Additional Chapters of Higher Mathematics for Masters in Civil and Geotechnical Engineering : учебное пособие по дополнительным разделам высшей математики для магистрантов по направлению «Строительство»

С.В. Кузнецов, Е.Л. Кошелева

ADDITIONAL CHAPTERS OF HIGHER MATHEMATICS 
FOR MASTERS IN CIVIL AND GEOTECHNICAL 
ENGINEERING

Учебное пособие
по дополнительным разделам высшеиматематики для магистрантов 
по направлению «Строительство» 

МИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ 
РОССИЙСКОЙ ФЕДЕРАЦИИ

ФГБОУ ВПО «МОСКОВСКИЙ ГОСУДАРСТВЕННЫЙ
СТРОИТЕЛЬНЫЙ УНИВЕРСИТЕТ»

М о с к в а  2017

2-е издание (электронное)

УДК 517
ББК 22.1
         К 89

Р е ц е н з е н т ы:
лауреат государственноипремии СССР, доктор физико-математических наук, 
профессор Р. А. Турусов, главныинаучныисотрудник 
Института химическоифизики РАН;
кандидат технических наук Г. А. Джинчвелашвили, 
профессор кафедры сопротивления материалов (ФГБОУ ВПО «МГСУ»)

Кузнецов, Сергей Владимирович.

Additional Chapters of Higher Mathematics for Masters in Civil and 
Geotechnical Engineering [Электронный  ресурс] : учебное пособие 
по дополнительным разделам высшей  математики для магистрантов по направлению «Строительство» / С. В. Кузнецов, 
Е. Л. Кошелева  ; М-во образования и науки Рос. Федерации, Моск. 
гос. строит. ун-т. — 2-е изд. (эл.). — Электрон. текстовые дан. 
(1 фай л pdf : 212 с.). — М. : Издательство МИСИ—МГСУ, 2017. — 
Систем. требования: Adobe Reader XI либо Adobe Digital Editions 
4.5 ; экран 10".

ISBN 978-5-7264-1731-8

Является спецкурсом математики для магистрантов. Необходимость  в учебном пособии возникла в связи с изучением различных 
курсов механики деформируемого твердого тела, которые ведут 
сотрудники Института прикладноимеханики РоссиискоиАкадемии 
наук и сотрудники МГСУ.

Рассматриваются разделы математики, касающиеся  таких вопросов, как топология, метрика и векторы пространства, ряды и интегралы Фурье, элементы теории матриц, различные методы решения  
дифференциальных уравнении.

Для магистрантов, иностранных студентов и аспирантов.

К 89

ISBN 978-5-7264-1731-8

Деривативное электронное издание на основе печатного 
издания: Additional Chapters of Higher Mathematics for Masters in 
Civil and Geotechnical Engineering : учебное пособие по дополнительным разделам высшей  математики для магистрантов по направлению «Строительство» / С. В. Кузнецов, Е. Л. Кошелева  ; М-во 
образования и науки Рос. Федерации, Моск. гос. строит. ун-т. — М. : 
Изд-во МИСИ—МГСУ, 2012. — 210 с. — ISBN 978-5-7264-0701-2.

УДК 517
ББК 22.1

В соответствии со ст. 1299 и 1301 ГК РФ при устранении ограничений, установленных 
техническими средствами защиты авторских прав, правообладатель вправе требовать 
от нарушителя возмещения убытков или выплаты компенсации.

©  Национальный  исследовательский

Московский  государственный  
строительный  университет, 2012

CONTENTS 

Preface………………………………………………………………………………………….5 

Chapter 1. Equation Chapter 1 Section 0. Topological, metric, functional, and 
vector spaces .................................................................................................................... 7 
1.1. Equation Chapter 1 Section 1. Basics of topological and metric spaces ............................ 7 
1.2. Equation Chapter 1 Section 2. (Real) trigonometric, hyperbolic, and some other 
functions and series ........................................................................................................ 16 
1.3. Equation Chapter 1 Section 3. Functions of complex variables ....................................... 30 
1.4. Equation Chapter 1 Section 4. Asymptotic expansions .................................................... 39 
1.5. Equation Chapter 1 Section 5. Generalized functions ...................................................... 43 
1.6. Equation Chapter 1 Section 6. Elements of vector algebra .............................................. 47 
Bibliography to Chapter 1 .............................................................................................. 54 

