01-Complex Numbers

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About the CourseComplex Numbers

OpeningCourse Outline

Opening

In mathematics you dont understand things. You just get used to them.

Johann von Neumann (1903 - 1957)Mathematician, Computer Scientist

Dr. Serkan Gunel 1 / 52


OpeningCourse Outline

Course Outline I

Complex Numbers

Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane


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MAT2001 Complex AnalysisLecture I : Complex Numbers I

Assit.Prof.Dr. Serkan Gunel

Dokuz Eylul UniversityDepartment of Electrical and Electronics Engineering

Izmir, TURKEY

[email protected]

21.09.2011



What is a Complex Number?

Consider the solution of

x2 +b2 = 0, b R (1)

Clearly, there is no x R that solves this equation. Because squares ofreals numbers are always positive!

Question:Can we extend the numbers such that this equation has a solution?


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Definition

Consider the solution of equation

z2 + 1 = 0 = z1,2= 1 (2)

Let us define1i =

1 (3)Then, two possible solutions are

z1,2= i

By same line of reasoning the solutions ofz2 +b2 = 0 are z1,2= ib.

1 = means by definition equal toDr. Serkan Gunel 5 / 52



Now, consider a more general quadratic polynomial of the form

c2z2 +c1z+ c0= 0, c0, c1, c2 R (4)

Then the solution would be:

z1,2 =c1

c21 4c2c0

2c2(5)

Let = c21 4c2c0. If 0

= c12c2

i

2c2=a +ib

Notice that

a= c12c2

R b=

2c2 R.


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Definition of Complex Number

Definition (Complex Number)Acomplex numberis any number of the form z= a +ibwhere a Randb Rand i is theimaginary unit defined as i2 = 1. Set of allcomplex numbers is denoted as C that is

C =

z| z= a +ib, a R, b R, i2 = 1 (6)

Definition (Real and Imaginary Parts)Letz= a +ib, a, b Rbe a complex number. Then a is thereal partandbis theimaginary partofz. We denote these as:

Re(z) = a (7a)

Im(z) = b (7b)




Remarks

In electrical engineering we use j instead ofi in order to denote

imaginary unit. Clearly, z= Re(z) +iIm(z).

Imaginary part ofz= a +ibis b, not ib!

Since any real number x can be considered as z= x+ i 0, R C.


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Equality of Complex Numbers

Definition (Equality of Complex Numbers)Letz1, z2

C, then z1= z2 if and only if

Re(z1) = Re(z2), and Im(z1) = Im(z2) (8)

RemarkNote that comparison of two complex numbers are not defined!Operators>,


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Properties of Arithmetic Operators

Forz1, z2, z3 Cfollowing properties hold

Addition is commutative: z1+z2= z2+z1 (13)

Multiplication is commutative: z1 z2 = z2 z1 (14)Addition is associative: z1+ (z2+z3) = ( z1+z2) +z3 (15)

Multiplication is associative: z1 (z2 z3) = ( z1 z2) z3 (16)Distributive Law: z1(z2+z3) = ( z1

z2) + (z1

z3) (17)

We also have:

Additive identity: z1+ (0 +i0) = z1 (18)

Additive inverse: z1+ (z1) = (0 +i0) (19)Multiplicative identity: z1 (1 +i0) = z1 (20)Multiplicative inverse: z1 (1

z1) = (1 +i0) (21)

Besides additive identity is multiplicative null element, i.e.z1 (0 +i0) = (0 +i0). Therefore complex numbers forms afieldwithdefined addition and multiplication operations!




Conjugate Operation

Definition (Complex Conjugate)Ifz= a +ib, thencomplex conjugateofz is z= a ib.PropertiesIfz, z1, z2 C then

z= z (22)

z1+z2 = z1+z2 (23)

z1 z2 = z1 z2 (24)z1

z2

=

z1

z2(25)

Re(z) = z+ z

2 , Im(z) =

z z2i

(26)

z1

z2=

z1z2

z2z2(27)


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Complex Plane

x-axis

Real axis

y-axisImaginary axis

z= a +ib

(a, b)

b

a

zplane

|z|

Another representation for a complexnumber z= a +ibis (a, b).

A complex number can also be viewedas a vector.

Indeed, C is avector spacewith

scalars are taken from R. Let, 1, 2 Rand z, z1, z2 C:

Clearly, associativity, commutativity,identity element, and inverse elementproperties of addition is already satisfied.We also have,

(z1+z2) = z1+z2,

(1+2)z= 1z+2z,

1 (2 z) = (1 2)z,1 z= z




Remarks

Since C is a vector space over scalar field R, they share the sameproperties of all (abstract) vector spaces.

