1
1 Functions of A Complex Variables I
Functions of a complex variable provide us some powerful and widely useful tools in
mathematical analysis as well as in theoretical physics.
e.g.:
• Some important physical quantities are complex variables (the wave-function , a.c.
impedance Z())
• Evaluating definite integrals.
• Obtaining asymptotic solutions of differentials equations.
• Integral transforms
• Many Physical quantities that were originally real become complex as simple theory
is made more general. The energy En  En0 + i (-1  the finite life time).
1.1 Complex Algebra
We here go through the complex algebra briefly.
A complex number z = (x,y) = x + iy
Where I = (-1)1/2. We will see that the ordering of two real numbers (x,y) is significant, i.e. in
general x + iy  y + ix
x is called the real part, labeled by Re z
y is called the imaginary part, labeled by Im z
For Cartesian components
z1  z 2  x1  x 2  iy1  y 2 
z1z 2  x1 x 2  y1 y 2   ix1 y 2  x 2 y1 
For polar components, we may write
z=r(cos  + i sin)
or
z=rei
Fig. 1.1
r – the modulus or magnitude of z
 - the argument or phase of z

r  z  x2  y2

1/ 2
  tan 1  y / x 
The choice of polar representation or Cartesian representation is a matter of convenience.
Addition and subtraction of complex variables are easier in the Cartesian representation.
Multiplication, division, powers, roots are easier to handle in polar form,
2
z1 z 2  r1r2 e i 1  2 
z / z  r / r e i1  2 
1
n
2
1
n in
2
z r e
Using the vector analogy, we have the triangle inequalities
z1  z 2  z1  z 2  z1  z 2
Using the polar form,
z1 z 2  z1 z 2
arg(z1z2) = arg(z1) + arg(z2)
From z, complex functions f(z) may be constructed. They can be written
f(z) = u(x,y) + iv(x,y)
in which v and u are real. For example if f(z)=z2, we have


f z   x 2  y 2  i 2 xy
The relationship between z and f(z) is best pictured as a mapping operation, we address it in
detail later.
Complex Conjugation: replacing I by –I, which is denoted by (*),
z *  x  iy
We then have
zz *  x 2  y 2  r 2
hence
 
z  zz *
Note:
12
Fig. 1.2
z  re i or rei   2n 
or
ln z  ln r  i
ln z  ln r  i  2n 
ln z is a multi-valued function. To avoid ambiguity, we usually set n=0 and limit the phase to
an interval of length of 2. The value of lnz with n=0 is called the principal value of lnz.
1.2 Cauchy – Riemann conditions
Having established complex functions, we now proceed to differentiate them. The derivative
of f(z), like that of a real function, is defined by
f  z  z   f  z 
f z  df
lim
 lim

 f  z 
z  0
z 0 z
z
dz
provided that the limit is independent of the particular approach to the point z. For real
variable, we require that lim f x   lim f x   f xo  . Now, with z (or zo) some point
x xo 
x xo 
in a plane, our requirement that the limit be independent of the direction of approach is very
restrictive.
3
Consider
z  x  iy ,
 f   u  i v ,
f u  iv

z x  iy
Let us take limit by the two different approaches as in the figure. First, with y = 0, we let
x0,
f
v 
 u
lim
 lim   i 
z 0 z x 0 x
x 
u
v

i
Fig. 1.3
x
x
Assuming the partial derivatives exist. For a second approach, we set x = 0 and then let
y 0. This leads to
f
u v
lim
 i

z  0  z
y y
If we have a derivative, the above two results must be identical. So,
u v
u
v
,


x y
y
x
These are the famous Cauchy-Riemann conditions. These Cauchy-Riemann conditions are
necessary for the existence of a derivative, that is, if f x  exists, the C-R conditions must
hold.
Conversely, if the C-R conditions are satisfied and the partial derivatives of u(x,y) and v(x,y)
are continuous, f z exists. (see the proof in the text book).
1.3 Cauchy’s integral Theorem
We now turn to integration. The integral of a complex variable over a contour in the
complex plane may be defined in close analogy to the integral of a real function integrated
along the real x – axis.
The contour z0z0 is divided into n intervals (Fig. 1.4). Let n   with
z j  z j  z j1  0 for j. Then
lim
n 
n
z0
j 1
z0
 f  j z j   f z dz
Fig. 1.4
4
provided that the limit exists and is independent of the details of choosing the points z j and j,
where j is a point on the curve between zj and zj-1. The right-hand side of the above equation
is called the contour (path) integral of f(z) (along the specific contour integral C from z 0 to
z0).
As an alternative, the contour may be defined by
z2
x2 y2
c z1
c x1 y1
x2 y2
 f z dz   ux, y   ivx, y dx  idy 
x2 y2
 udx  vdy i  vdx  udy

