Post on 31-Mar-2015
Chapter 2. Analytic Functions
Weiqi Luo (骆伟祺 )School of Software
Sun Yat-Sen UniversityEmail : weiqi.luo@yahoo.com Office : # A313
School of Software
Functions of a Complex Variable; Mappings Mappings by the Exponential Function Limits; Theorems on Limits Limits Involving the Point at Infinity Continuity; Derivatives; Differentiation Formulas Cauchy-Riemann Equations; Sufficient Conditions for
Differentiability; Polar Coordinates; Analytic Functions; Harmonic Functions; Uniquely
Determined Analytic Functions; Reflection Principle
2
Chapter 2: Analytic Functions
School of Software
Function of a complex variable Let s be a set complex numbers. A function f defined on
S is a rule that assigns to each z in S a complex number w.
12. Functions of a Complex Variable
3
S
Complex numbers
f
S’
Complex numbers
z
The domain of definition of f The range of f
w
School of Software
Suppose that w=u+iv is the value of a function f at z=x+iy, so that
Thus each of real number u and v depends on the real variables x and y, meaning that
Similarly if the polar coordinates r and θ, instead of x and y, are used, we get
12. Functions of a Complex Variable
4
( )u iv f x iy
( ) ( , ) ( , )f z u x y iv x y
( ) ( , ) ( , )f z u r iv r
School of Software
Example 2 If f(z)=z2, then
case #1:
case #2:
12. Functions of a Complex Variable
5
2 2 2( ) ( ) 2f z x iy x y i xy 2 2( , ) ; ( , ) 2u x y x y v x y xy
z x iy
iz re 2 2 2 2 2( ) ( ) cos 2 sin 2i if z re r e r ir
2 2( , ) cos 2 ; ( , ) sin 2u r r v r r
When v=0, f is a real-valued function.
School of Software
Example 3 A real-valued function is used to illustrate some important concepts later in this chapter is
Polynomial function
where n is zero or a positive integer and a0, a1, …an are complex constants, an is not 0; Rational function the quotients P(z)/Q(z) of polynomials
12. Functions of a Complex Variable
6
2 2 2( ) | | 0f z z x y i
20 1 2( ) ... n
nP z a a z a z a z
The domain of definition is the entire z plane
The domain of definition is Q(z)≠0
School of Software
Multiple-valued function A generalization of the concept of function is a rule that
assigns more than one value to a point z in the domain of definition.
12. Functions of a Complex Variable
7
f
S
Complex numbers
z
S’
Complex numbers
w1
w2
wn
School of Software
Example 4 Let z denote any nonzero complex number, then z1/2 has
the two values
If we just choose only the positive value of
12. Functions of a Complex Variable
8
1/2 exp( )2
z r i
Multiple-valued function
r
1/2 exp( ), 02
z r i r
Single-valued function
School of Software
pp. 37-38
Ex. 1, Ex. 2, Ex. 4
12. Homework
9
School of Software
Graphs of Real-value functions
13. Mappings
10
f=tan(x)
f=ex
Note that both x and f(x) are real values.
School of Software
Complex-value functions
13. Mappings
11
( ) ( ) ( , ) ( , )f z f x yi u x y iv x y
x
y
u
v
Note that here x, y, u(x,y) and v(x,y) are all real values.
z(x,y) w(u(x,y),v(x,y))
mapping
School of Software
Examples
13. Mappings
12
1 ( 1)w z x iy
x
y
z(x,y)
u
v
w(x+1,y)
w z x yi
u
v
w(x,-y)
Translation Mapping
Reflection Mapping
y
z(x,y)
School of Software
Example
13. Mappings
13
( ) exp( ( ))2
iw iz i re r i
y
rθ
z
u
v
rθ
w
2
Rotation Mapping
School of Software
Example 1
13. Mappings
14
2w z 2 2 , 2u x y v xy
Let u=c1>0 in the w plane, then x2-y2=c1 in the z plane
Let v=c2>0 in the w plane, then 2xy=c2 in the z plane
School of Software
Example 2
The domain x>0, y>0, xy<1 consists of all points lying on the upper branches of hyperbolas
13. Mappings
15
x=0,y>0
x>0,y=0
2 2;
2 2 1
u x y
v xy xy
School of Software
Example 3
13. Mappings
16
2 2 2iw z r e In polar coordinates
School of Software
The exponential function
14. Mappings by the Exponential Function
17
,z x iy x iyw e e e e z x iy
ρeiθ ρ=ex, θ=y
School of Software
Example 2
14. Mappings by the Exponential Function
18
w=exp(z)
School of Software
Example 3
14. Mappings by the Exponential Function
19
w=exp(z)=ex+yi
School of Software
pp. 44-45
Ex. 2, Ex. 3, Ex. 7, Ex. 8
14. Homework
20
School of Software
For a given positive value ε, there exists a positive value δ (depends on ε) such that
when 0 < |z-z0| < δ, we have |f(z)-w0|< ε
meaning the point w=f(z) can be made arbitrarily chose to w0 if we choose the point z close enough to z0 but distinct from it.