C is a field with defined addition and multiplication operators.Therefore, the most important result is we can use all linearalgebraic concepts used in real spaces in C, too.

You can use matrix algebra with matrices whose elements are takenfrom C and apply the same rules. Determinant rules, Cramers rule,etc. apply to complex numbers, too.


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Complex Plane

x-axis

Real axis

y-axisImaginary axis

z= x+ iy

(x, y)

y

x

zplane

|z|

Definition (Modulus, Absolute Value)Themodulusorabsolute valueof acomplex number z= x+ iy, is the realnumber

|z| = x2 +y2 (28)

PropertiesFor all z, z1, z2 C

|z| 0,|z| = 0 z= 0 (29)|z|2 =z z, |z| =

z z (30)

|z1 z2| = |z1| |z2| (31)

z1

z2

=|z1||z2|

(32)

|z1+z2| |z1| + |z2| (33)




Addition and subtraction in complex plane

Re

Im

z1

z2|z2|

z= z1+z2

|z1+

z2|

|z

1

|

z2

w= z1 z2


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Conjugation in complex plane

Re

Im

z= x+ iy

z= x iyz= x iy

z= x+ iy

z+ z

z z




Example - Complex Linear Algebraic System of Equations

Solve

z1+iz2+z3 = 1

iz1 z3 = 0z2+ (i 1)z3 = 0

Rearranging in matrix form we have

Az= b 1 i 1i 0 1

0 1 (i 1)

z1z2

z3

=

10

0

det(A) =

1 i 1i 0 10 1 (i 1)

= 1

0 11 (i 1) (i)

i 11 (i 1) + 0

i 10 1


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Example - Complex Linear Algebraic System of Equations

det(A) = 1 0 11 (i 1)

(i) i 11 (i 1)

+ 0 i 10 1

= 1 (0 (i 1) (1 1)) +i (i (i 1) 1 1) = 2 2i

Using Cramers rule we have

1 =

1 i 10 0 10 1 (i 1)

= 1 0 11 (i 1)

= 1 z1 = 1 = 1

2 2i =1

4(1 +i)

2 =

1 1 1i 0 10 0 (i 1)

= 1 i 10 (i 1)

= 1 i z2 = 2 =1 i

2 2i = i

2

3 =

1 i 1i 0 00 1 0

= 1 i 00 1

= i z3 = 3 = i2 2i =

1

4(1 i)




Polar form of a complex number

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Re

Im

r=|z

|

z

x= rcos()

y

=r

sin

()

Let z= x+ iy, define

r = |z| =

Re

2(z) +Im2(z) (34)

=

x2 +y2

= arctanIm(z)Re(z)

(35)

= arctany

x

(36)

x= rcos() (37)

y= rsin() (38)

z= rcos() +i rsin()

=r(cos() +isin()) (39)

=r cis() (40)

=r (41)


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Argument, Principle Argument

Definition (Argument)Letz= r(cos() +isin()) = r , then is called the argumentof thecomplex number. We denote the argument as:

arg(z) = + 2k, k= 0, 1, 2, . . . (42)

Remarkarg(z) is multi-valued.

Definition (Principle Argument)The argument of a complex number z that lies in the interval < is called theprincipal valueof arg(z) orthe principalargument ofz, and denoted as Arg(z). Clearly,

< Arg(z) (43)arg z= Arg(z) + 2k, k= 0, 1, 2, . . . (44)




Multiplication in Polar Form

Letz1 = r1(cos(1) +isin(1)) and z2= r2(cos(2) +isin(2)) . Then

z1z2= [r1(cos(1) +isin(1))] [r2(cos(2) +isin(2))] (45)

=r1r2[(cos(1)cos(2) sin(1)sin(2)) +i(cos(1)sin(2) + sin(1)cos(2))] (46)

=r1r2(cos(1+2) +isin(1+2)) (47)=r1r2 1+2 (48)

= |z1| |z2| 1+2 (49) cos(A B) = cos(A) cos(B) sin(A) sin(B)

sin(A B) = cos(A) sin(B) sin(A) cos(B)In other words

|z1z2| = |z1| |z2| (50)arg(z1z2) = arg(z1) + arg(z2) (51)


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Division in Polar Form

Letz1= r1(cos(1) +isin(1)) andz2 = r2(cos(2) +isin(2)) . Then

z1

z2=

r1(cos(1) +isin(1))

r2(cos(2) +isin(2)) (52)