c
x1 y1
c x1 y1
with the path C specified. This reduces the complex integral to the complex sum of real
integrals. It’s somewhat analogous to the case of the vector integral.
An important example:
z
n
dz
c
where C is a circle of radius r>0 around the origin z=0 in the direction of counterclockwise.
In polar coordinates, we parameterize z  re i and dz  ire i d , and have
1
2i

c
r n 1
z dz 
2
n
2
 expin  1 d
0
0
{
1
for n  -1
for n  - 1
which is independent of r. These integrals are examples of Cauchy’s integral formula which
we consider in the next section.
•Cauchy’s integral theorem
If a function f(z) is analytical (therefore single-valued) [and its partial derivatives are
continuous] through some simply connected region R, for every closed path C (Fig. 1.5) in R,
 f z dz  0
c
Fig. 1.5
Stokes’s theorem proof
Proof: (under relatively restrictive condition: the partial derivative of u, v are continuous,
which are actually not required but usually satisfied in physical problems)
 f z dz   udx  vdy  i  vdx  udy 
c
c
c
5
These two line integrals can be converted to surface integrals by Stokes’s theorem


c
s
( A  d l    A  d s ).



Using A  Ax x  A y y and ds  dxdyz ,
We have
 Ax dx  Ay dy    A  d l     A  d s
c
c
s
 A y A x 
dxdy
  


x

y


If we let u = Ax, and v = -Ay, then
udx  vdy    v  u dxdy
 x y 


c
=0
[since C-R conditions
v
u

]
x
y
Setting u = Ay and v = Ax, we have
vdx  udy    u  v dxdy  0
 x y 



 f z dz  0
As for a proof without using the continuity condition, see the text book.
The consequence of the theorem is that for analytic functions the line integral is a function
only of its end points, independent of the path of integration,
z2
z1
z1
z2
 f z dz  F z 2   F z1     f z dz
•Multiply connected regions
The original statement of our theorem demanded a simply connected region. This
restriction may easily be relaxed by the creation of a barrier, a contour line. Consider the
multiply connected region of Fig.1.6 In which f(z) is not defined for the interior R
Fig. 1.6
Cauchy’s integral theorem is not valid for the contour C, but we can construct a C for which
the theorem holds. If line segments DE and GA arbitrarily close together, then
A
E
G
D
 f z dz    f z dz
6
Fig. 1.7
Now let ABDC1 and EFG-C2, we obtain



 f  z dz


f z dz 





C
 ABD DE GA EFG 
 ABDEFGA 


      f z dz  0
ABD EFG 






 f z dz   f z dz
C1
C2
1.4 Cauchy’s Integral Formula
If f(z) is analytic on and within a closed contour C then
f z dz
 2if z 0 
z  z0

C
In which z0 is some point in the interior region bounded by C. Note that here z-z0 0 and the
integral is well defined.
Although f(z) is assumed analytic, the integrand (f(z)/z-z0) is not analytic at z=z0
unless f(z0)=0. If the contour is deformed as in Fig.1.8 Cauchy’s integral theorem applies.
So we have
f z dz
f z 

dz  0
z  z0
z  z0


C
C2
Fig. 1.8
Let z  z 0  re i , here r is small and will eventually be made to approach zero


C2
f z dz
dz 
z  z0

C2

f z 0  re i
re
i
rie  d
i

(r0)  if z 0  d  2if z 0 
C2
Here is a remarkable result. The value of an analytic function is given at an interior point at
z=z0 once the values on the boundary C are specified.
What happens if z0 is exterior to C?
7
In this case the entire integral is analytic on and within C, so the integral vanishes.
1
2i
 f z 0 ,

 0,
f z dz
 z  z0
C
z 0 interior
z 0 exterior
•Derivatives
Cauchy’s integral formula may be used to obtain an expression for the derivation of f(z)
f z 0  

