1. Functions of Complex Variable and Integral Transforms
Gai Yunying
Department of Mathematics
Harbin Institutes of Technology

2. 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 ( ), later his theory was made more rigorous and extended by such mathematicians as Peter Dirichlet ( ), Karl Weierstrass ( ), and Georg Friedrich Riemann ( ).

3. The first part includes Chapter 1-6.
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

4. 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 , which is satisfied by no real number , for example Cardano noticed that if these “complex numbers” were treated as ordinary numbers with the added rule that , they did indeed solve the equations.

5. The important expression is now given the widely accepted designation .
It is customary to denote a complex number:
The real numbers and are known as the real and imaginary parts of , respectively, and we write
Two complex numbers are equal whenever they have the same real parts and the same imaginary parts, i.e.
and

6. In what sense are these complex numbers an
extension of the reals?
We have already said that if is a real we also write to stand for a In other words, we are this
regarding the real numbers as those complex
numbers , where
If, in the expression the term We call a pure imaginary number.

7. 1.2 Four fundamental operations
The addition and multiplication of complex numbers are the same as for real numbers. If
Formally, the system of complex numbers is an example of a field.

8. Multiplication Rules: i. ; ii. ; iii. ; iv. for .
The crucial rules for a field, stated here for reference only, are:
Additively Rules:
i ;
ii ;
iii ; iv
Multiplication Rules:
i ;
ii ;
iii ;
iv for

9. Distributive Law:
Theorem 1. The complex numbers form a field.
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 given by the complex number.

10. Figure 1.1 Vector representation of complex numbers
Because the points correspond to real numbers, the horizontal or axis is called the real axis the vertical axis (the axis) is called the imaginary axis.
Figure Vector representation of complex numbers

11. Figure 1.2 Polar coordinate representation of complex numbers
The length of the vector is defined as
and suppose that the vector makes an angle
with the positive direction of the real axis, where
. Thus Since and
, we thus have
Figure Polar coordinate representation of complex numbers
This way is writing the complex number is called the polar coordinate( triangle )representation.

12. The length of the vector is denoted
and is called the norm, or modulus, or absolute value of . The angle is called the argument or amplitude of the complex numbers and is denoted
It is called the principal value of the argument.
We have

13. Polar representation of complex numbers simplifies the task of describing geometrically the product of two complex numbers.
Let and
Then
Theorem and

14. and is a positive integer, then .
As a result of the preceding discussion, the second equality in Th3 should be written as
. “ ” meaning that the left and right sides of the equation agree after addition of a multiple of to the right side.
Theorem (de Moivre’s Formula). If
and is a positive integer, then
Theorem 5. Let be a given (nonzero) complex number with polar representation , Then
the th roots of are given by the complex numbers

15. Example Solve for
Solution:

16. Figure 1.3 Complex conjugation
If , then , the complex conjugate of , is defined by .
Figure Complex conjugation
Theorem 6. i.
ii.
iii for
iv and hence is , we have
v if and only if is real
vi and
vii

17. Figure 1.4 Triangle inequality
Theorem 7. i.
ii. If , then
that is, and
iii and ;
Figure 1.4 Triangle inequality
iv. v.
vi.
vii.

18. 1.4 Riemann sphere
For some purposes it is convenient to introduce a point “ ” in addition to the points
Figure Complex sphere

19. Formally we add a symbol “ ” to to obtain the extended complex plane and define operations with
by the “rules”

20. 2. Complex numbers sets Functions of complex variable
2.1Fundamental concepts
(1) neighborhood of a point :
(2) A deleted neighborhood of a point :
(3) A point is said to be an interior point of If there exists
(4) A set is open iff for each , is an interior point of

21. 2.2 Domain Curve
An open set is connected if each pair of points
and 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
An open set that is connected is called a domain.
A curve, if
, then is continuous and if
then is called a simple curve.
If and
is called a smooth curve (a piecewise smooth curve).

22. 2.3 Mappings and continuity
A domain is called the simply connected iff, for every simply closed curve in , the inside of also lies in , or else it is called the multiple connected domain.
2.3 Mappings and continuity
Let be a set. We recall that a mapping
is merely an assignment of a specific point
to each , being the domain of
. When the domain is a set in and when the range (the set of values assumes) consists of complex numbers, we speak of as a complex function of a complex variable.

23. We can think of as a map ;
therefore becomes a vector-valued function of two real variables.
For , we can let
and define and
Thus and are merely the components of thought of as a vector function.
Hence we may write uniquely
, where and are real-valued functions defined on .

24. Def 1. Let be defined on a deleted neighborhood of . The
means that for every , there is a such that , and
imply that
We also define, for example, to mean that for any , there is an such that
implies that

25. The limit as is taken for an arbitrary
approaching but not along any particular direction.
Figure is close to when is close to

26. Also, if is defined at the points and , then
The limit is unique. The following properties of limits hold:
If and , then i.
ii.
iii if
Also, if is defined at the points and
, then
iv.

27. Proof: It is easy by using the following inequalities
Th1. Let then
and
Proof: It is easy by using the following inequalities

28. Def 2. Let be an open set and let
be a given function. We say is continuous at
iff
and is continuous on is is continuous at each .
From (i), (ii), and (iii) we can immediately deduce that if and are continuous on , then so are the sum and the product , and so is if
for all Also if is defined and continuous on the range of , then the composition
, defined by , is continuous by (iv).

29. Theorem 2. Let is continuous at and are continuous at .
EX1. If , the limit does not exist.
For, if it did exist, it could be found by letting the point approach the origin in any manner. But when is a nonzero point on the real axis (Fig 1.7),
Figure 1.7

30. and when is a nonzero point on the imaginary axis,
Thus, by letting approach the origin along the real axis, we would find that the desired limit is As approach along the imaginary axis would, on the other hand, yield the limit Since a limit is unique, we must conclude that does not exist.
EX2. Find that (1) ; (2) ;
(3)

31. Solution:
(1) , ;
(2) ;
(3)