Chapter 9. Conformal Mapping

Weiqi Luo (骆伟祺 )School of Software

Sun Yat-Sen UniversityEmail ： weiqi.luo@yahoo.com Office ： # A313

School of Software

Preservation of Angles Scale Factors Local Inverses Harmonic Conjugates

2

Chapter 9: Conformal Mapping

School of Software

Preservation of Angles Let C be a smooth arc, represented by the equation

and let f(z) be a function defined at all points z on C. The equation

is a parametric representation of the image Г of C under the transformation w=f(z).

101. Preservation of Angles

3

( ), ( )z z t a t b

[ ( )], ( )w f z t a t b

School of Software

Suppose that C passes through a point z0=z(t0) (a<t0<b) at which f is analytic and f’(z0)≠0. According to the chain rule, if w(t)=f[z(t)], then

101. Preservation of Angles

4

0 0 0'( ) '[ ( )] '( )w t f z t z t 0 0 0arg '( ) arg '[ ( )] arg '( )w t f z t z t

0 0 0

0 0arg '( )f z

School of Software

Consider two intersectant arcs C1 and C2

101. Preservation of Angles

5

1 0 1 2 0 2

0 1 1 2 2 1 2 1 2

from C1 to C2 from Г1 to Г 2

Note that both magnitude and sense are the same.

School of Software

Conformal A transformation w=f(z) is said to be conformal at a

point z0 if f is analytic there and f’(z0)≠0.

Note that such a transformation is actually conformal at each point in some neighborhood of z0. For it must be analytic in a neighborhood of z0; and since its derivative f ’ is continuous in that neighborhood, Theorem 2 in Sec. 18 tells us that there is also a neighborhood of z0 throughout which f (z) ≠ 0.

101. Preservation of Angles

6

School of Software

Conformal Mapping A transformation w = f (z), defined on a domain D, is

referred to as a conformal transformation, or conformal mapping, when it is conformal at each point in D. That is, the mapping is conformal in D if f is analytic in D and its derivative f has no zeros there.

101. Preservation of Angles

7

School of Software

Example 1 The mapping w = ez is conformal throughout the entire z

plane since (ez)’ = ez ≠ 0 for each z.

Consider any two lines x = c1 and y = c2 in the z plane, the first directed upward and the second directed to the right.

101. Preservation of Angles

8

School of Software

Isogonal mapping A mapping that preserves the magnitude of the angle

between two smooth arcs but not necessarily the sense is called an isogonal mapping.

Example 3 The transformation , which is a reflection in the real

axis, is isogonal but not conformal. If it is followed by a conformal transformation, the resulting transformation

is also isogonal but not conformal.

101. Preservation of Angles

9

w z

( )w f z

School of Software

Critical Point Suppose that f is not a constant function and is analytic at a

point z0 . If, in addition, f ’(z0) = 0, then z0 is called a critical point of the transformation w = f (z).

Example 4. The point z0 = 0 is a critical point of the transformation

w = 1 + z2, which is a composition of the mappings

Z = z2 and w = 1 + Z. A ray θ = α from the point z0 = 0 is evidently mapped onto the ray from the point w0 = 1 whose angle of inclination is 2α, and the angle between any two rays drawn from z0 = 0 is doubled by the transformation.

101. Preservation of Angles

10

School of Software

Scale factor From the definition of derivative, we know that

102. Scale Factors

11

0

00

0

( ) ( )| '( ) | | lim |

z z

f z f zf z

z z

0

0

0

| ( ) ( ) |lim

| |z z

f z f z

z z

Now |z-z0| is the length of a line segment joining z0 and z, and|f(z)-f(z0)| is the length of the line segment joining the point f(z0) and f(z) in the w plane.

Exercise 7, Sec. 18

0| '( ) | 1f z Expansion: 0| '( ) | 1f z Contraction:

School of Software

Example When f(z)=z2, the transformation

is conformal at the point z=1+i, where the half lines

intersect.

