Ch. 13 Complex Numbers and Functions. Complex Differentiation

Ch. 13 Complex Numbers and Functions. Complex Differentiation

서울대학교 조선해양공학과 서유택 2017.11

※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다.

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13.1 Complex Numbers and Their Geometric Representation

 Complex Numbers: An ordered pair (x, y) of real numbers x and y → z = (x, y) = x + iy

 x = Re z: Real Part of z

 y = Im z: Imaginary Part of z

 i = (0, 1): Imaginary Unit (단위 허수)

 Pure Imaginary: z = iy (x = 0)

 Addition, Multiplication, Subtraction, Division. Notation z = x + iy

 Addition:

 Multiplication:

 Subtraction:

 Division:

  







1 1 1 2 1 2 2 1 1 2

2 2 2 2

2 2 2 2 2 2 2 2 2 2

x iy x iy

x iy x x y y x y x y

x iy x iy x iy x y i x y

 

     

    



 

 

 





 

 

 





 

 

 



13.1 Complex Numbers and Their Geometric Representation

 Complex Plane (복소평면): The geometrical representation of complex numbers as points in the plane.

 Two perpendicular coordinate axes, the horizontal x-axis, called the real axis, and the vertical y-axis, called the imaginary axis.

The number 4 – 3i in the complex plane The complex plane

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13.1 Complex Numbers and Their Geometric Representation

 Complex Conjugate Number (공액복소수)

: Complex conjugate of a complex number z = x + iy







z

 

   

1 1

Re , Im

2 2

z x z z z y z z

     i 













2 2

, z z

z z z z

z z

 

     

  Complex conjugate numbers

i z

i

z₁  43 , ₂  25 Ex)

 

1

Im 1 (4 3 ) (4 3 ) 3

z 2 i i

 i    

) 6 20 ( 12 8 ) 5 2 )(

3 4 ( )

(z₁z₂   i  i   i   726i  726i, i

i i

i z

z_{1 2} (43 )(25 ) 815 (206)  726

 Argument (편각):

: the directed angle from the positive x- axis to OP in the right figure. Angles are measured in radians and positive in the counterclockwise sense.

 Absolute value or Modulus (절대값 또는 크기) : distance of the point z from the origin

13.2 Polar Form of Complex Numbers. Powers and Roots

 Polar Form (극형식): z = r(cosθ + isinθ), x = rcosθ & y = rsinθ

2 2

z  r x  y  zz

arg arctan y

z x



2 2 2 2

( )( )

( )

zz x iy x iy

x iy x y

  

 

 

   

 

y

x

Imaginary axis

Real axis

P z  x  iy

Complex plane, polar form of a complex number

r z 



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13.2 Polar Form of Complex Numbers. Powers and Roots

 Polar Form: z = r(cosθ + isinθ), x = rcosθ & y = isinθ

y

x

Imaginary axis

Real axis

P z  x  iy

Complex plane, polar form of a complex number

r z 



• Principal value (주값, Arg z) of arg z

) ,

2 ,

1 (

2 Arg

arg z  z  n  n    

3 z  i Ex)

Arg arctan 1 3 6

arg 2 ( 1, 2, )

6

z n n



 

 

    

Arg z

 

  

13.2 Polar Form of Complex Numbers. Powers and Roots

 Triangle Inequality (삼각형 부등식)

 Triangle Inequality:

 Generalized Triangle Inequality:

Triangle inequality

1 2 1 2

z  z  z  z

1 2 n 1 2 n

z    z z  z  z   z

Ex)

1 2

1 2

1 2

1 , 2 3 ,

1 4 17 4.123 2 13 5.020

z i z i

z z i

z z

    

     

   

1 2 1 2

z z z z

   

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13.2 Polar Form of Complex Numbers. Powers and Roots

8/1 45

) sin

(cos

and

) sin

(cos

1

1

r  i  z r  i 

z    

• Multiplication

1 2 1 1 1 2 2 2

1 2 1 2 1 2 1 2 1 2

(cos sin ) (cos sin )

