Download Document

講者： 許永昌 老師
1
Contents







Preface
Guide line of Ch6 and Ch7
Addition and Multiplication
Complex Conjugation
Functions of a complex variable
Example: Electric Circuit
Cauchy-Riemann Conditions for differentiable
function.
 Derivatives of Elementary Functions.
 Analytic functions.
2
Preface
 Functions of complex variables can be used:
1.
If f(z)=u+iv, and f(z) is an analytic function, we can get 2u=2v=0 & uv=0
Electrostatic potential vs. E.
Create a curved coordinate.
1.
2.
Real numbers  Complex numbers
2.
1.
1.
2.
Fundamental theorem of algebra:
Any polynomial of order n has n (in general) complex zeros.
Real functions, infinite real series, and integrals usually can be generalized naturally to complex
numbers simply by replacing a real variable x.
Propagation versus Evanescence
3.
1.
Helmholtz equation: (2+k2)u=-r.
Integrals:
4.
1.
2.
3.
4.
5.
6.
Evaluating definite integrals.
Inverting power series
Infinite product representations of analytic functions
Obtaining solutions of differential equations for large values of some variable.
Investigating the stability of potentially oscillatory systems
Inverting integral transforms
Many physical quantities that were originally real become complex as a simple physical
theory is made more general.
5.
1.
2.
Index of refraction.
Impedance.
3
Guideline of Ch6 & Ch7
 Basic complex algebra:
 +, ,









Polar form
Functions of a complex variable
The Cauchy-Riemann condition for analytic functions
Cauchy integral & Cauchy integral formula.
Morera’s Theorem & Liouville’s Theorem
Taylor expansion and Laurent expansion
Conformal mapping
Poles, branch points and branch cut lines.
Residue Theorem
 Product expansion of entire function
 Count the number of poles and zeros.
 The leading term of an asymptotic expansion
*Reference: J. Bak, D.J. Newman, Complex Analysis
4
Complex algebra (請預讀P319~P321)
 Reference: http://en.wikipedia.org/wiki/Complex_number
 幾乎整個Ch6 全包了。
 A complex variable is an ordered pair of two real variables
 z(x ,y )
:


z1+z2=(x1, y1)+(x2, y2)(x1+ x2, y1+ y2)=z2+z1.
z1+(z2 +z3)=(z1+z2)+z3.
Commutative
Associative
 Multiplication:  inner product
 z1z2=(x1, y1)(x2, y2)(x1x2-y1y2,x1y2+x2y1)= z2z1.
Commutative
 az1=(ax1,ay1). aR
[
: How about (a,0) (x1, y1)?]
 (z1z2)z3 = z1(z2z3)
Associative
 Distributive:
 z1(z2+z3)=(x1(x2+x3)-y1(y2+y3),x1(y2+y3)+y1(x2+x3)=z1z2+z1z3
 Others:
 Zero: 0(0,0), z+0=z.
 -z
 One: 1(1, 0).
 z-1.
 Imaginary unit: i  (0, 1) and i2= -1.
5
Complex algebra (continue)
y
Im{z}
 Therefore,
 z   x, y 
  x, 0    0, y 
 x  1, 0   y   0,1
 Addition 
 Multiplication 
 x 1  y  i
r

 z | z   x, 0  behave as real numbers 
 x  iy
x
Re{z}
 r   cos   i sin  
 Polar form 
 r  ei

where x and y are real numbers.


r  x 2  y 2 , cos  
x
x y
2
2
and sin  
y
x y
2
2
, respectively.
i1 2 
z1  z2  r1r2e
6
Exercise
 Find the roots of
 y(x)=x2+x+1.
 The polar form of 2+2i.

i
i1 2 
z  r  e and z1 z2  r1r2e
, why?
 ei=cos + isin, why?
7
Complex Conjugation (請預讀P321~P323)
 The complex conjugate of z is denoted by z*, where
 z*=x-iy if z=x+iy.
 Properties: (
)
(Modulus): |z| x 2  y 2  z *  z
(Argument):
arg  r  ei   


arg(z1z2)=arg(z1)+arg(z2).
arg(z*)=-arg(z)
 (z*)*=z.
 (z1  z2)*=z1*z2*.

1 z*
 2
z z
8
Exercise
 Example 6.1.2
3i
 The Polar form of
.
4 - 2i
9
Triangle inequality (請預讀P324)
 z1 - z2  z1  z2  z1  z2
 Please prove it.
10
Functions of a Complex Variable (請預
讀P325)
 w(z)=u(x, y)+iv(x, y).
part and
part:
 實部：Rw(z)=Re{w(z)}=u(x, y).
 虛部：I w(z)=Im{w(z)}=v(x, y).
 Code: z_to_uv.m
5
4
y
0.4
z-plane
0.3
v
3
0.2
2
0.1
1
0
w-plane
5
x
10
0
1.21.41.61.8
u
2
2.22.4
11
Multivalued functions (請預讀P326)
 nth root:
 z=rei ei2pm mZ

The source of multivalue.
z1/n= r 1/n ei /n  ei2pm/n , m=0 ,1 , …, n-1.
ei2pm
 Logarithm:
 z=rei ei2pm mZ

lnz=lnr + i( +2p m), mZ
 Q: How to solve this problem?
 A: Cut line. (Will be discussed in Ch6.6)
Cut line
12
Example (請預讀P327)
 Electric Circuit:
 Based on Kirchhoff’s loop rule

dI Q
IR  L   V0 cos wt
dt C

L
I
Q
R
-Q
C
V0coswt
The steady-state solution:
:
 I=Re{I0eiwt},
If we assume that I=I1coswt+I2sinwt,
 Q=Re{I0/(iw) eiwt},
can we get the same answer?
i
w
t
 dI/dt=Re{iw I0e },
 V=Re{V0eiwt},
 We get I0{R+iwL-i/(wC)}=ZI0=V0,  I0=V0/Z .

