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Файл:Additional Chapters of Higher Mathematics for Masters in Civill and Geotechnical Engineering. Учебное пособие по дополнительным разделам высшей матема
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Chapter 1. Topological, metric, functional, and vector spaces
xy
x
[
31
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1.3.1. Complex variables
Definition 1.3.1 (Complex number)
1) A complex number
,
where
Number
part, these parts are also denoted by
2) Geometric representation for a complex variable (Euler’s formula):
In Fig.1.1 it is denoted:
are real numbers, and
is called the real part of the complex number z , while y is called the imaginary
z is a pair
zxiy=+ , (1.3.1)
(1.3.2)
1i =−
and
Re( )z
Figure 1.1. Geometric representation of a complex
variable
respectively.
Im( )z
(cos sin )
and
zr i=ϕ+ϕ (1.3.3)
rxy≡+
arcsin( / ), 0; 2yrϕ= ϕ∈ π
22
(1.3.4)
(1.3.5)
)
Chapter 1. Topological, metric, functional, and vector spaces
y
p
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1.3.2. Properties of complex variables
Definition 1.3.2 (Complex conjugate number)
Complex conjugate number
is a pair
z
or in terms of the geometric representation
(cos sin )zr i=ϕ−ϕ (1.3.7)
Definition 1.3.3 (Module of complex number)
Module of a complex variable:
zxyrzz≡+==
Basic properties:
()( )( )
zz xx yy i xy x y
=−++=
12 12 12 12 21
12 12 1 2
()( )zz xx iy y+= + + + (1.3.9)
=ϕ+ϕ+ϕ+ϕ
zxiyr
1 111
zxiyr
2222
+
= = ϕ−ϕ + ϕ−ϕ ≠
cos( ) sin( ) , 0
()
+
zxi
=− , (1.3.6)
22
(1.3.8)
(1.3.10)
cos( ) sin( )
rr i
()
12 12 12
12 12 2
iz
(1.3.11)
ppp
() cos()sin(),
zxiyr pip p=+ = ϕ+ ϕ ∈ (1.3.12)
Remark 1.3.1 (Moivre’s formula)
Formula (1.3.12) is known as Moivre’s formula (obtained in 1707), notation similar to (1.3.12)
is due to Euler (1748). This formula will be generalized in the next section for complex powers
.
Definition 1.3.4 (Root of complex number)
n -th Roots of a complex number:
ϕπ ϕπ
22
⎛⎞
nn
zr i k n
=+++ =−
⎛⎞⎛⎞
cos sin , 0,... 1
⎜⎟⎜⎟
⎜⎟
⎝⎠
nn nn
⎝⎠⎝⎠
kk
()
(1.3.13)
Chapter 1. Topological, metric, functional, and vector spaces
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Examples 1.3.1
A. Geometric representation of i
ii
ππ
cos sin
=+
(1.3.14)
22
Geometric representation of i , according to formula (1.3.13):
B.
ππ
Geometric representation of
C.
D.
Geometric representation of
=
i
1+
1
+=
3
1+
⎧
⎪
⎪
⎨
⎪
⎪
⎩
⎧
⎨
⎩
+
cos sin
cos sin
i
44
ππ
55
i
+
(1.3.15)
44
, according to formula (1.3.13):
cos0 sin0 1
ii+=+
(1.3.16)
cos sin 1
π+ π=−
, according to formula (1.3.13):
⎧
⎪
+=
cos0 sin 0 1
i
⎪
ππ
2213
3
⎪
+= + =− +
1cos sin
⎨
⎪
⎪
cos sin
⎪
⎩
ii
3323
ππ
4413
ii
+=−−
3323
(1.3.17)
Thus, we got three cubic roots of 1+, and only one of them is real, while two others are
complex according to formulas (1.3.17).
1.3.3. Functions of complex variables
Elements of the theory of analytic functions of complex variables have a lot of
connections to the theory of vibrations. Below, we give only brief introduction to the
analysis of analytic functions of complex variables.
Definition 1.3.5 (Complex function)
Complex function is a function of a complex variable, defined by the following equation:
where
() (, ) (, )
z uxy ivxy we
≡+ =
i
θ
, (1.3.18)
Chapter 1. Topological, metric, functional, and vector spaces
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i
zxiyze
=+ = (1.3.19)
Definition 1.3.6 (Complex analytic function; Holomorphic function)
Complex analytic function (regular or holomorphic) is a function that can be represented by a
power series convergent in the corresponding vicinity.
Theorem 1.3.1
a) Any analytic function (in a suitable vicinity) is differentiable in that vicinity, and
∂∂ ∂∂
uv uv
==
∂∂ ∂∂
,
yyx
Definition 1.3.7 (Derivative with respect to complex variable)
ϕ
(1.3.20)
The derivative with respect to the complex variable of an analytic function
∂∂∂
zxy
∂∂∂
and
zxy
∂∂∂
Remark 1.3.2 (Cauchy-Riemann equations)
Relations (1.3.20) are known as Cauchy – Riemann equations (Cauchy, 1814; Riemann,
A.