Chapter 2. Equation Chapter 2 Section 0. Fourier series, wavelets, and integral 
transforms ..................................................................................................................... 57 
2.1. Equation Chapter 2 Section 1. Fourier series ................................................................... 57 
2.2. Equation Chapter 2 Section 2. Wavelet analyses ............................................................. 69 
2.3. Equation Chapter 2 Section 3. Fourier integral transforms and discrete Fourier 
transforms ....................................................................................................................... 78 
2.4. Equation Chapter 2 Section 4. Laplace, Laplace-Carson,  and Mellin  integral 
transforms ....................................................................................................................... 91 
2.5. Equation Chapter 2 Section 5. Other integral transforms ................................................. 97 
Bibliography to Chapter 2 ............................................................................................ 104 

Chapter 3. Equation Chapter 3 Section 0. Theory of matrices ....................................... 107 
3.1. Equation Chapter 3 Section 1. Elements of matrix algebra ............................................ 107  
3.2. Equation Chapter 3 Section 2. Eigenproblems ............................................................... 114 
3.3. Equation Chapter 3 Section 3. Simple and semisimple matrices ................................... 123 
3.4. Equation Chapter 3 Section 4. Non-semisimple matrices .............................................. 128 
3.5. Equation Chapter 3 Section 5. Matrix classes ................................................................ 134 
3.6. Equation Chapter 3 Section 6. Functions of semisimple matrices ................................. 141  
3.7. Equation Chapter 3 Section 7. Functions of non-semisimple matrices .......................... 150  
Bibliography to Chapter 3 ............................................................................................ 154 

Chapter 4. Equation Chapter 4 Section 0. Ordinary differential equations .................. 157 
4.1. Equation Chapter 4 Section 1. Basic concepts ............................................................... 158 
4.2. Equation Chapter 4 Section 2. Linear differential equations with constant coefficients 169 
4.3. Equation Chapter 4 Section 3. Closed form solutions for linear differential equations 
with variable coefficients ............................................................................................. 184 
4.4. Equation Chapter 4 Section 4. Closed form solutions for non-linear  differential 
equations
 193 
4.5. Equation Chapter 4 Section 5. Numerical methods for solving Cauchy problem of 
ordinary differential equations...................................................................................... 199 
Bibliography to Chapter 4 ............................................................................................ 208 

Preface 

The present book originated as lecture notes of our courses in different fields of 
mechanics and engineering, revealing that typical master students either completely 
forget or do not know some of the basic concepts of higher mathematics that are needed 
for proper understanding the specific material in mechanics. Depending on the nature of 
the course and the student average level in mathematics, we had to devote several 
lectures just to cover students' shortage knowledge in mathematics.  
The decision to write a lecture course on specific topics of higher mathematics 
that are admittedly indispensable for master students' was supported by our colleagues 
from the Institute for Problems in Mechanics of Russian Academy of Sciences 
(Moscow, Russia) and INSA de Lyon (Lyon, France).  
The book is divided into chapters covering topics on topology, metric, normed, 
and functional spaces. We wrote a brief introduction to the theory of distributions, 
elements of complex analysis, and several sections on wavelet approximations. There is 
also a chapter on integral transforms including Fourier, Laplace, Mellin, and some other 
integral and discrete transforms. We wrote a rather detailed exposition of the theory of 
matrices, including functions of matrices and a special but important case of nonsemisimple degeneracy. The course also contains a chapter on ordinary differential 
equations, including introduction to Hamiltonian formalism and a survey of the relevant 
numerical methods.  
The authors are acknowledged to all our colleagues and students who helped us 
in preparing the lecture course and the manuscript. 

Authors 

Sergey Kuznetsov 
Elena Kosheleva 

5

Chapter 1. Equation Chapter 1 Section 0
Topological, metric, functional, and 
vector spaces  

This chapter presents the basic mathematical facts and concepts needed for the theory of 
vibrations, namely: basic properties of the elementary functions, complex variable 
method, methods of linear algebra, and some basic facts of the theory of ordinary 
differential equations. The reader familiar with these topics can easily pass to the 
subsequent chapters.  

