pdfs week1lecture2

8/13/2019 PDFs Week1Lecture2

1/11

Analysis of a Complex KindWeek 1

Lecture 2: Algebra and Geometry in the Complex Plane

Petra Bonfert-Taylor

Lecture 2: Algebra and Geometry in C Analysis of a Complex Kind


2/11

The Complex Plane

Complex numbers: expressions of the form z=x+iy, where

x is called thereal partof z;x=Re z, andy is called theimaginary partof z;y=Im z.

Set of complex numbers: C (thecomplex plane).

Real numbers: subset of the complex numbers (those whose ima

is zero).

The complex plane can be identified with R2.



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4/11

The Modulus of a Complex Number

Definition

Themodulusof the complex number z=x+iyis the length of the ve

|z|=

x2 +y2.



5/11

Multiplication of Complex Numbers

Motivation: (x+iy) (u+iv) =xu+ixv+iyu+i2yv

So we define:

Definition

(x+iy) (u+iv) = (xu yv) +i(xv+yu) C.

Example:

(3 + 4i)(1 + 7i) = (3 28) +i(21 4) =31 + 17

The usual properties hold:

(z1z2)z3 =z1(z2z3)(associative)z1z2 =z2z1 (commutative)z1(z2+z3) =z1z2+z1z3 (distributive)



6/11

So What isi?

i=0 + 1i,

so

i2 = (0 + 1i)(0 + 1i) = (0 0 1 1) +i(0 1 + 1 0) =1

i3 =i2 i=1 i=i

i4 =i2 i2 = (1)(1) =1i5 =i4 i=i

i6 =1 . . .



7/11

How Do You Divide Complex Numbers?

Suppose thatz=x+iyandw=u+iv. What is z

w (forw=0)?

z

w =

x+iy

u+iv

= (x+iy)(u iv)

(u+iv)(u iv)

= (xu+yv) +i(xv+yu)

u2 +v2 +i(uv+vu)

= xu+yv

u2 +v2 +iyu xv

u2 +v2 .

In particular:1

z =

1

x+iy =

x iy

x2 +y2, as long asz=0.


Th C l C j


8/11

The Complex Conjugate

Note the importance of the quantityx iyin the previous calculation!

Definition

Ifz=x+iy thenz=x iy is thecomplex conjugateof z.

Properties:

z=z

z+w=z+ w

|z|= |z|zz= (x+iy)(x iy) =x2 +y2 =|z|2

1

z =

z

zz =

z

|z|2


M P i f h C l C j


9/11

More Properties of the Complex Conjugate

When isz=z?

z+z= (x+iy) + (x iy) =2x, so

Re z=z+ z

2 , similarly Im z=

z z

2i .

|z w|= |z| |w|zw

=

z

w, (w=0)

|z|= 0 if and only ifz=0.


S I liti


10/11

Some Inequalities

|z| Re z |z|

|z| Im z |z|

|z+ w| |z| + |w|(triangle inequality)

|z w| |z| |w|(reverse triangle inequality)


Th F d t l Th f Al b


11/11

The Fundamental Theorem of Algebra

Theorem

If a0,

a1, . . . ,

anare complex numbers with an=0, then the polynomia

p(z) =anzn +an1z

n1 + +a1z+ a0

has n roots z1, z2, . . . zn inC. It can be factored as

p(z) =an(z z1)(z z2) (z zn).

We will be able to prove this theorem later in this course!

Consider the polynomialp(x) =x2 + 1 in R. It has no real roots!But in C it can be factored: z2 + 1= (z+ i)(z i)!


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