1851). However, these relations were also known to D’Alembert (1752) and Euler
(1797).
Combining (1.3.20) – (1.3.22) we get
B.
and
∂∂∂
∂∂∂
⎛⎞
1
ff
=−
⎜⎟
2
⎝⎠
1
ff
⎛⎞∂∂∂
=+
⎜⎟
2
⎝⎠
∂∂∂
∂∂∂
uv
==
zxy
uv
ii
==
(1.3.21)
i
(1.3.22)
i
, (1.3.23)
. (1.3.24)
zyx
()
z
is
Relations (1.3.23) and (1.3.24) are valid, provided
is analytic function.
Chapter 1. Topological, metric, functional, and vector spaces
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C.
It follows from Definition 1.3.7 that
zz
∂∂
==
0, 0
∂∂
zz
Theorem 1.3.2 (Closed loop integral of analytic function)
If
is an analytic function in an open region
(a closed curve is called Jordan, if it is a continuous curve and it does not contain any multiple
points, except initial and end points) the following integral vanishes:
() 0fzdz
∫
Γ
Trigonometric representations for the exponential function of complex variable:
A. Euler representation
x
exp( ) (cos sin )
ze yi y=+, (1.3.27)
. (1.3.25)
D ⊂
=
, then for any closed Jordan curve
(1.3.26)
DΓ⊂
where as before
(cos sin )zxiyr i=+ = ϕ+ ϕ
. (1.3.28)
B. Relations (1.3.27), (1.3.28) yield
cos
r
exp( ) (cos sin sin sin )
ze r i r
=ϕ+ϕ
ϕ
()()
. (1.3.29)
Theorem 1.3.3 (Generalization of Moivre’s formula)
ii
ϕϕ
Let
zre z re
11 2 2
12
==
,
zr e
be two arbitrary complex numbers, then
zr r ir irr
222122122212
=
11
cos sin cos ln sin
ϕ−ϕ ϕ+ϕ ϕ+ ϕ
. (1.3.30)
Proof
Generalizing Moivre’s formula is obtained by applying expressions (1.3.27) – (1.3.29).
Relations between hyperbolic and trigonometric functions of complex variable:
sin( ) sinh( )zi iz=−
, (1.3.31)
cos( ) cosh( )ziz= , (1.3.32)
tan( ) tanh( )zi iz=− , (1.3.33)
cot( ) coth( )zi iz= , (1.3.34)
Chapter 1. Topological, metric, functional, and vector spaces
36
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Other relations between exponential, hyperbolic, and trigonometric functions:
iz
exp ln cos ln sin ln ,
a iza za i za a
==+ ∈
( ) () ()
i
exp ln cos ln sin ln ,
xix xixx
==+ ∈ (1.3.41)
( ) () ()
arcsin arsh ln 1zi iz iiz z=− =− + −
arccos arch ln 1zi z izi z=− =− + −
arctan arth ln
sinh( ) sin( )ziiz=−
cosh( ) cos( )ziz=
tanh( ) tan( )ziiz=− , (1.3.37)
coth( ) cot( )zi iz= , (1.3.38)
exp( ) cos( ) sin( )iz z i z=+
()
()
zi iz
=− =−
()
, (1.3.35)
, (1.3.36)
, (1.3.39)
(1.3.40)
+
+
2
()
()
1
iiz
⎛⎞
⎜⎟
21
⎝⎠
(1.3.42)
2
(1.3.43)
+
(1.3.44)
iz
−
Remark 1.3.3
Equations (1.3.31), (1.3.32) are sometimes called Euler’s formulas.
Examples 1.3.2
iiz
arccot arcoth ln
x
iix
=+ ∈
cos sin ,
==−
zi iz
xx
ππ
22
−π
ie
= , Principle value, (1.3.47)
ln( ) ln argzziz=+
()
/2i
⎛⎞
⎜⎟
21
iz
⎝⎠
, Principle value, (1.3.46)
. (1.3.48)
1
−
(1.3.45)
+
Chapter 1. Topological, metric, functional, and vector spaces
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1.3.4. Elements of the theory of residues
Definition 1.3.6 (Isolated singular point)
A point
this point, in which
z is called an isolated singular point for a function ()
0
()
Theorem 1.3.4 (Laurent series)
If ()
z is analytic in a ring bounded by two concentrating circles, then in at any point of the
annular region
can be represented by the series
()
z
where
is the center of the annular region, and coefficients
z
0
In (1.3.50)
encircles point
is (i) a closed curve lying in an annular region, where
Γ
. The integral over Γ in (1.3.50) is taken counterclockwise. Series (1.3.49) is
z
0
called the Laurent series.
is analytic everywhere, except only
z
k
=∞
() ( )
zazz
=−
∑
k
k
=−∞
adz
1()
=
k
i
π
2
fz
∫
zz
−
()
Γ
k
, (1.3.49)
0
. (1.3.50)
k
1
+
0
z if there exists a vicinity of
.
z
0
are defined by
a
k
is analytic, and (ii) Γ
()
z
Corollary 1
The coefficient
Definition 1.3.8 (Residue)
The coefficient
Quite often this coefficient is denoted as
in the Laurent series is obtained by
1a−
a
(if it does not vanish) is called the residue of a function
1a−
1
=
1
−
2
π
()
zdz
∫
i
Γ
(, )resfza
≡
01
. (1.3.51)
.