[(cos cos sin sin ) (sin cos cos sin )]

z z r i r i

r r i

   

       

  

   

2 1

2 1

2 1

2 1

2 1

2 1

sin cos

cos sin

) sin(

sin sin

cos cos

) cos(

































 

 Multiplication in Polar Form

   

1 2 1 2

cos

sin

z z  r r       i     

1 2 1 2 1 2

z z  r r  z z





arg z z      arg z  arg z

13.2 Polar Form of Complex Numbers. Powers and Roots

) sin

(cos

), sin

(cos

2 2

2 2

1 1

1 1









i r

z

i r

z









4 4

4

3 3

3 3

sin cos

), sin

(cos









i z

i r

z









z 2 2

1 z z



z 1



r

r

1

r r

2

1



 

x y



z

z z



z



r

1 r

4

3



 

x y



 Multiplication in Polar Form

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13.2 Polar Form of Complex Numbers. Powers and Roots

10/ 145

) sin

(cos

and

) sin

(cos

1

1

r  i  z r  i 

z    

• Division

2 2 1

1

z

z

z  z

2 1

1

z

z z  z



2 2

1 2

2 1

1

arg arg arg

arg z

z z z

z

z z   



 



 

 Division in Polar Form

1 1

2 2

z z

z z

 

1

1 2

2

arg z arg arg

z z

 z  

   

1 1

1 2 1 2

2 2

cos sin

z r

z  r       i     

   

1 2 1 2 cos 1 2 sin 1 2

z z  r r    i   

13.2 Polar Form of Complex Numbers. Powers and Roots

) sin

(cos

), sin

(cos

2 2

2 2

1 1

1 1









i r

z

i r

z







 ¹

z

r

x y



z 2



r



^z

^{/ z}

2

1



 

2 1

r r

 Division in Polar Form

   

1 1

1 2 1 2

2 2

cos sin

z r

z



     

    

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13.2 Polar Form of Complex Numbers. Powers and Roots

12/ 145

x

 Ex 3 illustrate the followings

1 2 1 2 1 2

z z  r r  z z

1 1

2 2

z z

z  z





arg z z      arg z  arg z

1

1 2

2

arg z arg arg

z z

z  

1

2 2 ,

3 z   i z   i

1 2

1 2

6 6 72 6 2

8 3 6 2

z z i

z z

   

  

1 2

1 2

/ 2 2 8 / 3 2 2 / 3 3

/ 8 / 3 2 2 / 3 z z i

z z

    

 

 

1 2

1 2

arg( ) arg 6 6

4 arg arg

4 2 4

z z i

z z



  

   

    





1 2

2 2 3

arg / arg

3 4

arg arg ( ) 3

4 2 4

z z i

z z



  

   

   

    

Sol)

13.2 Polar Form of Complex Numbers. Powers and Roots

• De Moivre’s formula:

(드 무아브르 공식)

 ^{cos   i sin } 

^{ cos n   i sin n }

 ^{cos sin} 

 ^{cos sin} 

n n

z    r   i     r n   i n 

 Ex 4 Integer Powers of z

   

1 2 1 2 cos 1 2 sin 1 2

z z  r r    i   

If n=2:  ^{cos   i sin } 

^{ cos 2   i sin 2 }

2 2

cos   2 cos sin i    sin   cos 2   i sin 2 

2 2

cos 2 cos sin sin 2 2 cos sin

  

  

 



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13.2 Polar Form of Complex Numbers. Powers and Roots

 Roots

 nth root of z: Each value w satisfying

z = w

 

2 2

cos sin , 0, 1, , 1

n n k k

z r i k n

n n

     

 

     

 ^{cos sin} 

z  r   i  ^{w  R}  ^{cos   i sin } 

 ^{cos sin}   ^{cos sin} 

n n

w  R n   i n    z r   i 

2 2 ( 0,1,... 1)

n n

R r R r

n k k k n

n

 