102. Scale Factors

12

2 2( ) 2w f z x y i xy

, ( 0) & 1,( 1)y x x x y

School of Software

Illustrations

102. Scale Factors

13

C1: y=x, x ≥ 0

C2: x=1,y ≥ 0

C3: y=0,x ≥ 0

Г1: u=0, v=2x2 ,x≥0

Г2: u=1-y2, v=2y

Г3: u=x2, v=0

2 2( ) 2

( , ) ( , )

w f z x y i xy

u x y iv x y

'( ) 2 2( )f z z x iy

'(1) 2 1& | '(1 ) | 2 2 1f f i

School of Software

Local Inverse

A transformation w = f (z) that is conformal at a point z0 has a local inverse there.

That is, if w0 = f (z0), then there exists a unique transformation z = g(w), which is defined and analytic in a neighborhood N of w0, such that g(w0) = z0 and f [g(w)] = w for all points w in N. The derivative of g(w) is, moreover,

Note that the transformation z=g(w) is itself conformal at w0.

103. Local Inverses

14

1'( )

'( )g w

f z

Refer to pp. 360-361 for the proof!

School of Software

Example If f(z)=ez, the transformation w=f(z) is conformal

everywhere in the z plane and, in particular at the point z0=2πi. The image of this choice of z0 is the point w0=1.

When points in the w plane are expressed in the form w = ρ exp(iφ), the local inverse at z0 can be obtained by writing g(w) = logw, where logw denotes the branch

103. Local Inverses

15

( ) log ln , ( 0, 3 )g w w i Why?

(1) 2 & '( ) 1/g i g w w

Not contain the origin

School of Software

Example (Cont’) If the point z0=0 is chosen, one can use the principal

branch

103. Local Inverses

16

( ) og ln , ( 0, )g w L w i

(1) ln1 0 0g i

School of Software

pp. 362-363

Ex. 1, Ex. 6

103. Homework

17

School of Software

Suppose

is analytic in a domain D, then the real-valued functions u and v are harmonic in that domain. That is

According to the Cauchy-Riemann equations

And v is called a harmonic conjugate of u.

104. Harmonic Conjugates

18

( ) ( , ) ( , )f z u x y iv x y

0 & 0xx yy xx yyu u v v

&x y y xu u u v

pp. 79 Theorem 1

School of Software

Properties If u(x,y) is any given harmonic function defined on a simply

connected domain D, then u(x,y) always has a harmonic conjugate v(x,y) in D.

Proof: Suppose that P(x, y) and Q(x, y) have continuous first-order partial derivatives in a simply connected domain D of the xy plane, and let (x0, y0) and (x, y) be any two points in D. If Py = Qx everywhere in D, then the line integral

from (x0, y0) to (x, y) is independent of the contour C that is taken as long as the contour lies entirely in D.

104. Harmonic Conjugates

19

( , ) ( , )C

P s t ds Q s t dt

School of Software

104. Harmonic Conjugates

20

0 0

( , )

( , )

( , ) ( , ) ( , )x y

x y

F x y P s t ds Q s t dt

( , ) ( , ) & ( , ) ( , )x yF x y P x y F x y Q x y

0 0

( , )

( , )

( , ) ( , ) ( , )x y

t s

x y

v x y u s t ds u s t dt

The integral is a single-valued function with the parameters x and y

Furthermore, we have

Since u(x,y) is harmonic 0 ( ) ( )xx yy y y x xu u u u

( , ) ( , )&v ( , ) ( , )x y y xv x y u x y x y u x y

+C,C R

Therefore v is the harmonic conjugate of u.

P Q

School of Software

Example Consider the function u(x, y) = xy, which is harmonic

throughout the entire xy plane. Find a harmonic conjugate of u(x,y).

Way #1: (pp.81)

104. Harmonic Conjugates

21

x yu y v 2

( )2

yv x

y xu v '( )x x2

( ) ,2

xx C C R

2 2

,2

x yv C C R

School of Software

Example (Cont’) Way #2:

104. Harmonic Conjugates

22

( , ) ( , )

(0,0) (0,0)

( , )x y x y

t sv x y u ds u dt sds tdt

( , )u s t st &s tu t u s

2 21 1( , )

2 2v x y x y

O

(x,y)

s

t

(x,0)

+C

School of Software

pp. 81

Ex. 1 (using method #2)

104. Homework

23