−
at point
()
z
.
z
0
Chapter 1. Topological, metric, functional, and vector spaces
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Corollary 2
If function
connected, if any closed curve contained in the domain can be shrunk to a point by a continuous
transformation), and a closed curve Γ is contained in this domain, then
Corollary 3
At the conditions of the previous corollary
for any
Definition 1.3.9 (Pole)
An
isolated singular point is called a pole for a function ()
series (1.3.49) contains a finite number of negative powers. The highest negative number is
called the
belonging to the region encircled by Γ.
z
0
order of the pole.
is analytic in a simply connected domain (the domain is called simply
()
z
1
2
π
1()
∫
2( )
π−
izz
Γ
i
∫
Γ
fz
() 0
fzdz
dz f z
0
=
=
. (1.3.52)
()
(1.3.53)
0
z if the corresponding Laurent
Definition 1.3.10 (Essentially singular point)
isolated singular point is called the essentially singular point for a function ( )fz
An
corresponding Laurent series (1.3.49) contains infinite number of negative powers.
Remark 1.3.4 (Residue theorem)
Despite their simplicity, formulas (1.3.51) – (1.3.53) are of high practical importance. If ( )
is an analytic function in a simply connected domain, except a finite number of singular points
,...,
zz
1
then
Sometimes Eq. (1.3.54) is called the
and a curve Γ is contained in this domain and encompasses all the singular points,
N
1
2
π
() res( , )
∫
i
Γ
N
zdz f z
=
∑
1
p
=
residue theorem.
if the
,
z
. (1.3.54)
Chapter 1. Topological, metric, functional, and vector spaces
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x
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x
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1.4. Equation Chapter 1 Section 4 Asymptotic
expansions
Herein, the main ideas and principles for deriving asymptotic relations are discussed. For
references see Bourbaki (2003), Copson (2004), and Erdelyi (2010).
1.4.1. Main definitions
Definition 1.4.1 (Asymptotic relations
A.
x
x≺≺
at
() ()
B.
() ()
x
x at
C.
() ()
x
x∼ at
x→
0
Remark 1.4.1
A. In (1.4.1), (1.4.2) “
limsup
exists.
≺≺,
, if
x→
0
xx
, if
x→
0
xx
, and ∼ for scalar functions of one variable)
()
lim sup 0
→
lim sup 0
→
fx
()
gx
0
()
gx
()
fx
0
. (1.4.1)
=
= . (1.4.2)
, if
()
lim sup 1
tt
→
fx
= . (1.4.3)
()
gx
0
” means the upper limit. It coincides with the ordinary limit, if it
B. Relation ( ) ( )
obvious relation
xgx∼ at
32 3
xx+ ∼
at
Definition 1.4.2 (Landau Omicron symbols o and
A.
() (())
xogx=
at
x→
0
does not mean that
x→
0
→+∞
leads to
)
O
, if
() () 0fx gx−→
32 3 2
()()xx x x+− =→+∞
. For example, an
at
→+∞
.
Chapter 1. Topological, metric, functional, and vector spaces
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40
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()
B.
lim sup 0
xx
→
()
( ) ( ( )) lim 1
==
fx Ogx
tt
→
0
ft
()
gt
at
0 lim sup
<<∞, (1.4.5)
xx
→
fx
= , (1.4.4)
()
gx
0
, if
x→
0
()
fx
()
gx
0
provided
Remark 1.4.2
A. Actually, symbol O (it reads as “big O”) bas introduced by the German mathematician
Bachman, while “little
B.
If the limit in the left hand side of (1.4.4) exists then () (())
C.
Sense of the big O Landau symbol can be easily understood by considering the following
truncated Taylor’s series:
In (1.4.6) the term
3
will be higher than any power of
The same truncated series can clarify sense of the little
lim ( ) 0
gx
tt
→
0
as
.
≠
o” was introduced by Landau.
2
exp( ) 1 ( ), 0
xOxx
xx
=+ + + →. (1.4.6)
3
1! 2!
3
means that an error in truncating the series is of the magnitude of
()Ox
. Note, that in (1.4.6) we cannot place either
0x →
.
o symbol:
2
exp( ) 1 ( ), 0
xoxx
xx
=+ + + → (1.4.7)
2
1! 2!
xogx= is equivalent to (1.4.1).
→+∞
or
→−∞
, as the error
It says that an error in truncating the series is smaller than
Example 1.4.1
,,
2
as
.
0x →
1.4.2. Examples
xn
ex x n→∞ ∀ (1.4.8)
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