ECE 6382
Functions of a Complex
Variable as Mappings
D. R. Wilton
ECE Dept.
8/24/10
A Function of a Complex Variable as a Mapping
 A function of a complex variable, w  f  z  , is usually viewed
as a mapping from the complex z
y
w  f  z
z
z
x
to the complex w plane.
v
w
w
u
Simple Mappings: Translations
 Translation :
waz
where a is a complex constant.
z
y
z
waz
w
v
w
z
x
u
a
 The mapping translates every point in the z plane
by the "vector" a .
Simple Mappings: Rotations
 Rotation :
i
we z  e
i
 re 
i
 re
i  
where  is a real constant.
z
y
v
i
we z
z
w

x
w
 The mapping rotates every point in
the z plane through an angle  .
z
u
Simple Mappings: Dilations
 Dilation (stretching) :
 
w  az  a rei
 (ar )ei
where a is a real constant.
z
y
z
w  az
x
w
v
w
z
u
 The mapping magnifies the magnitude z of a point z
in the complex plane by a factor a .
 Multiplying z by a complex number A  A arg A is equivalent to
stretching it by a factor A and rotating it through an angle arg A
Simple Mappings: Inversions
 Inversion:
1
1
ei
w
 i 
z
r
re
Im
Im
z
z  x0 iy
1
1 x0
1z
Re
Re
x0
The straight line z  x0  iy maps to a circle
1z
1
Inversion circle - preserving property
 Inversions have a "circle preserving" property, i.e., circles
always map to circles (straight lines are "circles" of infinite radius).
 Points outside the unit circle map to points inside the unit circle
and vice versa.
Inversion – Line-to-Circle Property
 To prove the straight line z  x0  iy maps to a circle :
Im
w  u  iv 
z  x0 iy
 u
1
1 x0
1z
Re
x0
Im
1 z*
x  iy
 2  20
z z
x0  y 2
x0
x02  y 2
1,
v
( x0 is a constant, y is a parameter)
y
x02  y 2
2 ,
y merely parameterizes the line, so eliminate it :
u x0
xv
 1 2 

,  y  0
3
v y
u
Substituting from 3 into 2
x0 v
u2
ux v
u
 v 
 2  2 20 2 2.
2
u
u x0  v x0
x v
x02   0 
 u 
2
Rearranging, u 2 x02  ux0  v 2 x02  0  ux0  12  v 2 x02  14

1 x0
1z
Re
x0
or

u  2 x1
0

2
 v   2 1x 
 0
2
1 ,
2x0


centered at  u,v    1 , 0 
2
x
 0 
Eq. of circle of radius
2

Inversion – Line-to-Circle Property, cont’d
Im
z  d  it
 z  d  it,    t   , is the equation of vertical line
passing through x  d on the x - axis .
Re
1d
d
1z
 Multiplying z by ei z rotates the line z  d  it by the angle  ,
but the circle is rotated in the opposite direction :
Im
1
1
w  i  ei
z
e z
z
d

1d
1z

Re
 Since every straight line in the z - plane may be
represented as
z  ei (d  it ),  <t < 
every straight line in the z - plane is mapped into
a circle via the transformation
1
1
w   e i
,  <t < 
z
d  it
Simple Mappings: Inversions, cont’d
1
1
ei
 Geometrical construction of the inversion w 
 i 
z
r
re
w v
y
u

z
1
w
z
z

x
w
v
1
z

w
w
u
A General Linear Transformation (Mapping) Is a
Combination of Translation, Rotation, and Dilation
 Linear transformation :
dilation
w  A  Bz  A  B ei Arg B rei 
 Br e
A
rotation
i   Arg B 
translation
where A ,B are complex constants .
z
y
z
w  A  Bz
w
v
Bz
Arg B
x
w
A
u
A General Bilinear Transformation (Mapping) Is a Succession
of Translations, Rotations, Dilations, and Inversions
 Bilinear (Fractional or Mobius) transformation : w 
A  Bz
C  Dz
where A ,B,C,D are complex constants .
 Note that if D  0,
w
A  BC D  B D  C  Dz 
A  Bz


C  Dz
C  Dz
If we let f    C  D ,
and
g   
 A  BC D 

A  BC D
 B D
C  Dz
(translation, dilation, rotation)
(inversion, dilation, rotation)
then the Mobius transformation may be written as
A  Bz
w
 B D  g  f  z 
C  Dz
which is just a sequence of translations, dilations, rotations, and an inversion

Since each transformation preserves circles, bilinear transformations also
have the circle - preserving property : circles in the z plane are mapped into circles in
the w plane, with straight lines included as circles of infinite radius.
Bilinear Transformation Example: The Smith Chart
 Let z  r  jx 
Z 
where Z 
Z0
  R    jX  
is the impedance at a
point on a transmission line of characteristic impedance Z 0 and


is the generalized reflection coefficient



Z
Z
  Z0
  Z0

z
z
 1
 1
or simply  
z 1
z 1
at point .
jx
z 1
 
z 1
z
r
Im 

Re 
For an interpretation of mobius transformations as projections on a sphere, see
http://www.youtube.com/watch?v=JX3VmDgiFnY
The Squaring Transformation

w  f  z   z 2  r 2 ei 2
y
z
v
w
z
x
w
u
z
 The transformation maps half the z - plane into the entire w - plane
 The entire z - plane covers the w - plane twice
 The transformation is said to be many - to - one
Another Mapping of the Squaring
Transformation

w  f  z   z 2  r 2 ei 2
z
180o
270o
3
Im
90o
9
2
4
1
1
360o
-360o
Re
1
-270o
-180o
2
3
0o
-90o
2
The Square Root Transformation

w  f  z  z  r e
z
bottom
sheet
    3
1 i   k 2
2
2
1
2
, k  0,1
y
v
z
w
w u
x
w
top sheet
    
