Download complex numbers

Functions of Complex
Variable and Integral
Transforms
Department of Mathematics
Harbin Institutes of Technology
Gai Yunying
Preface
There are two parts in this course.
The first part is Functions of complex variable(the
complex analysis). In this part, the theory of analytic
functions of complex variable will be introduced.
The complex analysis that is the subject of this
course was developed in the nineteenth century, mainly
by Augustion Cauchy (1789-1857), later his theory was
made more rigorous and extended by such
mathematicians as Peter Dirichlet (1805-1859), Karl
Weierstrass (1815-1897), and Georg Friedrich Riemann
(1826-1866).
Complex analysis has become an indispensable and
standard tool of the working mathematician, physicist,
and engineer. Neglect of it can prove to be a severe
handicap in most areas of research and application
involving mathematical ideas and techniques.
The first part includes Chapter 1-6.
The second part is Integral Transforms: the
Fourier Transform and the Laplace Transform.
The second part includes Chapter
7-8.
1
Chapter 1 Complex Numbers and
Functions of Complex Variable
1. Complex numbers field, complex plane and
sphere
1.1 Introduction to complex numbers
As early as the sixteenth century Ceronimo
Cardano considered quadratic (and cubic) equations
such as x 2  2 x  2  0, which is satisfied by no real
number x , for example 1  1 . Cardano noticed that
if these “complex numbers” were treated as ordinary
numbers with the added rule that
1  1  1 , they
did indeed solve the equations.
1 is now given the
widely accepted designation i  1 .
The important expression
It is customary to denote a complex number:
z  x  iy
The real numbers x and y are known as the real and
imaginary parts of z , respectively, and we write
Re z  x, Im z  y
Two complex numbers are equal whenever they have
the same real parts and the same imaginary parts, i.e.
z1  z2  x1  x2 and y1  y2 .
In what sense are these complex numbers an
extension of the reals?
We have already said that if a is a real we also write
to stand for a
a  0i
. In other words, we are this
regarding the real numbers as those complex
numbers a  bi , where b  0.
If, in the expression a  bi the term a  0 . We call
a pure imaginary number.
1.2 Four fundamental operations
The addition and multiplication of complex
numbers are the same as for real numbers.
( x1  iy1 )  ( x2  iy2 )  ( x1  x1 )  i( y1  y2 )
( x1  iy1 )( x2  iy2 )  ( x1x2  y1 y2 )  i( x1 y2  x2 y1 )
If x2  iy2  0,
x1  iy1 ( x1  iy1 )( x2  iy2 ) ( x1 x2  y1 y2 )  i( x2 y1  x1 y2 )


x2  iy2 ( x1  iy2 )( x2  iy2 )
x22  y22
Formally, the system of complex numbers is an
example of a field.
The crucial rules for a field, stated here for reference
only, are:
Additively Rules:
i. z  w  w  z ;
ii. z  ( w  s )  ( z  w)  s ;
iii. z  0  z ;
iv. z  ( z )  0.
Multiplication Rules:
i. zw  wz ;
ii. ( zw) s  z ( ws ) ;
iii. 1 z  z ;
iv. z ( z 1 )  1 for z  0 .
Distributive Law:
z ( w  s )  zw  zs
Theorem 1. The complex numbers
field.
form a
If the usual ordering properties for reals are to
hold, then such an ordering is impossible.
1.3 Properties of complex numbers
A
complex
number
may
be
thought
of
geometrically as a (two-dimensional) vector and
pictured as an arrow from the origin to the point
in
2
given by the complex number.
Because the points ( x,0)  2 correspond to real
numbers, the horizontal or x  axis is called the real
axis the vertical axis (the y  axis) is called the
imaginary axis.
Imaginary axis  y  axis
z  a  ib
0
Real axis  x  axis
Figure 1.1 Vector representation of complex numbers
The length of the vector (a, b)  a  ib is defined as
r  a 2  b 2 and suppose that the vector makes an angle
 with the positive direction of the real axis, where
     . Thus tan   b / a . Since a  r cos and
y
b  r sin  , we thus have
a  ib
a  bi  r cos  (r sin  )i
 r (cos  isin  )
This way is writing
the complex number
is called the polar
coordinate( triangle )
representation.
r