=r1(cos(1) +isin(1))

r2(cos(2) +isin(2)) cos(2) isin(2)

cos(2) isin(2) (53)

=

r1

r2

cos(1) cos(2) + sin(1)sin(2)

i(cos(1)sin(2)

sin(1) cos(2))

cos2(2) + sin2(2)(54)

=r1

r2(cos(1 2) +isin(1 2)) (55)

=r1

r21 2 =|z1||z2| 1 2 (56)z1z2

=|z1||z2| (57)arg

z1

z2

= arg(z1) arg(z2) (58)




Remarks

Addition and subtraction are easier in Cartesian form, howevermultiplication and division are easier in polar form.

Steinmetz notation z= r is usually preferred in electricalengineering.

z= r cis() notation is not preferred much. Ifz= 0, arg(z) is not defined!

Arg(z1z2) = Arg(z1) + Arg(z2) and Arg(z1/z2) = Arg(z1) Arg(z2),in general !

With addition, subtraction, multiplication and division operationsacting on complex numbers we can generate all geometricoperations on plane. Scaling, Translation, Rotation.


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1 2 3 4 0

15

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z

ii z

w

z2

Multiplication by i rotates z, 90

counter clockwise (ccw)!




Powers of Complex Numbers

Letz= r(cos() +isin()) = r , then

z2 =r r += r2 2 (59)z3 =zz2 =r r2 + 2= r3 3 (60)

... (61)

zn =zzn1 =r rn1 + (n 1)

zn =rn n (62)

=rn (cos(n) +isin(n)) (63)


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de Moivres Formula

Letz= (cos() +isin()) = 1 , then

zn = (cos() +isin())n

= 1n n (64)

= (cos(n) +isin(n)) (65)

(cos() +isin())n = (cos(n) +isin(n)) (66)




Negative Powers of Complex Numbers

Letz= r(cos() +isin()), then

z1 =1

z

=1 0

r

=1

r = r1(cos()

isin()) (67)

z2 = 1

z2 =

1 0

r2 2= r2(cos(2) isin(2)) (68)

... (69)

zn = 1

zn =

1 0

rn n =rn(cos(n) isin(n)) (70)


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Roots of Complex Numbers

We define nth root of a complex number z as the complex number wthat satisfies:

z= wn (71)

Letz= r(cos() +isin()) and w= (cos() +isin())

z= wn (72)

r(cos() +isin()) = n (cos(n) +isin(n)) (73)

= r= n

+ 2k= n k= 0, 1, 2, . . .

Solving for and yields

= n

r (74)

= + 2k

n , k= 0, 1, 2, . . . n 1 (75)

Therefore we have n roots

wk= n

r

cos

+ 2k

n

+isin

+ 2k

n

, k= 0, 1, 2, . . . , n 1

(76)




Definition (Principle nth Root)The unique root of a complex number z(obtained by using the principal

value of arg(z) with k= 0) is naturally referred to as the principal nth

rootofz.The choice of Arg(z) and k= 0 guarantees us that when zis a positivereal number r, the principal n th root is n

r.


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Roots of Unity

Letz= 1, then

z= w3 = 1 0

wk= n

r

cos

+ 2k

n

+ isin

+ 2kn

k= 0, 1, 2

wk= 3

1

cos

2k

3

+ isin

2k

3

k= 0, 1, 2

0.5 1 1.5 2 0

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zw0

w1

w2

120

120

120

w0 = 1 0, Principle cube root

w1 = cos

23

+isin

23

= 1 120,

w2 = cos

43

+isin

43

= 1 120




Quadratic Equations

Consider the solution (roots) of the following quadratic equation

az2 +bz+ c= 0, a, b, c

C (77)

By direct substitution we can verify that the solution is

z=b+ b2 4ac1/2

2a (78)

Sinceb2 4ac C, we have two roots as in real analysis.


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Example

Solve

z2 + (1 i)z 3i= 0.Applying the quadratic root formula

z=b+ b2

4ac1/2

2a =(1

i) + ((1

i)2 + 4

3i)1/2

2

witha = 1, b= (1 i), c= 3i. = b2 4ac= (1 i)2 + 4 3i= 14i= 1490

To get the roots of we apply the complex root formula:

w2 = = 1490wk=

14 90+2k2 , k= 0, 1




= w0=

1445 =

7(1 i)w1=

14 135 =

7(

1 +i)

z1=b+w0

2a =

(1 i) + 7(1 i)2

=7 1

2 (1 i)

z2=b+w1

2a =

(1 i)

7(1 i)2

=1 7

2 (1 i)


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Lines in Complex Plane

We can define lines on complexplane in several ways:

C1= {z| Re((m+i)z) +b= 0, b R}(79)where m is the slope and bis the

translation.