15. Limits
21
00lim ( )
z zf z w
School of Software
The uniqueness of limit If a limit of a function f(z) exists at a point z0, it is unique.
Proof: suppose that
then
when
Let , when 0<|z-z0|<δ, we have
15. Limits
22
0 00 1lim ( ) & lim ( )
z z z zf z w f z w
0 1/ 2 0, 0, 0
1 0 0 1| | | ( ( ) ) ( ( ) ) |w w f z w f z w
0 1| ( ) | | ( ) |2 2
f z w f z w
0 1min( , )
0| ( ) | / 2;f z w 0 00 | |z z
1| ( ) | / 2;f z w 0 10 | |z z
School of Software
Example 1 Show that in the open disk |z|<1, then
Proof:
15. Limits
23
( ) / 2f z iz
1lim ( )
2z
if z
| || 1| | 1|| ( ) | | |
2 2 2 2 2
i iz i i z zf z
0, 2 , . .s t
0 | 1| ( 2 )z when
| 1|0 | ( ) |
2 2
z if z
School of Software
Example 2 If then the limit does not exist.
15. Limits
24
( )z
f zz
0
lim ( )z
f z
0
0lim 1
0x
x i
x i
( ,0)z x
(0, )z y0
0lim 1
0y
iy
iy
≠
School of Software
Theorem 1
Let
and
then
if and only if
16. Theorems on Limits
25
00lim ( )
z zf z w
0 00
( , ) ( , )lim ( , )
x y x yu x y u
0 00
( , ) ( , )lim ( , )
x y x yv x y v
( ) ( , ) ( , )f z u x y iv x y z x iy
0 0 0 0 0 0;z x iy w u iv
and
(a)
(b)
School of Software
Proof: (b)(a)
16. Theorems on Limits
26
0 00
( , ) ( , )lim ( , )
x y x yu x y u
0 00
( , ) ( , )lim ( , )
x y x yv x y v
&
00lim ( )
z zf z w
1 2/ 2 0, 0, 0 . .s t
0| ( , ) |2
u x y u
When 2 20 0 10 ( ) ( )x x y y
2 20 0 20 ( ) ( )x x y y 0| ( , ) |
2v x y v
1 2min( , ) Let 2 20 0 00 ( ) ( ) , . .0 | |x x y y i e z z When
0 0 0| ( ) | | ( ( , ) ( , )) ( ) |f z w u x y iv x y u iv
0 0| ( , ) | | ( , ) |2 2
u x y u v x y v
0 0| ( , ) ( ( , ) ) |u x y u i v x y v
School of Software
Proof: (a)(b)
16. Theorems on Limits
27
0 00
( , ) ( , )lim ( , )
x y x yu x y u
0 00
( , ) ( , )lim ( , )
x y x yv x y v
&
00lim ( )
z zf z w
0, 0 . .s t 0| ( ) |f z w When 00 | |z z
0 0 0| ( ) | | ( , ) ( , ) ( ) |f z w u x y iv x y u iv
0 0| ( ( , ) ) ( ( , ) ) |u x y u i v x y v
Thus 0 0| ( , ) | ;| ( , ) |u x y u v x y v
0 0 0| ( , ) | | ( ( , ) ) ( ( , ) ) |u x y u u x y u i v x y v
0 0 0| ( , ) | | ( ( , ) ) ( ( , ) ) |v x y v u x y u i v x y v
When (x,y)(x0,y0)
School of Software
Theorem 2
Let and
then
16. Theorems on Limits
28
00lim ( )
z zf z w
00lim ( )
z zF z W
00 0lim[ ( ) ( )]
z zf z F z w W
00 0lim[ ( ) ( )]
z zf z F z w W
0
00
0
( )lim[ ] , 0
( )z z
wf zW
F z W
School of Software
16. Theorems on Limits
29
( ) ( , ) ( , ), ( ) ( , ) ( , )f z u x y iv x y F z U x y iV x y
0 0 0 0 0 0 0 0 0; ;z x iy w u iv W U iV
( ) ( ) ( ) ( )f z F z uU vV i vU uV
Let
00lim ( )
z zf z w
00lim ( )
z zF z W
When (x,y)(x0,y0); u(x,y)u0; v(x,y)v0; & U(x,y)U0; V(x,y)V0;
0 0 0 0( )u U v VRe(f(z)F(z)):
0 0 0 0( )v U u VIm(f(z)F(z)):w0W0
00 0lim[ ( ) ( )]
z zf z F z w W
00lim ( )
z zf z w
00lim ( )
z zF z W
&
School of Software
16. Theorems on Limits
30
0
limz z
c c
0
0limz z
z z
0
0lim ( 1,2,...)n n
z zz z n
20 1 2( ) ... n
nP z a a z a z a z