   

  

      

, 2 ( 0) duplicated!

if k n same as k

n n

 

 

     

13.2 Polar Form of Complex Numbers. Powers and Roots

 Roots

 nth root of unity (단위 n제곱근, z = 1  r = 1,  ^{= 0):}

 

2 2

1 cos sin , 0, 1, , 1

n k k

i k n

n n

 

   

 

2 2

cos sin , 0, 1, , 1

n n k k

z r i k n

n n

     

 

     

3 1 1

1 1, 3

2 2 i

   ₄

1    1, i

1

n = 3 n = 4 n = 5

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 Circles

and Disks

. Half-Planes

 Unit Circle: |z| = 1

 Open Circular Disk (열린 원판): |z-a| < ρ

 Closed Circular Disk (닫힌 원판): |z-a| ≤ ρ

 Neighborhood (근방) of a

: An open circular disk, ρ-Neighborhood of a

 Open Annulus (열린 환형): ρ₁< |z-a|< ρ₂

 Closed Annulus (닫힌 환형): ρ₁≤ |z-a|≤ ρ₂

 (Open) Upper Half-Plane (상반 평면):

: The set of all point z = x + iy such that y > 0

 Lower half-plane (하반 평면): y < 0

 Right half-plane (우반 평면): x > 0

 Left half-plane (좌반 평면): x < 0

Unit Circle

Circle in the complex plane

Annulus in the complex plane

13.3 Derivative

. Analytic Function

13.3 Derivative. Analytic Function

 [Reference] Concepts on Sets in the Complex Plane

 Point Set (점집합): Collection of finitely many or infinitely many points.

 S is Open (열려있다): S has a neighborhood (근방) consisting entirely of points that belong to S. 원 또는 정사각형 내부 점들, 우반 평면 (Re z = x > 0) 의 점들은 열린 집합 (open set)을 구성

 S is Connected (연결되었다): Any two of its points can be joined by a chain of finitely many straight-line segments all of whose points belong to S.

 Domain (영역): An open connected set Q: Connected or not ?

Connected Not-connected

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13.3 Derivative. Analytic Function

 [Reference] Concepts on Sets in the Complex Plane

 Complement of S (여집합): Set of all points of the complex plane that do not belong to S.

 S is Closed (닫혔다): Its complement is open.

 Boundary point of S (경계점): A point every neighborhood of which contains both points that belong to S and points that do not belong to S.

 Boundary (경계): Set of all boundary points.

 Region (영역): Set consisting of a domain plus some or all of its boundary points.

13.3 Derivative. Analytic Function

 Complex Function

:

 Complex Variable (복소변수): z

 Domain (정의역): set S

 Range (치역): set of all values of a function

Ex. 1) Function of a Complex Variable

Let w = f (z) = z²+3z. Find u and v and calculate the value of f at z = 1+3i.

 

     

2 2

2

Re 3 , 2 3

1 3 1 3 3 1 3 1 9 6 3 9 5 15

u f z x y x v xy y

f i i i i i i

     

            

   

 

w   ,  ,

Sol)

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 Limit (극한), Continuity (연속성)

 Limit l as z approaches a point z₀ from any direction in the complex plane

⇔

 Continuous

Function f (z) is continuous at z = z₀

⇔

13.3 Derivative. Analytic Function

   

0

lim 0

z z f z f z

 

 

0

lim

z z

f z l





 Derivative

 f (z) is defined in a neighborhood of z₀ and z may approach from any direction in the complex plane.

 Differentiability at z₀: along whatever path z approaches, f ′(z) always approaches a certain value and all these values are equal.