1
2i
d  1 f z dz 


dz 0  2i z  z 0 

d 

f z dz
 f z dz dz0  z  z0   2i  z  z0 2
1
1
Moreover, for the n-th order of derivative
f z dz
n!
f n   z 0  
2i z  z 0 n1
We now see that, the requirement that f(z) be analytic not only guarantees a first derivative
but derivatives of all orders as well! The derivatives of f(z) are automatically analytic. Here,
it is worth to indicate that the converse of Cauchy’s integral theorem holds as well (Morera’s
theorem, see the text book)

Examples:
1. If f z  
 an z n is analytic on and within a circle about the origin, find a .
n 0

j
f z   j! a j   a n  z n j
n
n  j 1
f  j  0  j! a j
f n  0 1

an 

n!
2i

f z dz
z n1
2. In the above case, f z   M on a circle of radius r about the origin, then
a n r n  M (Cauchy’s inequality)
Proof:
an 
1
2

z r
f z dz
z n1
 M r 
2r
2r n1

M
rn
where M r   Max z r f r  .
3. Liouville’s theorem: If f(z) is analytic and bounded in the complex plane, it is a constant.
Proof: For any z0, construct a circle of radius R around z0,
8
f  z 0  
f  z dz
  z  z 0 2
1
2i

R
M 2R
2 R 2
M
R
Since R is arbitrary, let R   , we have
f z  0, i.e, f (z)  const .

Conversely, the slightest deviation of an analytic function from a constant value implies that
there must be at least one singularity somewhere in the infinite complex plane. Apart from
the trivial constant functions, then, singularities are a fact of life, and we must learn to live
with them, and to use them further.
1.5 Laurent Expansion
Taylor Expansion

Suppose we are trying to expand f(z) about z=z0, i.e., f z    a n z  z 0  , and we have
n
n 0
z=z1 as the nearest point for which f(z) is not analytic. We construct a circle C centered at
z=z0 with radius z   z0  z1  z0 (Fig.1.9)
Fig. 1.9
From the Cauchy integral formula,
1 f zdz
1
f zdz
f z  



2i C z  z
2i C z  z 0   z  z 0 
1
f zdz


2i C z  z 0 1  z  z 0  z  z 0 
Here z is a point on C and z is any point interior to C. For |t| <1, we note the identity
1
 1 t  t2  
1 t

t n
n 0
So we may write
1
f z  
2i
z  z 0 n f z dz 
  z   z0 n1
C n 0



1
 z  z 0 n

2i n0
f z dz 
 z   z0 n1
C
9

   z  z 0 n
n 0
f n   z 0 
n!
which is our desired Taylor expansion, just as for real variable power series, this expansion is
unique for a given z0.
•Schwarz reflection principle
n
From the binomial expansion of gz   z  x 0  for integer n (as an assignment), it is easy to
see, for real x0

g * z   z  x 0 
  z
n *
*
 x0

n
 
 g z*
This leads to the Schwarz reflection principle:
If a function f(z) is (1) analytic over some region including the real axis and (2) real
when z is real, then
 
f * z   f z *
We expand f(z) about some point (nonsingular) point x0 on the real axis because f(z) is
analytic at z=x0.

f n   x 0 
 z  x 0 n
f z  
n!
n 0

Since f(z) is real when z is real, f(n)(x0) must be real.

n   x 
n f
*
0

f z  
z *  x0
 f z*
n
!
n 0


 
•Analytic continuation
We consider a Taylor series of an analytic function of defined in some restricted
region S1 (Fig.1.10). Then f is analytic inside the circle of convergence C1, whose radius is
given by the distance r1 from the center C1 to the nearest singularity of f at z1. If we choose a
point inside C1, that is farther than r1 from z1 and make a Taylor expansion of f about it (z2 in
Fig.1.10), then the circle of convergence C1 will usually extend beyond C1. In the overlap
region of both circles, f is uniquely defined. In the region of C2, f(z) is uniquely defined by
the Taylor series about the center of C2 and analytic there, although the Taylor series about
the center of C1 is no longer convergent there. This process is called analytic continuation. It
defines the analytic function f in terms of its original definition in C1 and all its continuations.
For example
f z   1 1  z 
which has a simple pole at z = -1 and is analytic elsewhere. For |z| < 1, the geometric series
expansion
1
 1 z  z2  
1 z

  z n
n 0
Suppose we expand it about z = i, so that
1
1
f z  

1  i  z  i  1  i 1  z  i  1  i 
10

2

1  z  i z  i 
1


 