It is easy to verify the limits
For the polynomial
00lim ( ) ( )
z zP z P z
We have that
School of Software
Riemannsphere & Stereographic Projection
17. Limits Involving the Point at Infinity
31
N: the north pole
School of Software
The ε Neighborhood of Infinity
17. Limits Involving the Point at Infinity
32
O x
y
R1
R2
When the radius R is large enough
i.e. for each small positive number ε
R=1/ε
The region of |z|>R=1/ε is called the ε Neighborhood of Infinity(∞)
School of Software
Theorem If z0 and w0 are points in the z and w planes,
respectively, then
17. Limits Involving the Point at Infinity
33
0
lim ( )z z
f z
iff0
1lim 0
( )z z f z
0lim ( )z
f z w
iff00
1lim ( )z
f wz
lim ( )z
f z
iff0
1lim 0
1( )
zf
z
School of Software
Examples
17. Limits Involving the Point at Infinity
34
School of Software
Continuity A function is continuous at a point z0 if
meaning that
1. the function f has a limit at point z0 and
2. the limit is equal to the value of f(z0)
18. Continuity
35
00lim ( ) ( )
z zf z f z
For a given positive number ε, there exists a positive number δ, s.t.
0| |z z When 0| ( ) ( ) |f z f z
00 | | ?z z
School of Software
Theorem 1 A composition of continuous functions is itself continuous.
18. Continuity
36
Suppose w=f(z) is a continuous at the point z0; g=g(f(z)) is continuous at the point f(z0)
Then the composition g(f(z)) is continuous at the point z0
School of Software
Theorem 2 If a function f (z) is continuous and nonzero at a point z0,
then f (z) ≠ 0 throughout some neighborhood of that point.
18. Continuity
37
Proof 0
0lim ( ) ( ) 0z z
f z f z
0| ( ) |0, 0, . .
2
f zs t
0| |z z When
00
| ( ) || ( ) ( ) |
2
f zf z f z
r
f(z0)f(z)
If f(z)=0, then 0
0
| ( ) || ( ) |
2
f zf z
Contradiction!
0| ( ) |f z
Why?
School of Software
Theorem 3 If a function f is continuous throughout a region R that is
both closed and bounded, there exists a nonnegative real number M such that
where equality holds for at least one such z.
18. Continuity
38
| ( ) |f z M for all points z in R
2 2| ( ) | ( , ) ( , )f z u x y v x y Note:
where u(x,y) and v(x,y) are continuous real functions
School of Software
pp. 55-56
Ex. 2, Ex. 3, Ex. 6, Ex. 9, Ex. 11, Ex. 12
18. Homework
39
School of Software
Derivative
Let f be a function whose domain of definition contains a neighborhood |z-z0|<ε of a point z0. The derivative of f at z0 is the limit
And the function f is said to be differentiable at z0 when f’(z0) exists.
19. Derivatives
40
0
00
0
( ) ( )'( ) lim
z z
f z f zf z
z z
School of Software
Illustration of Derivative
19. Derivatives
41
0 00
0
( ) ( )'( ) lim
z
f z z f zf z
z
0limz
dw w
dz z
0 0( ) ( )w f z z f z
0
00
0
( ) ( )'( ) lim
z z
f z f zf z
z z
O u
v
f(z0)
f(z0+Δz)
Δw
0z z z
Any position
School of Software
Example 1 Suppose that f(z)=z2. At any point z
since 2z + Δz is a polynomial in Δz. Hence dw/dz=2z or f’(z)=2z.