Ex. 3) Differentiability (미분가능성). Derivative

The function f (z) = z² is differentiable for all z and has the derivative f ′(z) = 2z

13.3 Derivative. Analytic Function

         

0

0 0 0

0 0

0

lim lim

z z z

f z z f z f z f z

f z

z z z

  

   

  

 

   

 

 

0 0 0

2 2

lim lim lim 2

z z z

z z z z z z z z z z z

f z z

z z z

     

 

          

         

Sol)

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 Differentiation Rule

 Chain rule and the power rule

 If f (z) is differentiable at z₀, it is continuous at z₀. Ex. 4) Show that is not differentiable.

is not differentiable.

f (z) do not have a derivative at any point z.

13.3 Derivative. Analytic Function

 





 

g g

  

  

 

    

        

 

 

f z   z x iy

     

f z z f z z z z z x i y

z z z x i y

            

     

 

z

Sol) ^{ x 0}

0

 y 0

 y

0

 x

 Path I: ∆y → 0 first and then ∆x→ 0:

 Path II: ∆x→ 0 first and then ∆y→ 0:

x i y 1 x i y

   

   x i y 1 x i y

    

  

13.3 Derivative. Analytic Function

 Analytic Functions

 Definition Analyticity (해석성)

 A function f (z) is said to be analytic at a domain D if f (z) is defined and differentiable at all points of D.

 The function f (z) is said to be analytic at a point z = z₀ in D if f (z) is analytic in a neighborhood of z₀.

 Also, an analytic function (해석함수) means a function that is analytic in some domain.

Ex. 5) Polynomials (다항식), Rational Functions (유리함수)

The nonnegative integer powers are analytic in the entire complex plane → So are polynomials .

Rational function is analytic except at the points where h(z)= 0.^{f z  g z}_{h z }^{ } 1, , , z z2

 

n

f z  c c zc z  c zn

“해석적 = 미분 가능”

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13.4 Cauchy-Riemann Equations. Laplace’s Equation

 Cauchy-Riemann Equations: u

= v

, u

= ‒v

 Theorem 1 Cauchy-Riemann Equations

 Let f (z) = u(x, y) + iv(x, y) be defined and continuous in some neighborhood of a point z = x + iy and differentiable at z itself.

 Then at that point, the first-order partial derivatives of u and v exist and satisfy the Cauchy-Riemann equation.

 Hence if f (z) is analytic in a domain D, those partial derivatives exist and satisfy Cauchy-Riemann equation at all points of D.

“해석적 (미분 가능)  Cauchy-Riemann equation 만족

13.4 Cauchy-Riemann Equations. Laplace’s Equation

 Proof

 Path I: ∆y → 0 first and then ∆x → 0

 Path II: ∆x → 0 first and then ∆y → 0

     

lim0 z

f z z f z

f z

  z

  

 



         

0

[ , , ] [ , , ]

limz

u x x y y iv x x y y u x y iv x y

f z

x i y

 

          

 

  

         

0 0

, , , ,

lim lim

x x

u x x y u x y v x x y iv x y

f z i

x x

   

     

  

 

 

f  z  u iv

         

0 0

, , , ,

lim lim

y y

u x y y u x y v x y y iv x y

f z i

i y i y

   

     

  

 

 

f  z  iu v

Differentiability at z₀: along whatever path z approaches, f′(z) always approaches a certain value and all these values are equal.

0

 x

0

 y 0

 y

0

 x

∴ u_x = v_y , u_y = ‒v_x

Seoul National

Univ. 26

13.4 Cauchy-Riemann Equations. Laplace’s Equation

( ) ( , ) ( , )

w  f z  u x y  iv x y ( , ) ( , )

( , ) ( , )

x y

y x

u x y v x y u x y v x y



Analytic

 

Theorem 1 Theorem 2

Example)

)

( )

( z x iy

f  

) 2

2

(

2

y i xy

x  



xy v

y x

u 



,  2



y

x

x v

u  2 

x

y

y v

u  2   

iy x

z

f ( )  

y v

x

u   

 ,

y

x

v

u  1   1 

x

y

v

u  0   )

( z z

f 

f ( z )  z

13.4 Cauchy-Riemann Equations. Laplace’s Equation

Ex. 2 Cauchy-Riemann Equations. Exponential Function

Cauchy-Riemann equation are satisfied and thus f (z) is analytic for all z.

