2)$