 The transformation maps the entire z - plane into one half of the w - plane
 The transformation is said to be one - to - many.
 The inverse of the square
function (i.e., the square root function) is not unique.
Connecting the Two Sheets at a Branch Cut to Form a
Riemann Surface

w  f  z 
z
1
z2
1 i k 2
 r 2e 2
, k  0,1
top sheet
y
branch point
    3
z
x
branch cut
    
bottom sheet
 If we agree to limit arg z to one sheet, never crossing the branch cut,
the inverse is unique. Physics usually dictates the branch choice and cut.
 The two sheets shown may be defined as
1)    arg z   (principal branch)
2)
  arg z  3
Only at a branch point are multiple
cycles encircling the point required
to return to the starting value
The Branch Cut Allows Us to Map the Square Root Function

1
2
3
2
i  k
w  f  z  z  r e
45o
67.5o
1
2
 , k  0,1,
    
225o
Im
22.5o
247.5o
3
2
3
Im
3
2
2
2
1
1
1
90o
-90o
1
Re
1
-67.5o
202.5o
-45o
top sheet, k = 0
2
3
270o
90o
0o
-22.5o
1
112.5o
z
1
2
Re
135o
2
3
180o
157.5o
bottom sheet , k = 1
Constant u and v Contours are Orthogonal
 Consider contours in the z   x, y  plane on which the real quantities u  x, y  and v  x, y 
are constant.
y
u
v  constant
z
v
w
v
u  constant
x
u
 The directions normal to these contours are along the gradient direction :
u
u
ˆx  yˆ
x
y
v
v
ˆ
v  ˆx  y
x
y
u 
 The gradients, and therefore the contours, are orthogonal (perpendicular) by the C. R. conditions :
 u
u   v
v 
ˆ    ˆx  yˆ 
u  v   ˆx  y
y   x
y 
 x
C.R.
cond's

 u
u   u
u 
u u
u u
ˆ
ˆ
ˆ
ˆ
x

y


x

y



0

 


x

y

y

x

y

x

y

x

 

Mappings of Analytic Functions are Conformal
(Angle-Preserving)
 Consider a pair of intersecting paths C 1, C 2 in the z  x  iy plane mapped
onto the w  u  iv plane.
y
v

z
C1
w
z0
 The condition f   z0   0 below is
C1

needed since Arg 0 is indeterminant!
w0

 C2
u
x
C2
E.g., consider w  f  z   z 2 at z  0 :
w  f   z  z  2 zz; since f   0   0,
 Arg w  z 0  Arg z  z 0  Arg z  2Arg z
Arg z
 If f   z0   0 at the intersection of their mapping, w0  f  z0  , then the angles
between the intersecting contours are preserved :
w1  f   z0  z 1
 Argw1  Arg f   z0   Arg z 1 , z 1 along C1
w 2  f   z0  z 2
 Argw 2  Arg f   z0   Arg z 2 , z 2 along C2
 Argw 2  Argw1  Arg z 2  Arg z 1
The approximation improves as z2 , z 1  0, with
Arg z 2  Arg z 1  
Argw 2  Argw1   and the above approximation yielding
 
Constant |w| and Arg w Contours are also
Orthogonal
 If w  Re i  the constant R and  contours are orthogonal.
 If z  f 1  w  is a mapping back to (a branch of) the z plane,
the mapping preserves the orthogonality.
45o
w
3
67.5
v
Im
22.5o
o
3
2
z
2
u
1
1
90o
-90o
Re
1
2
3
0o
1
 Magnitude, phase plot of f  z   z 2 :
-67.5o
-45o
-22.5o
The Logarithm Function
 Since w  e z  e z i 2 k , the inverse function z  ln w cannot be unique.
I.e., given w, z is the complex number such that e z  w . But this
number cannot be unique, since z  z  i 2 k, k  1, 2 , ,
also has the property that e z  e z i 2 k  e z  w.
 Let w  ln z . To evaluate w , note that we have
z  z ei  e w  e wi 2 k  eu iv i 2 k  eu ei (v  2 k ) so that
z  eu  u  ln z , v    2 k , k  0, 1, 2 ,
and hence
w  u  iv  ln z  i   2 k 
, k  0, 1, 2 ,
which has a branch point z  0 with an infinite number of branches.
y
z
Branch
point
y
x
Branch cut
Riemann surface
for the log function
Infinite # of sheets
x
Arbitrary Powers of Complex Numbers


 Since z  eln z  ln e z ,
z a  ealn z  e
aln z  ai  2 k 
also has an infinite number of values (branches), unless ak  integer for some k,
i.e., unless a  ar  i ai is real ( ai  0) and rational.
Riemann surfaces for za
y
z
Branch
point
x
Branch cut
Infinite # of sheets
y
x