( )

(cos sin ) ( , ) ( , ) f z  e  e y i  y  u x y  iv x y

y e

y x v y e

y x

u ( , ) 

cos , ( , ) 

sin

y x

x

e y v

u  cos 

x x

y

e y v

u   sin  

 Theorem 2 Cauchy-Riemann Equations

 If two real-valued continuous functions u (x, y) and v (x, y) of two real variables x and y have continuous first partial derivatives that satisfy the Cauchy-Riemann equation in some domain D,

 then the complex function f (z) = u (x, y) + iv (x, y) is analytic in a domain D. Cauchy-Riemann equation 만족  해석적 (미분 가능)

Seoul National

Univ. 28

13.4 Cauchy-Riemann Equations. Laplace’s Equation

 EX. 3 An Analytic Function of Constant Absolute Value Is Constant The Cauchy-Riemann equations also help in deriving general

properties of analytic functions. Show if f (z) is analytic in a domain D and |f (z)| = k = const in D, then f (z) = const in D.

By assumption, By differentiation,

Use Cauchy-Riemann equation

Use Cauchy-Riemann equation

 

 

2 2 2

2 2 2

i 0 0 0

ii 0 _{x y} 0 const

k u v u v f

k u v u u u

       

       

2 2 2 2 2

f  u iv u v k

0, 0

x x y y

uu vv  uu vv 

0, 0

x y y x

uu vu uu vu

    

Sol) ^{x( , )}_{( , )} ^{y( , )}_{( , )}

y x

u x y v x y u x y v x y



 

 

2 2

2 2

0, 0

0

x y y x

y

uvu v u u u uvu

u v u

     

  

(1) ( v), (2)u

0 const

x y

v  v    v

=const

 f

(1) (2)

 

2 2

2 2

0, 0

0

x y y x

x

u u uvu uvu v u

u v u

    

  

(1)u, (2)v

13.4 Cauchy-Riemann Equations. Laplace’s Equation

Ex. 4) Verify that is harmonic in the whole complex plane and find a harmonic conjugate function v of u.

1 2

,

2   

 x u y

u

2 ,

2  



xx

u

u

2

   0

 u u

u

So u is harmonic in the whole complex plane.

y y

x

u  ²  ² 



Let f (z) = u(x, y) + iv(x, y) is analytic in a domain D (u_x = v_y , u_y = ‒v_x), then both u and v satisfy Laplace’s equation

∇²u = u_xx + u_yy = 0, ∇²v = v_xx + v_yy = 0 in D and have continuous second partial derivatives in D.

Proof) ∇²u = u_xx + u_yy = v_yx ‒v_xy= 0, ∇²v = v_xx + v_yy = ‒ u_yx + u_xy = 0

Sol)

Solutions of Laplace’s equation are called harmonic function (조화함수).

Thus, u and v are harmonic functions.

v is called harmonic conjugate function v of u.

Seoul National

Univ. 30

13.4 Cauchy-Riemann Equations. Laplace’s Equation

Ex. 4 ) Verify that is harmonic in the whole complex plane and find a harmonic conjugate function v of u.u  x²  y²  y

By the Cauchy-Riemann Equations

x u

v_y  _x  2

1

 2 



 u y

v_{x y}

Integrating first equation with respect to y

) (x h dy u

v 



Differentiating v with respect to x,

.

2 dx

y dh v_x  

Comparing this and v_x = -u_y,

1

 dx

dh  h x( )  x c c

x xy

v   

 2 (c is any real constant.) The corresponding analytic function

) 2

( )

(

2

2 y y i xy x c

x

iv u z

f

















ic iz z  

 ² Sol)



Let f (z) = u(x, y) + iv(x, y) is analytic in a domain D (u_x = v_y , u_y = ‒v_x), then both u and v satisfy Laplace’s equation

∇²u = u_xx + u_yy = 0, ∇²v = v_xx + v_yy = 0 in D and have continuous second partial derivatives in D.

