Chapter 13.   Complex Numbers and Function & Complex Differentiation


2018학년도 2학기

       반도체 공학과 공학수학2 (GEDB005) 강의교안  (2018학년도용)

    주교재: Erwin Kreyszig, Engineering Mathematics10th Edition

    부교재: 최신공학수학 I 과 II, 한빛출판사 및 (영문) 다변량 Calculus (이상구, 김응기 et al)   (http://www.hanbit.co.kr/EM/sage/)

    강의시간: E12 (금 12:00 - 14:45), 반도체관 400126호 담당교수: 김응기 박사 

 

주차

주교재

부교재

9

13.1: Complex Numbers. Complex Plane

13.2: Polar Form of Complex Numbers. Powers and Roots

13.3: Derivative. Analytic Function

13.1 복소수

13.2 복소함수

web

http://www.hanbit.co.kr/EM/sage/2_chap13.html 

 


   Week 9


Chapter 13. Complex Numbers and Function

Complex Differentiation



13.1 Complex Numbers and Their Geometric Representation

 

complex number  is an ordered pair  of  and .

     .

            is the real part of 

            is the imaginary part of 


Two complex numbers  and  are equal.

                     


(1)                         is the imaginary unit.


Addition, Multiplication. Notation 

        

Addition of two complex numbers  and  is

(2)                     .


Multiplication of two complex numbers  and  is

(3)                   .


For ,

           and .


The complex numbers “extend” the real numbers.

We can thus write

           .   Similarly    

because by  and the definition of multiplication we have

           .

Complex numbers  are 

(4)          .


Electrical engineers often write  instead if  because they need  for the current.


                   is pure imaginary.

Also,  and  give

(5)         .      the definition of multiplication.



Addition of two complex numbers  and  is 

       .


Multiplication of two complex numbers  and  is

     .


Example 1  Real Part, Imaginary, Sum and Product of Complex Numbers

Let  and .

     .

     ,

     .


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

8

3

9

-2

I + 17

11*I + 78


SubtractionDivision


Subtraction is defined as the inverse operations of addition.

Division is defined as the inverse operations of multiplication..


Difference of two complex numbers  and  is 

(6)                  .


Quotient of two complex numbers  and  is

(7)           

By multiplying numerator and denominator of  by 


Example  2  Difference and Quotient of Complex Numbers

      and  .

      and

     .


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

5*I - 1

43/85*I + 66/85


Complex Plane


Two perpendicular coordinate axes,

The horizontal axis is the real axis.

The vertical axis is the imaginary axis.

The real axis and imaginary axis have the same unit of length.

This is a Cartesian coordinate system.


A given complex number  as the point  with coordinate .

The -plane in which the complex numbers are represented is the complex plane.

     그림입니다.

        The complex plane


The point represented by  in the complex plane  The point  in the complex plane


Visualization of addition and subtraction

     그림입니다.                   

     Addition of complex numbers      Subtraction of complex numbers



Complex Conjugate Numbers


 is the complex conjugate of a complex number .

   Obtained by reflecting the point  in the real axis.


Ex.  

     그림입니다.

     Fig.  Complex conjugate numbers



Complex conjugate is important because it permits us to switch complex to real.


     .

     

.            real part  of 

.            imaginary part  of 


 is real, ,       by the definition of .


(9)             .


Example  3  Illustration of  and 

Let  and . Then by ,

     .

The multiplication formula in  is verified by

     ,

     .


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

4

3

-3*I + 4

-5*I + 2

-26*I - 7

-26*I - 7

True 



13.2 Polar From of Complex Numbers. Powers and Roots


The polar coordinates  of a point are related to the rectangular coordinates  by the equations

                           

                      

                      

     


To convert from polar coordinates  to rectangular coordinates ,

(1)                         ,   .


Polar form of  is

(2)                  .

 is the absolute value or modulus of .

Hence

(3)                       .

                                              

        그림입니다.       그림입니다.

           Complex plane, polar form            Distance between two

           of a complex number                 points in the complex plane

 

 is the distance of the point  from the origin.

 is the distance between  and .


 is the argument of .

(4)                                .

 is the directed angle from the positive -axis to .

All angles  are measured in radians and positive in the counterclockwise sense.

Define the principal value  of  by the double inequality

(5)                                  .

Positive real       

Negative real       

For a given , the other values of  are ().



Example 1  Polar Form of Complex Numbers. Principal Value 

     .

     ,

     (the principle value).

      and .

     그림입니다.


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

sqrt(2)

1/4*pi

sqrt(2)*e^(1/4*I*pi)

I + 1


Triangle Inequality

The triangle inequality of the complex

(6)                                .

        그림입니다.

         Fig. 326 Triangle inequality


Generalized triangle inequality

(6*)                       

Absolute value of a sum cannot exceed the sum of the absolute values of the terms.