19. Derivatives
42
2 2
0 0 0
( )lim lim lim(2 ) 2z z z
w z z zz z z
z z
School of Software
Example 2 If f(z)=z, then
19. Derivatives
43
w z z z z z z z
z z z z
Case #1: Δx0, Δy=0
( , ) (0,0)z x y In any direction
O Δx
Δy
Case #1
Case #2
0
0lim 1
0x
z x i
z x i
Case #2: Δx=0, Δy0
0
0lim 1
0x
z i y
z i y
Since the limit is unique, this function does not exist anywhere
School of Software
Example 3 Consider the real-valued function f(z)=|z|2. Here
19. Derivatives
44
2 2| | | | ( )( )w z z z z z z z zz zz z z
z z z z
Case #1: Δx0, Δy=0
0 0
0lim ( ) lim ( )
0x x
z x iz z z z x z z z
z x i
Case #2: Δx=0, Δy0
0 0
0lim ( ) lim ( )
0y y
z i yz z z z i y z z z
z i y
0z z z z z dw/dz can not exist when z is not 0
School of Software
Continuity & Derivative Continuity Derivative
Derivative Continuity
19. Derivatives
45
For instance, f(z)=|z|2 is continuous at each point, however, dw/dz does not exists when z is not 0
0 0 0
00 0 0
0
( ) ( )lim[ ( ) ( )] lim lim( ) '( )0 0z z z z z z
f z f zf z f z z z f z
z z
Note: The existence of the derivative of a function at a point implies the continuity of the function at that point.
School of Software
Differentiation Formulas
20. Differentiation Formulas
46
[ ( ) ( )] '( ) '( )d
f z g z f z g zdz
[ ( ) ( )] ( ) '( ) '( ) ( )d
f z g z f z g z f z g zdz
2
( ) '( ) ( ) ( ) '( )[ ]
( ) [ ( )]
d f z f z g z f z g z
dz g z g z
1[ ]n ndz nz
dz
0; 1; [ ( )] '( )d d d
c z cf z cf zdz dz dz
( ) ( ( ))F z g f z
0 0 0'( ) '( ( )) '( )F z g f z f z
dW dW dw
dz dw dz
Refer to pp.7 (13)
School of Software
Example To find the derivative of (2z2+i)5, write w=2z2+i and
W=w5. Then
20. Differentiation Formulas
47
2 5 4 2 4 2 4(2 ) (5 ) ' 5(2 ) (4 ) 20 (2 )d
z i w w z i z z z idz
School of Software
pp. 62-63
Ex. 1, Ex. 4, Ex. 8, Ex. 9
20. Homework
48
School of Software
Theorem Suppose that
and that f’(z) exists at a point z0=x0+iy0. Then the first-order partial derivatives of u and v must exist at (x0,y0), and they must satisfy the Cauchy-Riemann equations
then we have
21. Cauchy-Riemann Equations
49
( ) ( , ) ( , )f z u x y iv x y
;x y y xu v u v
0 0 0 0 0'( ) ( , ) ( , )x xf z u x y iv x y
School of Software
Proof:
21. Cauchy-Riemann Equations
50
Let 0 0 0;z x iy z x i y
0 0
0 0 0 0 0 0 0 0
( ) ( )
[ ( , ) ( , )] [ ( , ) ( , )]
w f z z f z
u x x y y u x y i v x x y y v x y
00
0 0 0 0 0 0 0 0
( , ) (0,0)
'( ) lim
( , ) ( , )] [ ( , ) ( , )lim
z
x y
wf z
zu x x y y u x y i v x x y y v x y
x i y
Note that (Δx, Δy) can be tend to (0,0) in any manner .
