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# Chapter 12 Complex Numbers and Functions - PowerPoint PPT Presentation

Chapter 12 Complex Numbers and Functions. 複數( complex numbers) 與複變數( complex variables). 複數( complex numbers) : z = a + i b , 其中 a 與 b 均為實數,. 複變數( complex variables) : z = x + i y , 其中 x 與 y 均為實變數,. 複數運算規則 :. 相等( equality) :. z 1 = z 2. x 1 = x 2 , y 1 = y 2.

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Chapter 12 Complex Numbers and Functions

z1 = z2

x1 = x2 , y1 = y2

z1 + z2 = (x1 , y2) + (x2 , y2) = (x1+ x2 , y1+y2)

z1 z2 = (x1 , y2) · (x2 , y2) = (x1 x2 - y1y2 , x1 y2 + x2y1)

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

y

(x,y)

Polar representation

y

r

z = r (cosθ + i sinθ)

x = r cosθ

y = r sinθ

θ

r : the modulus or magnitude of z

x

z = r eiθ

x

O

θ : the argument or phase of z

z = x + i y

Euler’s Formula :

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

De Moivre’s (隸美弗) Formula

Q : 試證明

A :

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Q : 試解 1. 2. 3.

A :

1.

2.

3.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

(x,y)

z

θ

x

θ

(x,-y)

Chapter 12 Complex Numbers and Functions

Complex function w(z) = u(x,y) + iv(x,y) where u(x,y) and v(x,y) are pure real

y

v

For example :

z - plane

w - plane

w(z) = z2 = (x + iy)2 = (x2 - y2) + i 2xy

2

2

mapping

1

1

x

u

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

100oC

Q : 某一楔形金屬板,其兩面之溫度固定為恆溫(如圖所

π/3

A :

0oC

v

y

π/3

100oC

π/3

x

u

0oC

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

v = θ + 2nπ

mapping

n = 1

n = 0

x

n = -1

u = lnr

n = -2

z - plane

w - plane

Chapter 12 Complex Numbers and Functions

z = r eiθ

θ : 主幅角(the principle argument)

w : 多值函數

Q : 試計算 之值

A : 假設

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Q : 試計算下列之值

A :

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Q : 1.試解 2.求之值

A : 1.

2.

or

n 為偶數

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

v

v

y

x

u

u

x

Chapter 12 Complex Numbers and Functions

z-plane

w-plane

y0

x0

z-plane

w-plane

θ0

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

v

x

u

Chapter 12 Complex Numbers and Functions

z-plane

w-plane

v

y

w-plane

z-plane

x

u

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

line  circle

y

z-plane

y = c1

1

2

3

4

x

v

w-plane

4

u

3

2

1

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

two-to-one correspondence

Upper half-plane of z, 0  θ < π  whole plane of w, 0  φ < 2π

Cover by two times

Lower half-plane of z, π θ < 2π  whole plane of w, 0  φ < 2π

For example :

two-to-one correspondence

,

w(z) = z2 = (x + iy)2 = (x2 - y2) + i 2xy

v

y

w-plane

z-plane

u

x

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

one-to-two correspondence

two-to-one correspondence

,

z = 0的點除外

z-plane

w-plane

,

,

w-plane

z-plane

How to make the function of w a singled-values function ?

one-to-one correspondence

y

z-plane

branch point singularities

x

cut line

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

many-to-one correspondence

w-plane

z-plane

the same point

any points

also

If

y

y

cut line

x

x

The Riemann surface for ln z

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

holomorphic

f (z) is analytic at z = z0

regular

y

δx0

z0

δy = 0

δx = 0

δy0

x

First

approach

δy = 0

δx0

δx = 0

Second

approach

δy0

Cauchy-Riemann conditions : if exists, then ,

if does not exist at z = z0, then z0 is labeled a singular point .

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Cauchy-Riemann conditions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

1.

2.

For any points

3.

4.

Except for branch points and cut lines

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Q : 試問下列函數在何處解析?

A :

Cauchy-Riemann conditions :

1.

1.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

A :

3.

4.

