Chapter 1 复变函数与积分变换(英文版)

Polar
representation
of
complex
numbers
simplifies the task of describing geometrically the
product of two complex numbers. Let z1 r1 (cos1 isin 1 ) and z2 r2 (cos 2 isin 2 ) .
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
a 0i
to stand for a
. 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 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
a bi r cos (r sin )i
This way is writing the complex number is called the polar coordinate( triangle ) representation.
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).
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 )
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.
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
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
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
Then
z1 z2 r1r2 ([cos1 cos2 sin 1 sin 2 ]
i[cos1 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
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:
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.
r sin
x

0
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 . Argz arg z 2k k 0, 1, 2, arg z 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
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.
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.