0
r sin 
x
r cos 
Figure 1.2 Polar coordinate
representation of complex numbers
The length of the vector z  a  ib is denoted | z |
and is called the norm, or modulus, or absolute value
of z . The angle  is called the argument or amplitude
of the complex numbers and is denoted   arg z .
  arg z  
Argz  arg z  2k k  0, 1, 2,
It is called the principal value of the argument.
We have
y

z  I or IV
arctan x

y

arg z  arctan  
z  II
x

y

z  III
arctan x  
Polar
representation
of
complex
numbers
simplifies the task of describing geometrically the
product of two complex numbers.
Let z1  r1 (cos1  isin 1 ) and z2  r2 (cos 2  isin  2 ) .
Then
z1 z2  r1r2 ([cos1  cos2  sin 1  sin  2 ]
i[cos1  sin  2  cos 2  sin 1 ])
 r1r2 [cos(1   2 )  isin(1   2 )]
Theorem 3. | z1 z2 || z1 |  | z2 | and arg( z1z2 )  arg z1  arg z2
As a result of the preceding discussion, the
second equality in Th3 should be written as
arg z1z2  arg z1  arg z2 (mod 2 ) . “ mod 2 ” meaning that
the left and right sides of the equation agree after
addition of a multiple of 2 to the right side.
Theorem 4.
(de Moivre’s Formula). If
z  r (cos  isin  ) and n is a positive integer,
then z n  r n (cos n  isin n ) .
Theorem 5. Let w be a given (nonzero) complex
number with polar representation w  r (cos  isin  ), Then
the n th roots of w are given by the n complex numbers
   2k
zk  r cos  
n
 n
n

  2k
  isin  
n

n

  , k  0,1,

, n  1.
Example 1. Solve z 3  1 for z .
Solution:
  2k
  2k 

z  1  | 1|  cos
 isin

3
3


3
3
1

3
 
i
 
cos 3  isin 3
2 2



 cos   isin 
  1

5
5  1
 isin
cos
  3 i
3
3  2 2

k 0
k 1
k 2
If z  a  ib , then z ,
the complex conjugate
of z , is defined by
z  a  ib .
Theorem 6.
i. z  z  z  z
ii. zz  z  z
y
z  a  ib

0
x

z  a  ib
Figure 1.3 Complex conjugation
iii. z / z  z / z for z  0
iv. zz | z |2 and hence is z  0 , we have
v. z  z if and only if z is real
vi. Re z  z  z / 2
vii. z  z .
and Im z  z  z / 2i
z 1  z / | z |2 .
Theorem 7.
i. | zz || z |  | z |
ii. If z  0 , then | z / z || z | / | z |
iii.  | z | Re z | z | and  | z | Im z | z |;
that is, | Re z || z | and | Im z || z | .
y
z2
iv. | z || z |
z1
v. | z  z || z |  | z |
z1  z2
x
0
vi. | z  z | | z |  | z |
vii. | z1w1 
Figure 1.4 Triangle inequality
 zn wn | | z1 |   | zn |  | w1 |   | w0 |
2
2
2
2
1.4 Riemann sphere
For some purposes it is convenient to introduce a
point “  ” in addition to the points z  .
N
P
Q
0
P
Figure 1.5
x
y
Q
Complex sphere
Formally we add a symbol “  ” to
extended complex plane
 by the “rules”
z 
z   
 
  
to obtain the
and define operations with
2. Complex numbers sets Functions of complex
variable
2.1Fundamental concepts
(1) A  neighborhood of a point z0: N  {z | z  z0 |  }.
(2) A deleted  neighborhood of a point z0:
{z 0 | z  z0 |  }.
(3) A point
If there exists
z0 is said to be an interior point of E.
N  ( z0 )  E .
(4) A set E is open iff for each z0  E , z0 is an
interior point of E .
2.2 Domain Curve
An open set S is connected if each pair of points
z1 and z2 in it can be joined by a polygonal line,
consisting of a finite number of line segments joined
end to end, that lies entirely in S .
An open set that is connected is called a domain.
if
 : z  z (t )  x(t )  iy(t ) (  t   )
x(t ), y(t )  C[ ,  ] , then  is continuous and ift1  t2  z (t1 )  z (t0 )
then  is called a simple curve.
A curve,
If z(t )  x(t )  iy(t )  0 and x(t ), y(t )  C[ ,  ].
 is called a smooth curve (a piecewise smooth
curve).
A domain D  is called the simply connected iff,
for every simply closed curve
in
,Dthe inside of