C2= {z| |z z1| = |z z2| , z1, z2 C}(80)

defines the set of points whosedistance to z1 and z2 is equal.

Re

Im

C1

mb

z1

z2 C2




Circles in Complex Plane

The set of points:

C= {z| |z zo|= , > 0, zo C}(81)

defines a circle in zplane whose radiusis and center is zo.

To see this let z= x+ iy andzo= xo+ iyo

|z zo|= |(x xo) + i(y yo)|=

=

(x xo)2 + (y yo)2 =

2 = (x xo)

2 + (y yo)2

Re

Im

zo

2 = (Re(z) Re(zo))2 + (Im(z) Im(zo))2 (82)


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Disks in Complex Plane

The set of points:

R= {z| |z zo| , >0, zo C}(83)

defines aclosed disk with radius and

center zoin zplane.

R= {z| |z zo| < , >0, zo C}(84)

defines aopen disk with radius andcenter zoin zplane.

Re

Im

zo




Definitions about Set of Points in C

Definition (Neighborhood)The set{z | |z zo| < } is called neighborhoodofzo.The{z | 0 < |z zo| < } is calleddeleted neighborhoodofzo or a

punctured disk. Clearly, it does not includez

o.Definition (Interior Point)A point zois said to be an interior point of a set Sof the complex planeif there exists some neighborhood ofzo that lies entirely within S.

Definition (Open Set)If every point zof a set S is an interior point, then S is said to be anopen set.


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Some Open Sets

Re

Im

Open Left-half Plane

Re(

z)

0

Re

Im

Im(z)

>0

Re

Im

Re(

z)

>

a

Re

Im

Open Unit Disc

1

Re

Im

Arbitrary open SetS

S




Definitions about Set of Points in C II

Definition (Boundary Point)If every neighborhood of a point zoof a set S contains at least one pointofSand at least one point not in S, then zo is said to be a boundarypointofS.

Definition (Boundary)The collection of boundary points of a set S is called theboundaryofS,and denoted withS.

Definition (Exterior Point)A point zthat is neither an interior point nor a boundary point of a setSis said to be anexterior pointofS; in other words, zo is an exterior pointof a set S if there exists some neighborhood ofzo that contains no pointsofS.


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Boundary, Interior, Exterior

Re

Im

S

SInte

rior

Exterior

Boundary Point




Definitions about Set of Points in C III

Definition (Connected Set)If any pair of points z1 and z2 in a set Scan be connected by a polygonalline that consists of a finite number of line segments joined end to end

that lies entirely in the set, then the setS

is said to be connected.Definition (Domain)An open connected set is called a domain.

Definition (Region)Aregionis a set of points in the complex plane with all, some, or none ofits boundary points. Since an open set does not contain any boundarypoints, it is automatically a region.


Notes

Notes

D fi i i d P i


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Connected Set, Disconnected Set

Re

Im

S

z1

z2

Connected Set

Re

Im

S1S2

z1

z2

Disconnected Set




Definitions about Set of Points in C IV

Definition (Closed Set)A region that contains all its boundary points is said to be closed.

Definition (Bounded Set)A set Sin the complex plane is boundedif there exists a real numberR> 0 such that|z|


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Bounded Set, Unbounded Set

Re

Im

R

|z|=

R

Sbou

nde

d

Sunbounded

R




Annulus

Let 0< 1< 2, then the set

R= {z | 1< |z zo| < 2 } (85)

is called an opencircular annulus.Similarly,

R= {z | 1 |z zo| 2 } (86)

is a closed annulus. zo1

2

Re

Im

R


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Definition and Properties


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Extended Complex Plane

There is a one to onecorrespondence between the pointon the real line and the unit circle.

x

y

R2

a1

(x1, y1)

a2

(x2, y2)

a3

(x3, y3)

(0, 1)

a1 (x1, y1)a2 (x2, y2)a3 (x3, y3)

(0, 1)

Similarly, we can map the complexplane to unit sphere.

(0, 0, 1)

|z| = (0, 0, 1);When|z| = included, the plane iscalledextended complex plane.


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01-Complex Numbers

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