Consider the horizontally and vertically directions
School of Software
Horizontally direction (Δy=0)
Vertically direction (Δx=0)
21. Cauchy-Riemann Equations
51
0 0 0 0 0 0 0 00
0
( , ) ( , ) [ ( , ) ( , )]'( ) lim
0x
u x x y y u x y i v x x y y v x yf z
x i
0 0 0 0 0 0 0 0
0
( , ) ( , ) [ ( , ) ( , )]limx
u x x y u x y i v x x y v x y
x
0 0 0 0( , ) ( , )x xu x y iv x y
0 0 0 0 0 0 0 00 0
( , ) ( , ) [ ( , ) ( , )]'( ) lim
0y
u x y y u x y i v x y y v x yf z
i y
2
0 0 0 0 0 0 0 0
0
{[ ( , ) ( , )] [ ( , ) ( , )]}lim
( )y
i u x y y u x y i v x y y v x y
i i y
0 0 0 0( , ) ( , )y yv x y iu x y
;x y y xu v u v Cauchy-Riemann equations
School of Software
Example 1
is differentiable everywhere and that f’(z)=2z. To verify that the Cauchy-Riemann equations are satisfied everywhere, write
21. Cauchy-Riemann Equations
52
2 2 2( ) 2f z z x y i xy
2 2( , )u x y x y ( , ) 2v x y xy
2x yu x v 2y xu y v
'( ) 2 2 2( ) 2f z x i y x iy z
School of Software
Example 2
21. Cauchy-Riemann Equations
53
2( ) | |f z z
2 2( , )u x y x y ( , ) 0v x y
If the C-R equations are to hold at a point (x,y), then
0x y y xu u v v
0x y
Therefore, f’(z) does not exist at any nonzero point.
School of Software
Theorem
be defined throughout some ε neighborhood of a point z0 = x0 + iy0, and suppose that
a. the first-order partial derivatives of the functions u and v with respect to x and y exist everywhere in the neighborhood;
b. those partial derivatives are continuous at (x0, y0) and satisfy the Cauchy–Riemann equations
Then f ’(z0) exists, its value being f ’ (z0) = ux + ivx where the right-hand side is to be evaluated at (x0, y0).
22. Sufficient Conditions for Differentiability
54
( ) ( , ) ( , )f z u x y iv x y
;x y y xu v u v at (x0,y0)
School of Software
Proof
22. Sufficient Conditions for Differentiability
55
z x i y Let
0 0( ) ( )w f z z f z u i v
0 0 0 0( , ) ( , )u u x x y y u x y 0 0 0 0( , ) ( , )v v x x y y v x y
0 0 0 0 1 2( , ) ( , )x yu u x y x u x y y x y
Note : (a) and (b) assume that the first-order partial derivatives of u and v are continuous at the point (x0,y0) and exist everywhere in the neighborhood
0 0 0 0 3 4( , ) ( , )x yv v x y x v x y y x y
Where ε1, ε2, ε3 and ε4 tend to 0 as (Δx, Δy) approaches (0,0) in the Δz plane.
School of Software
22. Sufficient Conditions for Differentiability
56
0 0 0 0 1 2( , ) ( , )x yu u x y x u x y y x y
0 0 0 0 3 4( , ) ( , )x yv v x y x v x y y x y
Note : The assumption (b) that those partial derivatives are continuous at (x0, y0) and satisfy the Cauchy–Riemann equations
-vx(x0,y0)
ux(x0,y0)
0 0 0 0 1 2( , ) ( , )x xu u x y x v x y y x y
0 0 0 0 3 4( , ) ( , )x xv v x y x u x y y x y
0 0 0 0 1 2 0 0 0 0 3 4( , ) ( , ) ( , ) ( , )x x x x
w u i v
z zu x y x v x y y x y v x y x u x y y x y
iz z
0 0 0 0 0 0 0 0 1 3 2 4( , ) ( , ) ( , ) ( , ) ( ) ( )x x x xu x y x iu x y y v x y x iv x y y i x i yi
z z z z
i2vx(x0,y0)
0 0 0 0 1 3 2 4( , ) ( , ) ( ) ( )x x
x yu x y iv x y i i
z z
School of Software
22. Sufficient Conditions for Differentiability
57
0 0 0 0 1 3 2 4( , ) ( , ) ( ) ( )x x
w x yu x y iv x y i i
z z z
| | 1;| | 1x y
z z
1 3 1 3 1 3| ( ) | | | | | | |x
i iz
2 4 2 4 2 4| ( ) | | | | | | |y
i iz
0 0 0 00
lim ( , ) ( , )x xz
wu x y iv x y
z
School of Software
Example 1
22. Sufficient Conditions for Differentiability
58
( ) cos sinz x i y x xf z e e e e y ie y
( , ) cosxu x y e y ( , ) sinxv x y e y
Both Assumptions (a) and (b) in the theorem are satisfied.