alsolies in , or else
D it is called the multiple connected
domain.
2.3 Mappings and continuity
Let G 
be a set. We recall that a mapping
f : G  is merely an assignment of a specific point
to each z  G , G being the domain of
f ( z) 
and when the range
f . When the domain is a set in
(the set of values f assumes) consists of complex
numbers, we speak of f as a complex function of a
complex variable.
f :G  2  2 ;
therefore f becomes a vector-valued function of two
real variables.
We can think of
f as a map
For f : G 
, we can let z  x  iy  ( x, y )

and define u ( x, y )  Re f ( z ) and v( x, y )  Im f ( z ) .
Thus u and v are merely the components of f
thought of as a vector function.
Hence we may write uniquely
f ( x  iy )  u ( x, y )  iv( x, y ) , where u and v are real-
valued functions defined on G .
Def 1. Let f be defined on a deleted neighborhood
of z0 . The
limit f ( z )  A
z  z0
means that for every   0 , there is a   0 such
that z  D( z0 , r ),
0 | z  z0 |  ,
z  z0 , and
| z  z0 | 
imply that | f ( z )  A |  .
We also define, for example,
lim f ( z )  A to
z 
mean that for any   0 , there is an R such that
| z | R implies that | f ( z )  A |  .
The limit as z  z0 is taken for an arbitrary
z approaching
z0 but not along any particular
direction.
v
y


z0  ( x0 , y0 )
0
Figure 1.6
A
w  f ( z)
x
0
f ( z ) is close to A when z is close to z0
u
The limit A is unique. The following properties of
limits hold:
If
lim f ( z )  A and
z  z0
lim g ( z )  B , then
z  z0
i. lim[ f ( z )  g ( z )]  A  B
z  z0
ii. lim[ f ( z ) g ( z )]  AB
z  z0
iii. lim[ f ( z ) / g ( z )]  A / B if B  0 .
z  z0
Also, if h is defined at the points
lim h( w)  c , then
wa
iv. lim h( f ( z ))  c
z  z0
f ( z ) and
Th1. Let f ( z )  u ( x, y)  iv( x, y), A  a  ib then
lim f ( z )  A  lim u ( x, y )  a and
z  z0
x  x0
y  y0
lim
v( x, y )  b.
x x
0
y  y0
Proof: It is easy by using the following inequalities
| u  a |,| v  b | (u  a) 2  (v  b) | u  a |  | v  b |
Def 2. Let A 
be an open set and let f : A 
be a given function. We say f is continuous at
z0  A iff
lim f ( z )  f ( z0 )
z  z0
and f is continuous on A is f is continuous at each
z0  A .
From (i), (ii), and (iii) we can immediately deduce
that if f and g are continuous on A , then so are the
sum f  g and the product fg , and so is f / g if
g ( z0 )  0 for all z0  A . Also if h is defined and
continuous on the range of f , then the composition
h f , defined by h f ( z )  h( f ( z )) , is continuous by
(iv).
Theorem 2. Let f ( z )  u ( x, y )  iv( x, y ) is continuous
at z0  x0  iy0  u ( x, y ) and v( x, y ) are continuous at
( x0 , y0 ) .
z
EX1. If f ( z )  , the limit lim f ( z ) does not exist.
z 0
z
For, if it did exist, it
y
could be found by letting
z  (0, y )
the point z  ( x, y ) approach
the origin in any manner.
But when z  ( x,0) is a
nonzero point on the real
axis (Fig 1.7), f ( z )  x  i0  1;
x  i0
0
z  ( x,0)
Figure 1.7
x
and when z  (0, y )
imaginary axis,
is a nonzero point on the
0  iy
f ( z) 
 1.
0  iy
Thus, by letting z approach the origin along the
real axis, we would find that the desired limit is 1 . As
approach along the imaginary axis would, on the other
hand, yield the limit 1 . Since a limit is unique, we
must conclude that lim f ( z ) does not exist.
z 0
iz  3
2z  i
EX2. Find that (1) lim
; (2) lim
;
z 1 z  1
z  z  1
iz 3  1
(3) lim
.
z i z  i
Solution:
z 1
iz  3
(1) lim
 0 ,  lim
;
z 1 iz  3
z 1 z  1
2/ z  i
2  iz
(2) lim
 lim
2 ;
z 0 1/ z  1
z 0 1  z
iz 3  1
 0.
(3) lim
z i z  i