1.1. Equation Chapter 1 Section 1 Basics         
of topological and metric spaces 

This paragraph is devoted to  The main reference books for this chapter are Bourbaki (1989, 
1998), Edwards (1995), and Hörmander (2003).  

1.1.1. Topological spaces 

Definition 1.1.1 (Topological space) 

Topological space T  is a space containing a set Λ  of its subsets (called topology of the space 
T ) with the following properties: 
I. ∅∈Λ ; 
II. If 
1
2
,
L
L ∈Λ , then 
1
2
L
L
∩
∈Λ  and 
1
2
L
L
∪
∈Λ ;
III. 
L

L
T
∈Λ
=
∪
 (union of all the subsets L  belonging to Λ  coincides with T ). 

Subsets L  are called “open” sets. 

Definition 1.1.2 (Open vicinity) 

An open vicinity of the point x
T
∈
 is any subset L∈Λ , containing x . 

7

Chapter 1. Topological, metric, functional, and vector spaces 
____________________________________________________________________________ 

 

Definition 1.1.3 (Open subset) 

A subset C
T
⊂
 is called open, if it belongs to the open set Λ . 

Definition 1.1.4  (Closed subset) 

A subset C
T
⊂
 is called closed, if it is a complement to an open set. 

Definition 1.1.5 (Everywhere dense subset)  

A subset S
T
⊂
 is called everywhere dense, if any open vicinity L∈Λ  contains at least one 
point from S . 

Definition 1.1.6 (Separable topological space) 

Topological space is called (closed) separable, if it contains a countable everywhere dense 
subset. 

Definition 1.1.7 (Homeomorphism) 

Let 
,
X Y  be two topological spaces.  

I. A map 
:
f
X
Y
→
 is called continuous, if 
1( )
f
V
X
−
⊂
 is open for any open V
Y
⊂
. 
II. A map 
:
f
X
Y
→
 is called homeomorphism if f  is a one-to-one correspondence and 

both f  and 
1
f −  are continuous functions. 

Remark 1.1.1 (Locally convex topological space) 

In the subsequent analyses all the topological spaces will be assumed to be locally convex. This 
means that their topologies can be defined by the corresponding sets of the convex subsets.  

1.1.2. Metric and normed spaces 

Definition 1.1.8 (Metric space) 

Topological space T  is called a metric space, if its topology is defined by a distance function 
:
d T
T
×
→ with the following properties: 
I. 
( , )
0
d x y =
, if and only if x
y
=
 
II. 
( , )
( , )
( , )
d x z
d x y
d y z
≤
+
 (inequality of triangle) 

8

Chapter 1. Topological, metric, functional, and vector spaces 
____________________________________________________________________________ 

 

Topology in the metric space is defined by a system of balls 
,x
V δ of radius 
0
δ >
 with origins at 

points x
T
∈
: 

 
,
0
( , )
x
x Tx
y
V
d x y
δ
∈
δ>
∀
∀δ
∈
⇔
< δ  
(1.1.1) 

 
Inequalities I and II from the preceding definition imply: 

Proposition 1.1.1 

 
( , )
0,
( , )
( , )
d x y
d x y
d y x
≥
=
 
(1.1.2) 

Definition 1.1.9 (Normed space) 

Topological vector space T  is called a normed space, if its topology is defined by a norm 
:
N T → with the following properties: 

 
( )
0
0
N
=
⇔
=
x
x
, 
(1.1.3) 

 

 
,
(
)
( )
( )
,
T
N
N
N
∈
+
≤
+
∀
x y
x
y
x
y
x y , 
(1.1.4) 

 

 
(
)
( )
T
t
N t
t N
t
∈
∈
=
∀
∀
x
x
x
x
. 
(1.1.5) 

 
 
It can be shown that conditions (1.1.3) – (1.1.5) ensure: 

Proposition 1.1.2 

 
( )
0
T
N
∈
∀
>
x
x
x
. 
(1.1.6) 

Definition 1.1.10 (Cauchy sequence) 

Cauchy sequence is an infinte countable sequence {
}
n
x
, whose elements become infinitely 
close with the incresing number, this means 

 
0
.
(
)
n
n
n n
n
n N
ε
ε
ε
ε>
>
∀ε ∃
∀
−
< ε
x
x
 
(1.1.7) 

9

Chapter 1. Topological, metric, functional, and vector spaces 
____________________________________________________________________________ 
Definition 1.1.11 (Complete normed space) 

Banach space is a complete normed space, which means that any Cauchy sequence converges to 
an element belonging to this space. 