'( ) cos sin ( )x xx xf z u iv e y ie y f z
School of Software
Example 2
22. Sufficient Conditions for Differentiability
59
2( ) | |f z z
2 2( , )u x y x y ( , ) 0v x y
2 0 0x yu v x x
2 0 0y xu v x y
Therefore, f has a derivative at z=0, and cannot have a derivative at any nonzero points.
School of Software
Assuming that z0≠0
23. Polar Coordinates
60
cos , sinx r y r u u x u y
r x r y r
u u x u y
x y
cos sinr x yu u u sin cosx yu u r u r
Similarly cos sinr x yv v v sin cosx yv v r v r If the partial derivatives of u and v with respect to x and y satisfy the Cauchy-Riemann equations
;x y y xu v u v
;r rru v u rv
School of Software
Theorem Let the function f(z)=u(r,θ)+iv(r,θ) be defined throughout
some ε neighborhood of a nonzero point z0=r0exp(iθ0) and suppose that
the first-order partial derivatives of the functions u and v with respect to r and θ exist everywhere in the neighborhood;
those partial derivatives are continuous at (r0, θ0) and satisfy the polar form rur = vθ, uθ = −rvr of the Cauchy-Riemann equations at (r0, θ0)
Then f’(z0) exists, its value being
23. Polar Coordinates
61
0 0 0 0 0'( ) ( ( , ) ( , ))ir rf z e u r iv r
School of Software
Example 1 Consider the function
23. Polar Coordinates
62
1 1 1 1( ) (cos sin ), 0i
if z e i z
z re r r
cos sin( , ) , ( , )u r v r
r r
Then
cos sin&r rru v u rv
r r
2 2 2 2 2
cos sin 1 1'( ) ( )
( )
ii i
i
ef z e i e
r r r re Z
School of Software
pp. 71-72
Ex. 1, Ex. 2, Ex. 6, Ex. 7, Ex. 8
23. Homework
63
School of Software
Analytic at a point z0
A function f of the complex variable z is analytic at a point z0 if it has a derivative at each point in some neighborhood of z0.
Analytic function
A function f is analytic in an open set if it has a derivative everywhere in that set.
24. Analytic Function
64
Note that if f is analytic at a point z0, it must be analytic at each point in some neighborhood of z0
Note that if f is analytic in a set S which is not open, it is to be understood that f is analytic in an open set containing S.
School of Software
Analytic vs. Derivative For a point
Analytic Derivative
Derivative Analytic
For all points in an open set
Analytic Derivative
Derivative Analytic
24. Analytic Function
65
f is analytic in an open set D iff f is derivative in D
School of Software
Singular point (singularity) If function f fails to be analytic at a point z0 but is analytic at
some point in every neighborhood of z0, then z0 is called a singular point.
For instance, the function f(z)=1/z is analytic at every point in the finite plane except for the point of (0,0). Thus (0,0) is the singular point of function 1/z.
Entire Function
An entire function is a function that is analytic at each point in the entire finite plane.
For instance, the polynomial is entire function.
24. Analytic Function
66
School of Software
Property 1 If two functions are analytic in a domain D, then their sum and product are both analytic in D their quotient is analytic in D provided the function in the
denominator does not vanish at any point in D
Property 2 From the chain rule for the derivative of a composite function, a
composition of two analytic functions is analytic.
24. Analytic Function
67
( ( )) '[ ( )] '( )d
g f z g f z f zdz
School of Software
Theorem If f ’(z) = 0 everywhere in a domain D, then f (z) must be
constant throughout D.
24. Analytic Function
68
( )du
gradu Uds
x=u ygradu i u j
0 & 0x y x yu u v v
'( ) 0x x y yf z u iv v iu
U is the unit vector along L
School of Software
Example 1 The quotient
is analytic throughout the z plane except for the singular points
25. Examples
69
3
2 2
4( )
( 3)( 1)
zf z
z z
3 &z z i
School of Software
Example 3 Suppose that a function and its
conjugate are both analytic in a given domain D. Show that f(z) must be constant throughout D.
25. Examples
70
( ) ( , ) ( , )f z u x y iv x y
( ) ( , ) ( , )f z u x y iv x y
( ) ( , ) ( , )f z u x y iv x y is analytic, then
,x y y xu v u v ( ) ( , ) ( , )f z u x y iv x y is analytic, then
Proof:
,x y y xu v u v
0, 0x xu v '( ) 0x xf z u iv
Based on the Theorem in pp. 74, we have that f is constant throughout D
School of Software
Example 4 Suppose that f is analytic throughout a given region D, and
the modulus |f(z)| is constant throughout D, then the function f(z) must be constant there too.