Impedance:
Z w   R  iw L 
1
iwC
13
Homework
 6.1.1
 6.1.2
 6.1.3
 6.1.6
 6.1.7
 6.1.15
 6.1.17
 6.1.20
14
Cauchy-Riemann Conditions (請預
讀P331~P334)
 Definition of
:
 f(z) is differentiable at z0 if the limit
lim
f  z  - f  z0 
z - z0
z  z0
Q: Can we say that if a
complex function
obeys this conditions,
it is differentiable at z0?
exists from any direction in the complex plane.
 df
dz
 lim
z0
z  z0
f  z  - f  z0 
z - z0
 It means that lim
x 0

 f '  z0 
f  z0   x  - f  z0 
x
If f(z)=u(x, y)+iv(x, y), we get
 lim
 y 0
f  z0  i y  - f  z0 
i y
u v 1  u v  v u
 i    i   - i .
x x i  y
y  y y
u v
u
v
Therefore,

and
 - .  Cauchy-Riemann conditions 
x y
y
x
15
Cauchy-Riemann Conditions
(continue)
 If a complex function obeys Cauchy-Riemann
conditions (fy=ifx) and
it is differentiable at z. (注意粗體字的部分)
 Proof:

f  f  x  x, y  y  - f  x, y  y   f  x, y  y  - f  x, y  
If f x is continuous, then 

C z
if it is differentiable
 C x  iC y,
i.e.
If f y is continuous, then 
where f ,
,
f
f
 x  y
x
y
1
f f
, , C , z 
x y
and u  x, y  , v  x, y  , x, y  , respectively.
f
f
 C  -i ,
x
y
 Let’s substitute these conditions into Eq. (1), we get
f 
f
df
 x  iy   C z, where C  is independent of z.
x
dz
16
Cauchy-Riemann Conditions
(continue)
 A counterexample of a function whose fx and fy are not
continuous but obey the Cauchy-Riemann Condition:
 Reference: J. Bak & D. J. Newman, Complex Analysis,
P33.
 xyz
 f  z    z 2
 0


z0
0
0
z0
-0.5
1
0.5
0

1
0.5
-0.5
Non-differentiable at z=0:
f  z  - f  0

0.5
0.5
-0.5
1
1
0.5
0
-0.5
-1
0.5
0
-1
0
-0.5
-0.5
-1
-1

which depends on the direction.
2
2
z
1


z if along y  x
fy(0)=fx(0)=0 which obeys the Cauchy-Riemann Conditions.
f x  x  i x  x0 
except   0.

xy

yz z  xy z - 2 x 3 yz
2
2
z

4
y  x
 2  i   1   2  - 2 1  i 
1   
2
2
 0  f x 0
17
Cauchy-Riemann Conditions
(continue)
f
 f z, z  z

*

 Where
f
z

z*
x
z
f
y

x y
z
z*

f
z  * z* (此條待考證)
z z
z*


x  z  z* / 2
z*

f
1
  f x - if y   f x ,
y x 2
f y if x
y i z* - z / 2
f
x f
y f
1
 *
 *
  f x  if y   0.
*
z z z z x y z z y x 2
 Therefore,
f '

f
z
and
z*
f
 0.
*
z z
這就是為何我們在看複變時，看到的 f 都是z的函數，幾乎沒
看過z*的。
18
Cauchy-Riemann Conditions (final)
Differentiable at z:
f ’(z) does exist
CauchyRiemann
Conditions
fx(z)=ify(z).
fx and fy are continuous at z.
19
Derivatives of elementary
functions (請預讀P334)
 We define the elementary functions by their Taylor
expansion with xz.


zn
e  ,
n 0 n !
z

sin z    -1
n 0

n
z 2 n 1
,
 2n  1!
ln 1  z     -1
n -1
n 1
zn
,
n

cos z    -1
n
n 0
z 2n
,
 2n  !
z 1
 In the convergent region,
z  z  - z n

dz n
 lim
 nz n -1 ,
dz z 0
z

de z
d  z n   z n -1
  
 ez ,
dz n 0 dz  n !  n 1  n - 1 !
n
2n
d sin z  d 
z 2 n 1  
n
n  2n  1 z
   -1
 cos z ,
    -1
dz
2
n

1
!
2
n

1
!




n  0 dz 
n

0



d ln 1  z 
dz
n

d 
1
n -1 z 
n -1 n -1
   -1

1
z

,


 
n  n 1
1 z
n 1 dz 

20
Analytic Functions (請預讀P335~P336)
 Analytic Function:
 f(z) is analytic at the point z0 if (1) f(z) is
and (2)
.

所以光一個點differentiable還不能說他在該點就 analytic。
 Entire function:
 f(z) is an entire function if it is analytic

1 2  *
 z - 1 z ,
2





.
: Counterexample:
z* and |z|2
Differentiable at z=?
fx, f y exist?
Analytic?
Entire function?
21
Homework
 6.2.1
 6.2.3
 6.2.5
22
Nouns
:
) of z:
) of z:
:
:
i  (0, 1) and i2= -1.
P320~P321
z*=x-iy.
|z|=(x2+y2)½ .
arg(z)= if z=rei.
|z1|-|z2||z1+z2||z1|+|z2|.
P326
of lnz:
P365
lnr+i, -p  p.
:
:
(
(
:
To Math. Phys.
at z0:
:
:
P169 in M. T. Vaughn, Intro.
P21 of this slide
P21 of this slide
23