Example 2  Triangle Inequality

If  and  then .


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

True


Multiplication and Division in Polar Form

Let  and .

Multiplication 

    

    

    

Absolute value of a product equals the product of the absolute values of the factors,

                   .

Argument of a product equals the sum of the arguments of the factors,

(9)                   (up to multiples of )



Division

   

   

(12)                        

Absolute value of a division equals the product of the absolute values of the factors,

(10)                                  .

Argument of a division equals the difference of the arguments of the factors,

(11)                    (up to multiples of )


Example 3  Illustration of Formulas 

Let  and .

       ,

       .

Then 

.

For the arguments we obtain

       ,

       

       .

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

z1*z2 =  -6*I - 6

z1/z2 =  2/3*I + 2/3

|z1| =  2*sqrt(2)

|z2| =  3

|z1*z2| =  6*sqrt(2)

|z1/z2| =  2/3*sqrt(2)

arg(z1) =  3/4*pi

arg(z2) =  1/2*pi

arg(z1*z2) =  -3/4*pi

arg(z1/z2) =  1/4*pi



Example 4  Integer Powers of . De Moivre’s Formula

Form  and  with  we obtain by induction for 

(13)                             .

Similarly,  with  and  gives for .

For , formula  becomes De Moivre’s Formula

(13*)                        .


We can use this to express  and  in terms of  and .

For ,

.

.



Roots

       Each value of  there corresponds one value of .

(14)                              an  root of 

Hence this symbol is multivalued, namely, -valued.

The  values of  can be obtained as follows. We write  and  in polar from

        and   .

Then the equation  becomes, by De Moivre’s formula (with  instead of )

      .

               where  is positive real.

               where  is an integer.


For   we get  distinct values of .

Further integers of  would give values already obtained.

For instance,  gives , hence the corresponding to , etc.


Consequently, , for , has the  distinct values

(15)              ,  where .

These  values lie on a circle of radius  with center at the origin and constitute the vertices of a regular polygon of  sides.

The values of  obtained by taking the principal value of  and  in  is called the principal value of .

Taking  in , we have  and . Then  gives

(16)                     ,    .

These  values are called the th roots of unity. They lie on the circle of radius  and center , called the unit circle.

 

Show that .

We write  and  in polar from

        and   .

      

            

            ,  

      

            

            

            

   그림입니다.

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[w == 1/2*I*sqrt(3) - 1/2, w == -1/2*I*sqrt(3) - 1/2, w == 1]




Evaluate

1

1/2*I*sqrt(3) - 1/2

-1/2*I*sqrt(3) - 1/2


Show that .

We write  and  in polar from

        and   .

      

            

            ,  

      

            

            

            

            

그림입니다.


Show that .

We write  and  in polar from

     and   .

   

         

         ,  

   ,  

Hence 

.

     

                                                            

If  denotes the value corresponding to  in , then the  values of  can be written

                                  .

More generally, if  is any  root of an arbitrary complex number , then the  values of  in  are

(17)                            ,

because multiplying  by  corresponds to increasing the argument of , by .

Formula  motivates the introduction of roots of unity and shows their usefulness.

 

Example

Calculate the square root of 5 of .

Solution

Taking , the square root of  of  is

                      .

    Hence 

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[x == (-sqrt(3) + I)^(1/5)*e^(2/5*I*pi), x == (-sqrt(3) + I)^(1/5)*e^(4/5*I*pi), x == (-sqrt(3) + I)^(1/5)*e^(-4/5*I*pi), x == (-sqrt(3) + I)^(1/5)*e^(-2/5*I*pi), x == (-sqrt(3) + I)^(1/5)]




Evaluate

1/2*(sqrt(3) + I)*2^(1/5)

(I*sin(17/30*pi) + cos(17/30*pi))*2^(1/5)

(I*sin(29/30*pi) + cos(29/30*pi))*2^(1/5)

(I*sin(41/30*pi) + cos(41/30*pi))*2^(1/5)

(I*sin(53/30*pi) + cos(53/30*pi))*2^(1/5)


Example

Prove that 

.

Solution

                              

                             

                              

                              

     

     


13.3. Derivative. Analytic Function



Unit Circle : 

       그림입니다.

General circle of radius  and center 

          .

The set of all  whose distance  from the center  equals .

    그림입니다.


 is interior (“open circular disk”)

 is interior plus the circle itself (“closed circular disk”)

 is exterior.



Open circular disk  is a neighborhood of .

                                  is a -neighborhood of .

Any set containing -neighborhood of  is a neighborhood of .


Closed circular disk : 

Exterior : .

Neighborhood of  : An open circular disk 


Open annulus (circular ring) : .

This is the set of all  whose distance from a is greater than  but less than .

Closed annulus : .

      그림입니다.


Open circular disk  is a neighborhood of  or, -neighborhood of .