Proof:
|f(z)| = c, for all z in D
where c is real constant.
If c=0, then f(z)=0 everywhere in D.
If c ≠ 0, we have
25. Examples
71
2( ) ( )f z f z c2
( ) , ( ) 0( )
cf z f z inD
f z
Both f and it conjugate are analytic, thus f must be constant in D. (Refer to Ex. 3)
School of Software
pp. 77~78
Ex. 2, Ex. 3, Ex. 4, Ex. 6, Ex. 7
25. Homework
72
School of Software
A Harmonic Function A real-valued function H of two real variables x and y is
said to be harmonic in a given domain of the xy plane if, throughout that domain, it has continuous partial derivatives of the first and second order and satisfies the partial differential equation
Known as Laplace’s equation.
26. Harmonic Functions
73
( , ) ( , ) 0xx yyH x y H x y
School of Software
Theorem 1 If a function f (z) = u(x, y) + iv(x, y) is analytic in a domain
D, then its component functions u and v are harmonic in D.
Proof:
26. Harmonic Functions
74
( ) ( , ) ( , )f z u x y iv x y is analytic in D
&x y y xu v u v Differentiating both sizes of these equations with respect to x and y respectively, we have
Theorem in Sec.52: a function is analytic at a point, then its real and imaginary components have continuous partial derivatives of all order at that point.
continuity
0 & 0xx yy xx yyu u v v
&xx yx yx xxu v u v
&xy yy yy xyu v u v &xx xy xy xxu v u v &xy yy yy xyu v u v
School of Software
Example 3 The function f(z)=i/z2 is analytic whenever z≠0 and since
The two functions
26. Harmonic Functions
75
2 2 2
2 2 2 22 2
( ) 2 ( )
( )( )
i i z xy i x y
z x yz z
2 2 2
2( , )
( )
xyu x y
x y
2 2
2 2 2( , )
( )
x yv x y
x y
are harmonic throughout any domain in the xy plane that does not contain the origin.
School of Software
Harmonic conjugate
If two given function u and v are harmonic in a domain D and their first-order partial derivatives satisfy the Cauchy-Riemann equation throughout D, then v is said to be a harmonic conjugate of u.
26. Harmonic Functions
76
If u is a harmonic conjugate of v, then
&x y y xu v u v
Is the definition symmetry for u and v?
Cauchy-Riemann equation &x y y xu v u v
School of Software
Theorem 2 A function f (z) = u(x, y) + iv(x, y) is analytic in a domain D
if and only if v is a harmonic conjugate of u.
Example 4
The function is entire function, and its real and imaginary components are
Based on the Theorem 2, v is a harmonic conjugate of u throughout the plane. However, u is not the harmonic conjugate of v, since is not an analytic function.
26. Harmonic Functions
77
2 2( , ) & ( , ) 2u x y x y v x y xy
2( )f z z
2 2( ) 2 ( )g z xy i x y
School of Software
Example 5 Obtain a harmonic conjugate of a given function.
Suppose that v is the harmonic conjugate of the given function
Then
26. Harmonic Functions
78
3 2( , ) 3u x y y x y
&x y y xu v u v
6x yu xy v
2 23 3y xu y x v
23 ( )v xy x
2 2 23 3 ( 3 '( ))y x y x
2 3'( ) 3 ( )=x x x x C
2 33v xy x C
School of Software
pp. 81-82
Ex. 1, Ex. 2, Ex. 3, Ex. 5
26. Homework
79
School of Software
Lemma Suppose that
a) A function f is analytic throughout a domain D;
b) f(z)=0 at each point z of a domain or line segment contained in D.
Then f (z) ≡ 0 in D; that is, f (z) is identically equal to zero throughout D.
27. Uniquely Determined Analytic Function
80
Refer to Chap. 6 for the proof.
School of Software
Theorem
A function that is analytic in a domain D is uniquely determined over D by its values in a domain, or along a line segment, contained in D.
27. Uniquely Determined Analytic Function
81
f(z)
D
g(z)
D
( ) ( )f z g z
School of Software
Theorem Suppose that a function f is analytic in some domain D
which contains a segment of the x axis and whose lower half is the reflection of the upper half with respect to that axis. Then
for each point z in the domain if and only if f (x) is real for each point x on the segment.
28. Reflection Principle
82
x
y
D
( ) ( )f z f z