2
1  i  1  i 1  i 

converges for z  i  1  i  2 (Fig.1.10)
Fig. 1.10
The above three equations are different representations of the same function. Each
representation has its own domain of convergence.
•Permanence of Algebraic Form
All our elementary functions, ez, sin z, and so on can be extended into the complex
plane. For example, they can be defined by power-series expansions such as

z2
zn
e 1 z 

2!
n!
n 0

z
for the exponential. Such definitions literally constitute an analytic continuation of the
corresponding real functions into the complex plane. This result is often referred as
permanence of the algebraic form.
•Laurent Series
We frequently encounter functions that are analytic in annular region (Fig.1.11).
Fig. 1.11
Drawing an imaginary contour line to convert our region into a simply connected region, we
apply Cauchy’s integral formula for C2 and C1, with radii r2 and r1, and obtain


1 
 f  z dz 
f z  

 z  z
2i 
C1 C2 
 
We let r2 r and r1 R, so for C1, z  z 0  z  z 0 while for C2, z  z 0  z  z 0 . We
expand two denominators as we did before


1 
f  z dz 
f  z dz 

f z  




2i   z   z 0 1  z  z 0   z   z 0 
z  z 0 1  z   z 0   z  z 0 
C2
C1




1
z  z 0 n

2i n0

f  z dz 
 z   z0 n1
C1


1
1

2i n0  z  z 0 n1
 z   z 0 
C2
n
f  z dz 
11

f z  

 a n  z  z 0 n
(Laurent Series)
n  
where
an 
1
2i
f z dz 
 z   z0 n1
C
Here C may be any contour with the annular region r < |z-z0| < R encircling z0 once in a
counterclockwise sense.
Laurent Series need not came from evaluation of contour integrals. Other techniques such as
ordinary series expansion may provide the coefficients.
Numerous examples of Laurent series appear in the next chapter. I give one example
here.
Let f z   zz  1 . If we choose z0 = 0, then r = 0 and R = 1, f(z) diverging at z = 1
1
1
an 
2i
 z  n1
1
dz 
1

z z   1 2i

 m0 z 
m
dz 
z  n2
If we employ the polar form,

1
rie i d
an  
2i m0 r n 2m e i n 2m 




1
 2i  n 2m,1
2i
m 0
for n  -1
 1
an  
for n  - 1
0
The Laurent expansion becomes



1
1
  1 z  z2  z3   
zn
z z  1
z
n 1
12
1.6 Mapping (Optional Reading)
In this section, we introduce some of geometric aspects of functions of complex variables.
In ordinary analytic geometry we may take y = f(x) and then plot y vs. x. However, the
problem here is more complicated, for z is a function of two variables x and y. We use the
notation
w = f(z) = u(x, y) + i v(x, y)
Then for a point in the z-plane, there may correspond specific values for u and v which yield
a point in the w-plane. As points in the z-plane are mapped into points in the w-plane, lines or
areas in the z-plane will be mapped into the lines or areas in the w-plane. We now see now
they map from the z-plane to the w-plane for a number of simple functions.
• Translations
w = z + z0
u = x + x0
v = y + y0
• Rotation
Fig. 1.12
w = z z 0,
using w =  e i, z = r e i, and z0 = r0 e i0
then
 e i = r r0 e i(+0)
or
 = r r0
=  + 0
The modulus r has been modified by the factor r0, and the argument  has been increased by
the additive constant 0. This represents a rotation of the complex variables through angle 0.
Fig. 1.13
• Inversion
w = 1/z
 e i = 1/ (r e i )= (1/r) e -i
which shows that = 1/r, = -.
The first part shows inversion clearly. The interior of the unit circle is mapped onto the
exterior and vice versa.
13
Fig. 1.14
In addition, the second part shows that the polar angle is reversed in sign.
To see how lines transform, we return to the Cartesian form:
u + i v = 1/(x +i y) = (x-i y)/(x2+y2)
x
u
x 2
,
2
u  v2
x y
y
v
y 2
v 2
,
2
u  v2
x y
A circle centered at the origin in the z-plane has the form
u
2
x2 + y2 = r2 =1/(u2+v2)2 =1/
which describes a circle in the w-plane also centered at the origin.
The horizontal line y = c1 transforms into
v
 c1
2
u  v2
or
v
1
1
u 2  v2  