13.4 Cauchy-Riemann Equations. Laplace’s Equation

Ex) Is u = x³ -3xy² harmonic? If your answer is yes, find a corresponding analytic function f (z) = u(x, y) + iv(x, y).

By Laplace’s Equation

Sol)

2 2

3 3 , 6 ,

6 , 6

harmonic function

x xx

y yy

u x y u x

u xy u x

  

   



2 2

3 3

6

y x

x y

v u x y

v u xy

  

  

2 3

3 ( )

v







6 ( ) 6

vx





 

( ) 

2 3

3

v x y y c

   

3 2 2 3

3 3

( ) ( 3 ) (3 )

( )

f z x xy i x y y c

x yi ic z ic

    

    



Let f (z) = u(x, y) + iv(x, y) is analytic in a domain D (u_x = v_y , u_y = ‒v_x), then both u and v satisfy Laplace’s equation

∇²u = u_xx + u_yy = 0, ∇²v = v_xx + v_yy = 0 in D and have continuous second partial derivatives in D.

Seoul National

Univ. 32

13.5 Exponential Function (지수함수)

 Exponential Function:

1. e^z = e^x for real z = x because cos y = 1 and sin y = 0 when y = 0 2. e^z is analytic for all z.

3. The derivative of e^z is e^z, that is, (e^z)′ = e^z

 Further Properties

 e^z is entire (analytic for all z  완전함수).

 Euler’s formula:

 Polar form of a complex number: z = r(cosθ + isinθ) = re^iθ, e^2πi = 1

 |e^z| = e^x≠ 0

  

 



 

exp cos sin

z x

e  z  e y i y

 

 

1 2 1 2

1 2

1 2 1 2

1 1 2 2

1 2 1 2 1 2 1 2

1 2 1 2

(cos sin ) (cos sin )

(cos cos sin sin ) (sin cos cos sin )

cos( ) sin( )

z z x x

x x

x x z z

e e e y i y e y i y

e y y y y i y y y y

e y y i y y e



 

  

   

    

cos sin , 1

z x iy iy iy

e  e e  e  y i y e 

 

f  z  u iv

(when, z = iy)

 Periodicity of e

with period 2  i

e^z+2πi= e^z for all z

All the values that w = e^z can assume are already assumed in the horizontal strip of width 2.

 Ex. 1 Function Values. Solution of Equations

Solve the equation e^z=3 + 4i using e^z = e^x (cos y + i sin y).

13.5 Exponential Function

5 ln 5 1.609

z x

e  e x  

Fundamental region of the exponential function e^z in the z-plane





z x

e  e y i y





  

Sol)

 

5, cos 3, sin 4 cos 0.6, sin 0.8 0.927 1.609 0.927 2 0, 1, 2,

x x x

e e y e y y y y

z i n i n

     

    

e^2πi = 1

Seoul National

Univ. 34

13.6 Trigonometric (삼각함수) and Hyperbolic Functions (쌍곡선함수). Euler’s Formula

 Complex trigonometric functions

 cos z and sin z are entire functions (analytic for all z).

 tan z and sec z are not entire functions and analytic except at the point where cos z is zero.

 cot z and csc z are not entire functions and analytic except where sin z is zero.

   

1 1

cos , sin

2 2

sin cos 1 1

tan , cot , sec , csc

cos sin cos sin

iz iz iz iz

z e e z e e

i

z z

z z z z

z z z z

 

   

   

x i

x

e

 cos  sin x i

x

e

 cos  sin

) 2(

cos x  1 e^ix e^ix

) 2 (

sin 1 e^ix e ^ix x  i  ^

This suggests definitions for complex values





z x

e  e y i y