Any set containing -neighborhood of  is a neighborhood of .

 

Half-Planes


The set of all points .

Upper half-plane : 

Lower half-plane : 

Right half-plane : 

Left half-plane : 



Point set 

Collection of finitely many of infinitely many points.

Examples : the solution of a quadratic equation, the points of a line, the [point in the interior of a circle.


Set  is open 

Every point of  has a neighborhood consisting entirely of points that belong to .

Example, the points in the interior of a circle or a square from an open set, and so do the points of the right half-plane .


Set  is connected 

Any two of its points can be joined by a broken line of points that belong to .


Domain

Open connected set is a domain.


Complement of a set  in the complex plane

Set of all points of the complex plane that do not belong to .


 is closed

Its complement is open.

Example, the points on and inside the unit circle from a closed set (“closed unit disk”) since its complement  is open.


Boundary point of a set 

A point every neighborhood of which contains both points that belong to  and points that do not belong to .

Example, the boundary points of an annulus are the points on the two bounding circles.

Clearly, if a set  is open, then no boundary point belongs to  if  is closed, then every boundary point belongs to .


Boundary of 

Set of all boundary points of a set .


Region

Set consisting of a domain plus some or all of its boundary points.

 

Complex function


 is a set of complex numbers. And a function  defined on  is a rule that assigns to every  a complex number (the value of  at ).

.

Here,  varies in  and  is a complex variable.

Set  is the domain of definition of  or, briefly, the domain of .

Set of all values of a function  is the range of .


Example 1  Function of a Complex Variable

Let . Find  and  and calculate the value of at .

Solution

   

    and .

   

    and .

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

x^2 + 2*I*x*y - y^2 + 3*x + 3*I*y

15*I - 5



Example 2  Function of a Complex Variable

Let . Find  and  and calculate the value of at .

Solution

    

    and .

   .

    and .


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

2*I*x - 2*y + 6*conjugate(x) - 6*I*conjugate(y)

-23*I - 5



Limit, Continuity

Limit

(1)                         .

                                have the limit  as  approaches a point 

(2)                            .

      그림입니다.

                        Fig.  Limit


Continuity

(3)                            and  is defined.

                                 Function  is continuous at 


Derivative

The derivative of a complex function  at a point  is written  and is defined by

(4)                     

                            derivative of  at a point 

Then  is differentiable at .

(4')                      .      ,  



Example 3  Differentiability, Derivative

Function  is differentiable for all       derivative 

.


The differentiation rules

Analytic function  and  and constant ,

      

Chain rules and the power rule ().


       is differentiable at        is continuous at .


Example 4   not Differentiable

It may come as a surprise that there are many complex functions that do not have a derivative at any point.

For instance,  is not differentiable.

To see this, we write  and obtain

(5)   

                         

 do not have a derivative at any point .

      그림입니다.

           Fig.  Paths in 



Definition  : Analyticity

A function  is said to be analytic in a domain  if  is defined and differentiable at all points of .

The function  is analytic at a point  in  if  is analytic in a neighborhood of .

Also, by an analytic function we mean a function that is analytic in some domain.

A  more modern term for analytic in  is holomorphic in .



Example 5  Polynomials, Rational Functions

The nonnegative integer powers  are analytic in the entire complex plane, Polynomials are

            

where  are complex constants.


Quotient two polynomials  and ,

           

is a rational function.

Rational function  is analytic except at the points .

 

 

   http://matrix.skku.ac.kr/sglee/ 


[한빛 아카데미] Engineering Mathematics with Sage:

[저자] 이상 구, 김영 록, 박준 현, 김응 기, 이재 화


Contents

 A. 공학수학 1 – 선형대수, 상미분방정식+ Lab

Chapter 01 벡터와 선형대수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-1.html 


Chapter 02 미분방정식의 이해 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-2.html 

Chapter 03 1계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-3.html    

Chapter 04 2계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-4.html 

Chapter 05 고계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-5.html 

Chapter 06 연립미분방정식, 비선형미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-6.html 

Chapter 07 상미분방정식의 급수해법, 특수함수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-7.html  

Chapter 08 라플라스 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-8.html 

 

B. 공학수학 2 - 벡터미적분, 복소해석 + Lab

Chapter 09 벡터미분, 기울기, 발산, 회전 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-9.html 

Chapter 10 벡터적분, 적분정리 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-10.html 

Chapter 11 푸리에 급수, 적분 및 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-11.html 

Chapter 12 편미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-12.html

Chapter 13 복소수와 복소함수, 복소미분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-13.html

Chapter 14 복소적분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-14.html

Chapter 15 급수, 유수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-15.html

Chapter 16 등각사상 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-16.html

 





Copyright @ 2018 SKKU Matrix Lab. All rights reserved.
Made by Manager: Prof. Sang-Gu Lee and Dr. Jae Hwa Lee
*This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03035865).