Complex Analysis/Complex Numbers/Introduction

For more details and problems with worked out solutions, please see the Wikibook on complex numbers.

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

$z=x+iy$,

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 $\mathrm{\Re }z$ and $\mathrm{\Im }z$, respectively. Note that for the number $z=3-2i$, $\mathrm{\Im }z=y=-2$, not $-2i$. 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 $z=a+ib$, $w=c+id$,

$z=w$ iff $a=c$ and $b=d$.

Also, we define the standard operators:

$z+w=(a+ib)+(c+id)=(a+c)+i(b+d)$

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

$zw=(a+ib)(c+id)=ac+iad+ibc+i^{2}bd=(ac-bd)+i(ad+bc)$,

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 $\overline{z}$ of a complex number z by

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

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

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

$x=r\cos \theta$, $y=r\sin \theta$

and z may be written in the polar form

$z=r(\cos \theta +i\sin \theta )$.

Here, r must be positive, and θ is unique modulo $2\pi$. 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 $r^2={\left|z\right|}^2=x^2+y^2=z\overline{z}$.

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

$z_1{z}_2$	$=(r_1\mathrm{cis}{\theta }_1)(r_2\mathrm{cis}{\theta }_2)$
	$=r_1{r}_2\left[(\cos {\theta }_1+i\sin {\theta }_1)(\cos {\theta }_2+i\sin {\theta }_2)\right]$
	$=r_1{r}_2[(\cos {\theta }_1\cos {\theta }_2-\sin {\theta }_1\sin {\theta }_2)+i(\sin {\theta }_1\cos {\theta }_2+\cos {\theta }_1\sin {\theta }_2)]$
	$=r_1{r}_2(\cos ({\theta }_1+{\theta }_2)+i\sin ({\theta }_1+{\theta }_2))$
	$=r_1{r}_2\mathrm{cis}({\theta }_1+{\theta }_2)$

by elementary trigonometric identities. Applying this formula can therefore simplify many calculations with complex numbers.

By induction using the previous identity, we can show that

$z^n=r^n\mathrm{cis}(n\theta )$,

which holds for all non-negative numbers.

Now that we have set up the basic concept of a complex number, we continue to topological properties of the complex plane.

Next pt:Análise Complexa/Índice/Números Complexos/Introdução