Remark 1.1.2 

The norm of the Banach space is quite often denoted by x  or x . 

Example 1.1.1 (
p
L -norm in a finite dimensional vector space with finite 
,
1
p
p
∈
≥
) 

Let T  be n -dimensional vector space, then 
p
L -norm (denoted by 
p
L
⋅
) is a function

1/

1

p

p
n
p
k
L
T
k
x
∈
=

⎛
⎞
∀
≡ ⎜
⎟
⎜
⎟
⎝
⎠
∑
x x
x
,
(1.1.8)

where 
,
1,...,
kx
k
n
=
 are coordinates of vector x  in a particular basis. Direct verification shows 
that conditions (1.1.3) – (1.1.5) are satisfied at any 
1
p ≥ . 

Remark 1.1.3 (Euclidian norm) 

The
2
L -norm (1.1.8) at 
2
p =
 is used most often: 

(
)
2

1/2
1/2
2

1

n

k
L
T
k
x
∈
=

⎛
⎞
∀
≡
≡
⋅
≡
⎜
⎟
⎜
⎟
⎝
⎠
∑
x x
x
x x
x
(1.1.9)

This norm is sometimes called as Euclidian norm. 

Example 1.1.2 ( L∞ -norm in a finite dimensional vector space) 

Such a norm is defined by 

max
k
L
k
x
∞ ≡
x
.
(1.1.10)

L∞ -norm is sometimes called uniform norm. It can be shown, that conditions (1.1.3) – (1.1.5) 
are satisfied. 

10

Chapter 1. Topological, metric, functional, and vector spaces 
____________________________________________________________________________ 
Example 1.1.3 (
p
L -norm in an infinite dimensional vector space at finite 
,
1
p
p
∈
≥
) 

Let T  be a vector space of the all integrable on some set X  functions, then 
p
L -norm (denoted 
by 
p
L
⋅
) is a map T
+
→ defined by

1/

( )
p

p

p
L
f T
X
f
f
f x
dx
∈

⎛
⎞
∀
≡ ⎜
⎟
⎜
⎟
⎝
⎠
∫
.
(1.1.11)

Example 1.1.4 ( L∞ -norm in an infinite dimensional vector space of numerical functions) 

sup
( )
L
x X

f
f x
∞
∈
≡
.
(1.1.12)

Such a norm is called the uniform norm. 

Remark 1.1.4 

A. Condition 
1
p ≥  in Examples 1.1.1 and 1.1.3 is needed to satisfy inequality (1.1.4), 
known also as Minkowski inequality. At 
1
p <  condition (1.1.4) fails.  

B. In any of functional spaces 
, 1
p
L
p
≤
≤ ∞ , space of continuous functions is dense in the 

corresponding 
p
L -topology. 

C. The following embedding of spaces 
p
L  takes place:  

,
q
p
L
L
q
p
⊂
>
,
(1.1.13)

and at q
p
>
 the topology 
q
L  is stronger than topology 
p
L  induced in 
q
L .  

Theorem 1.1.1 (Hölder’s inequality) 

A. Let f  be an integrable function, then a set I  of real p , 1
p
≤
≤ ∞ , at which 
p
L -norms 

p
L
f
 are finite, is either empty, or a closed interval. In the latter case log(
)
p
L
f
 is a

convex function of 1/ p .   
B. If integrable function f  has a finite support, then the interval I  is either empty, or has 
1
p =  as the starting point. In the latter case 
p
L
f
 is the increasing function of p .

C. Let 
P
f
L
∈
 and 
q
g
L
∈
, where 1
p
≤
≤ ∞ , 1
q
≤
≤ ∞  and  

1
1
1
p
q
+
= ,
(1.1.14)

11