2
c1 2c1 
2c1 2
2

1 
1
 
u   v 
2c 1 
2c1 2

which describes a circles in the w-plane of radius (1/2c1) and centered at u=0, v=-1/2c1
2
Fig. 1.15
14
• Branch Points and Multivalent Functions
The above three transforms involve one to one correspondence of points in the z-plane to
points in the w-plane. In fact, a variety of transforms are possible, e.g., two-to-one and manyto-one.
Consider the transformation
w = z2
which leads to
 = r2,  = 2.
Clearly, the transformation is nonlinear, and the phase is doubled. This means that
upper half-plane of z, 0 ≤  <  whole plane of w,
0 ≤  < 2.
The lower half-plane of z maps into the already covered entire plane of w, thus covering the
w-plane a second time. This is our two-to-one correspondence, two points: z0 and z0ei=-z0,
corresponding to the single points w = z02.
In Cartesian representation
u + iv = (x + i y)2
leading to
u = x2 - y2
v=2xy
Hence the lines u = c1, v = c2 in the w-plane correspond to x2 - y2 = c1, 2xy = c2, rectangular
(and orthogonal), hyperbolas in the z-plane. (Fig.1.16)
Fig. 1.16
15
It will be shown in the next section that if lines in the w-plane are orthogonal the
corresponding lines in the z-plane are also orthogonal, as long as the transformation is
analytic.
The inverse of the above transformation is
w = z 1/2
leading to
 = r1/2, 2 = 
We now have two points in the w-plane ( and  + ) correspond to one point in the z-plane
(except z = 0). The important point here is that we can make the function w a single-valued
function instead of a double-valued function if we agree to restrict as 0 <  < 2. This may be
done by agreeing never to cross the line  = 0 in the z-plane.
Fig. 1.17
Such a line of demarcation is called a cut line.
The cut line joins the two branch point singularities at 0 and , where the function is clearly
not analytic. Any line from z = 0 to infinity would serve equally well. The purpose of the cut
line is to restrict the argument of z. The points z and z exp(2 coincide in the z-plane but
yield different points w and -w in the w-plane. Hence in the absence of a cut line the function
w = z1/2 is ambiguous. Alternatively, we can also glue two sheets of the complex z-plane
together along the cut line so that so that arg(z) increases beyond 2 along the cut line and
steps down from 4 on the second sheet to the start on the first sheet. This construction is
called the Riemann surface of w = z1/2.
The transformation
w = ez
leads to
 = ex,  = y
If 0 ≤ y < 2 ( or - ≤ y < ), then  covers the same range; but this is the whole w-plane.
Further, any point x + i (y+2n) with n integer, maps into the same point in the z-plane. We
have a many (infinitely many)-to-one map.
Finally, we discuss,
w = ln z
By expanding it, we obtain
u + iv = ln r + i ( + 2n)
16
u = ln r
v =  + 2n
We have an infinitely many-to-one correspondence.
The cut line joins the branch point at the origin with infinity. As  increases past 2 we glue a
new sheet of the complex z-plane along the cut line etc. Going around the unit circle in the zplane is like the ascent of a person walking up a spiral staircase(Fig.1.18), which is the
Riemann surface of w = ln z.
Fig. 1.18
1.7 Conformal Mapping (Optional Reading)
In the previous section hyperbolas were mapped into straight lines and straight lines were
mapped into circles. Yet in all these transformations one feature stayed constants, which is
due to the fact that all of them are analytic.
For a analytic function w = f(z), we have
df dw
w

 lim
dz dz z 0 z
If df/dz 0,
w
w
arg lim
 lim arg
z 0 z
z 0
z
 lim arg w  lim arg z
z 0
z 0
df
 arg

dz
where , the argument of the derivative, may depend on z but is a constant for a fixed z,
independent of the direction of approach. To see the significance of this, consider two curves,
Cz in the z -plane and Cw in the w-plane.
Fig. 1.19
From the above equation,
=  +
as long as w is analytic and the derivative is non-zero.
17
Since this result holds for any line through z., it will hold for a pair of lines. Then for the
angle between two lines,
2 -1 = (2 + ) - ( 1 + ) = 2 - 1,
which shows that the included angle is preserved under an analytic transformation. Such
angle-preserving transformation are called conformal mapping.