MAT 3730Complex Variables

Section 1.4

The Complex Exponential

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Preview

Extension of the exponential function to the complex numbers

The Euler’s Formula The De Moivre’s Formula

(du mwA´vru )

The Complex Exponential

?zx ee

CR

xe

xdtt

x

x

x

ln of inverse

0 ,1

ln1

The Complex Exponential

?zx ee

CR

xe

xdtt

x

x

x

ln of inverse

0 ,1

ln1

function" lexponentia

real theas properties

of kind same theHave"

:IDEA

Basic Property

iyxiyx

zzzz

eee

Ryx

eee

,for ,particularIn

2121

Basic Property

iyxiyx

zzzz

eee

Ryx

eee

,for ,particularIn

2121

real exponentialdefine toneed

Definition of eiy

There are 2 ways to look at the definition of

1. Through the Maclaurin Series

2. Through the property

iye

zz eedz

d

Definition of eiy

There are 2 ways to look at the definition of

1. Through the Maclaurin Series

2. Through the property

iye

zz eedz

d

Through the Maclaurin Series

432

!4

1

!3

1

!2

11 , xxxxeRx x

Suppose we want eiy to have the same Maclaurin series, then

Through the Maclaurin Series

432

!4

1

!3

1

!2

11 , xxxxeRx x

Suppose we want eiy to have the same Maclaurin series, then

2 3 41 1 11 ( ) ( ) ( )

2! 3! 4!iye iy iy iy iy

Through the Maclaurin Series

2 3 41 1 11 ( ) ( ) ( )

2! 3! 4!

cos sin

iye iy iy iy iy

y i y

The Euler’s Formula

yiyeiy sincos

Definition of Complex Exponential

)sin(cos yiyee xz

iyxz

Example 1

Zke ik for 12

Example 2

i

ee

ee

ii

ii

2sin

2cos

Properties of Complex Exponential

1 2 1 2

1 2 1 2

1.

2. /

3. , for

4. 1,

z z z z

z z z z

nz nz

iy

e e e

e e e

e e n

e y

1,3&4

Polar Form (Revisit)

)arg( and , where

sincos

zzr

re

irzi

Example 3

(du mwA´vru) Example 4 De Moivre’s Formula

cos sin cos sin n

i n i n n N

Example 5

Express cos3 in terms of cos and sin .

Next Class

Read Section 1.5 We will look at how to find:Powers zn

Roots z1/m