Chapter 9. Conformal Mapping
Weiqi Luo (骆伟祺)
School of Software
Sun Yat-Sen University
Email：weiqi.luo@yahoo.com Office：# A313
Chapter 9: Conformal Mapping




Preservation of Angles
Scale Factors
Local Inverses
Harmonic Conjugates
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101. Preservation of Angles
 Preservation of Angles
Let C be a smooth arc, represented by the equation
z  z (t ),(a  t  b)
and let f(z) be a function defined at all points z on C.
The equation
w  f [ z (t )],(a  t  b)
is a parametric representation of the image Г of C under
the transformation w=f(z).
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101. Preservation of Angles
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
w '(t0 )  f '[ z(t0 )]z '(t0 ) arg w '(t0 )  arg f '[ z(t0 )]  arg z '(t0 )
0   0  0
 0  arg f '( z0 )
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101. Preservation of Angles
 Consider two intersectant arcs C1 and C2
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.
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101. Preservation of Angles
 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.
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101. Preservation of Angles
 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.
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101. Preservation of Angles
 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.
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101. Preservation of Angles
 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 w  z , which is a reflection in the real
axis, is isogonal but not conformal. If it is followed by a
conformal transformation, the resulting transformation
w  f ( z ) is also isogonal but not conformal.
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101. Preservation of Angles
 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.
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102. Scale Factors
 Scale factor
From the definition of derivative, we know that
f ( z )  f ( z0 )
| f '( z0 ) || lim
|
z  z0
z  z0
| f ( z )  f ( z0 ) |
 lim
z  z0
| z  z0 |
Exercise 7, Sec. 18
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.
Expansion:|
f '( z0 ) | 1
Contraction:|
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f '( z0 ) | 1
School of Software
102. Scale Factors
 Example
When f(z)=z2, the transformation
w  f ( z)  x  y  i 2xy
2
2
is conformal at the point z=1+i, where the half lines
y  x,( x  0) & x  1,( y  1)
intersect.
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102. Scale Factors
 Illustrations
w  f ( z )  x 2  y 2  i 2 xy
 u( x, y)  iv( x, y)
C1: y=x, x ≥ 0
Г1: u=0, v=2x2 ,x≥0
C2: x=1,y ≥ 0
Г2: u=1-y2, v=2y
C3: y=0,x ≥ 0
Г3: u=x2, v=0
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f '( z)  2 z  2( x  iy)
f '(1)  2  1&| f '(1  i) | 2 2  1
School of Software
103. Local Inverses
 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,
1
g '( w) 
f '( z )
Note that the transformation z=g(w) is itself conformal at w0.
Refer to pp. 360-361 for the proof!
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103. Local Inverses
 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
g (w)  log w  ln   i ,(   0,     3 )
Why?
Not contain the origin
g (1)  2 i & g '(w)  1/ w
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103. Local Inverses
 Example (Cont’)
If the point z0=0 is chosen, one can use the principal
branch
g (w)  L og w  ln   i ,(   0,      )
g (1)  ln1  i0  0
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103. Homework
pp. 362-363
Ex. 1, Ex. 6
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104. Harmonic Conjugates
 Suppose
f ( z)  u( x, y)  iv( x, y)
is analytic in a domain D, then the real-valued functions
u and v are harmonic in that domain. That is
uxx  uyy  0& vxx  vyy  0
pp. 79 Theorem 1
According to the Cauchy-Riemann equations
ux  uy & uy  vx
And v is called a harmonic conjugate of u.
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104. Harmonic Conjugates
 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 firstorder 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
 P( s, t )ds  Q(s, t )dt
C
from (x0, y0) to (x, y) is independent of the contour C that is
taken as long as the contour lies entirely in D.
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104. Harmonic Conjugates
The integral is a single-valued function with the parameters x and y
( x, y )
F ( x, y ) 

P( s, t )ds  Q( s, t )dt
( x0 , y0 )
Furthermore, we have
Fx ( x, y)  P( x, y) & Fy ( x, y)  Q( x, y)
P
Since u(x,y) is harmonic
Q
uxx  uyy  0  (uy ) y  (ux ) x
( x, y )
v ( x, y ) 

ut ( s, t ) ds  us ( s, t )dt
+C,C  R
( x0 , y0 )
vx ( x, y)  uy ( x, y)&v y ( x, y)  ux ( x, y)
Therefore v is the harmonic conjugate of u.
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104. Harmonic Conjugates
 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)
y2
ux  y  v y
v
  ( x)
2
x2
 ( x)    C , C  R
u y  vx
x   '( x)
2
 x2  y 2
v
 C, C  R
2
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104. Harmonic Conjugates
 Example (Cont’)
Way #2:
us  t & ut  s
u ( s, t )  st
( x, y )
v ( x, y ) 

( x, y )
ut ds  us dt 
(0,0)
t
(x,0)
 sds  tdt
(0,0)
1 2 1 2
v ( x, y )   x  y
2
2
(x,y)
O

+C
s
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104. Homework
pp. 81
Ex. 1 (using method #2)
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