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
About the CourseComplex Numbers
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
Dr. Serkan Gunel 2 / 52
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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
21.09.2011
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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?
Dr. Serkan Gunel 4 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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.
Dr. Serkan Gunel 6 / 52
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Notes
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 7 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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.
Dr. Serkan Gunel 8 / 52
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Notes
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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!
Dr. Serkan Gunel 11 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 12 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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
Dr. Serkan Gunel 13 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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.
Dr. Serkan Gunel 14 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 15 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
Addition and subtraction in complex plane
Re
Im
z1
z2|z2|
z= z1+z2
|z1+
z2|
|z
1
|
z2
w= z1 z2
Dr. Serkan Gunel 16 / 52
Notes
Notes
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
Conjugation in complex plane
Re
Im
z= x+ iy
z= x iyz= x iy
z= x+ iy
z+ z
z z
Dr. Serkan Gunel 17 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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
Dr. Serkan Gunel 18 / 52
Notes
Notes
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 19 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 20 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 21 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 22 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 23 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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.
Dr. Serkan Gunel 24 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
1 2 3 4 0
15
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45
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z
ii z
w
z2
Multiplication by i rotates z, 90
counter clockwise (ccw)!
Dr. Serkan Gunel 25 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 26 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 27 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 28 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 29 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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.
Dr. Serkan Gunel 30 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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
Dr. Serkan Gunel 31 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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.
Dr. Serkan Gunel 32 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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
Dr. Serkan Gunel 33 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
= 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)
Dr. Serkan Gunel 34 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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
Dr. Serkan Gunel 35 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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)
Dr. Serkan Gunel 36 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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
Dr. Serkan Gunel 37 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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.
Dr. Serkan Gunel 38 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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
Dr. Serkan Gunel 39 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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.
Dr. Serkan Gunel 40 / 52
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About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
Boundary, Interior, Exterior
Re
Im
S
SInte
rior
Exterior
Boundary Point
Dr. Serkan Gunel 41 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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.
Dr. Serkan Gunel 42 / 52
Notes
Notes
D fi i i d P i
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8/13/2019 01-Complex Numbers
22/24
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
Connected Set, Disconnected Set
Re
Im
S
z1
z2
Connected Set
Re
Im
S1S2
z1
z2
Disconnected Set
Dr. Serkan Gunel 43 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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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8/13/2019 01-Complex Numbers
23/24
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
Bounded Set, Unbounded Set
Re
Im
R
|z|=
R
Sbou
nde
d
Sunbounded
R
Dr. Serkan Gunel 45 / 52
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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
Dr. Serkan Gunel 46 / 52
Notes
Notes
Definition and Properties
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8/13/2019 01-Complex Numbers
24/24
About the CourseComplex Numbers
Definition and PropertiesArithmetic Operations, Complex ConjugateComplex PlanePolar form of a complex numberPower and Roots of Complex NumbersSet of Points in Complex Plane
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.
Dr. Serkan Gunel 47 / 52
Notes
Notes