Chapter 15 Series & Residue
15.1 Sequences and Series
A sequence
cfw_zn is
a function whose domain is a set of positive
integers,
y
1
2
3
n
n
4
If for every > 0 , there exists an integer N such
that
zn L <
whenever
n>N
Then
zN
z
2
z
1
L
cfw_zn is said t
Chapter 14 Complex Integration
14.1 Line integral in the complex plane
b
For definite integral of a real function
f ( x)dx , it means
a
b
N
f ( x)dx lim 0 f ( xk )xk
x
k =1
k
a
x0
xk
a
xk 1 xk
x
b = xN
xk
Where xk xk xk 1 ;
x0 = a ; xN = b
xk some x bet
Chapter 16
Laurent series.
Residue integration
16.1 Laurent Series
Singular point
If f (z) is not analytic at z0.
z0 is called a singular point of f (z).
If f (z) is analytic for every z in the disk z z0 < excepted z0 , z0
is called an isolated singular p
Illustration of Gibbs phenomenon
Consider a function f ( x ) :
f(x)
1 when x < 1
f ( x) =
0 otherwse
x
-1
1
The Fourier integral of f ( x ) is:
f ( x) =
=
2 sin cos x
0
d
1 2sin cos x
0
d =
1 sin( + x) + sin( x)
0
d
lim 1 a sin (1 + x)
lim 1 a sin (1 x)
Chapter 13 Complex Numbers
Complex Analytic Functions
13.1 Complex number. Complex plane
Def: A number of form: z = x + i y
(1)
is called a complex number, where x,y are real,
i = 1
x = Re[ z ]
x is called the real part of z:
y is called the imaginary par
Chapter 11
Fourier Series, Integrals, and Transforms
11.0 Orthogonal function, Generalized Fourier Series
Inner product of functions
If f ( x ) and g ( x ) are two real functions defined in the
interval x [ a ,b ]
the inner product of f ( x ) and g ( x )
Chapter 9 Vector Integral Calculus. Integral Theorems
9.1 Line Integral
Consider a space curve C : r = x ( s ) i + y ( s ) j + z ( s ) k
s: arc length parameter
and a scalar function
= (x , y ,z )
the line integral of along C from point a to b is
defined
Chapter 8
Vector Calculus
8.1 Vector function
A Vector whose components are function of either one or several
variables.
e.g. (i) V = V ( x , y , z ) i +V ( x , y , z ) j +V ( x , y , z ) k
1
2
3
where V ,V ,V are scalar function of space.
1 2 3
(ii) r (
Chapter 12 Partial Differential Equations
12.1 Basic concepts
Partial Differential Equation: An equation involving one or more
partial derivatives of an (unknown) function of two or more
independent variables
Order : the order of the highest derivative in