5.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Q : 試將拉式運算子 以共軛座標表示之

A:

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

y

Consider the sum :

z0’=zn

z2

Let n  with for all j

z1

ζ2

z0

If the sum exists and is independent of the

details of choosing the points zj and j .

ζ1

x

then

f (z)沿著特定路徑C (由z = z0到 z = z0’)的線積分

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

?

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Stokes’s theorem :

Let u = Vx and v = -Vy

Let v = Vx and u = Vy

If f (z) is analytic

Cauchy-Riemann condition

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

C1

C1

C2

C3

C2

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

77交大控制

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

C1

contour line

z0

C2

As r  0

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

z0 interior

z0 exterior

f(z)的微分可利用歌西積分公式來表示

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

z0 interior

z0 exterior

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

z0 interior

z0 exterior

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

A complex sequence : z1, z2, z3 , z4,…. zn,…

A complex sequence z1, z2,… is said to converge to the number L if ,given ε > 0, there is some positive integer N such that whenever n  N.

zN+3

ε

L

zN

zN+2

zN+1

Cauchy sequence

Theorem :

Let zn = xn +iyn. Then, znA + iB if and only if xn  A and yn  B

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Given a complex sequence : z1, z2, z3 , z4,…. zn,…

The complex series :

The sum :

Theorem :

Let zn = xn +iyn. Then, if and only if and

Theorem :

If converges, then

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

convergence and

convergence

convergence

divergence but

1. 對滿足 的 z 而言, 此級數為絕對收斂

2. 對滿足 的 z 而言, 此級數為發散

3. 對滿足 的 z 而言, 無法判定收斂性

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

where

ρ

z0

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

ρ

z0

z

C

z為C內部的一點, 而s在圓上

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

f(z)在z0點處的複變泰勒級數(Complex Taylor series)

 冪級數

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Let and

For all z

if

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

C2

r2

f(z)在 區域中為解析

z0

C1

r1

z

C

f(z)在C2中並非都解析

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

m = -n

C : 區域

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

C : 區域

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

C : 區域

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

for

for

for

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

1. 當展開式中沒有(z-z0)的負冪次項,則稱z0為可移除奇異點(removable singularity).

2. 當展開式中有無窮多個(z-z0)的負冪次項,則稱z0為本質奇異點(essential singularity).

3. 當展開式中有(z-z0)的負冪次項一個以上,則稱z0為極點(pole).

4. 當展開式中(z-z0)的負冪次項部分稱為主要部份(principal part).

5. 當展開式中(z-z0)的負冪次項部分只到第k項,則稱z0為k階極點(kth order pole).

6. 當展開式中 沒有主要部分,且 則稱z0為k階零點(kth order zero).

7. b1稱為f(z)在z0的殘(留)數(residue).

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

8. 當z0為f(z)的一個k階零點,則z0為1/ f(z)的一個k階極點,反之亦然.

9. 當展開式中(z-z0)的負冪次項只一個,則稱z0為簡單極點(simple pole).

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

h(z)在z = 0處解析,且h(0)  0, 故f(z)在z = 0處為一個二階極點

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

C

z0

r

f(z)在 區域中為解析函數,但除了z0, z1,…, zn點為不為解析,當C為包含上述奇異點的單連封閉曲線時,則

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

z = 1處為三階極點

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

z = 0處為簡單極點, z = 2i, -2i處為簡單極點

z = 2i, -2i處為簡單極點

z = 0處為簡單極點

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Where f is both finite and single-valued for all values of 

Let

The path of integration is the unit circle

By the residue theorem

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

CR

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

q(z)之次方數  p(z)之次方數+2

q(z)沒有實根,以避免函數p(z)/q(z)在實軸上出現極點

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

counterclockwise

clockwise

CR

net

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

CR

net

0

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

and

Consider

CR

a > 0即可

Jordan’s Lemma

: q(z)之次方數  p(z)之次方數+1即可

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

sinx與x均為奇函數

0

1

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

f(z)是一個常數函數

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

C

d

Z0

z0為C內之任一點,因此得證!

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

Example

Sol.

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 12 Complex Numbers and Functions

i

-3

-2

-

2

3

0

-i

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung