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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
x
x
s
41
____________________________________________________________________________
This is the very important asymptotic relation that can be easily proved by applying L’Hopital’s
rule for uncertainty of limits:
xnxx
In terms of the Landau symbols (1.4.8) can be written in a form
edee
lim lim
→∞ →∞
xx
===∞. (1.4.9)
nnn
xdx
!
n
but, in view of (1.4.5)
(), ,
Example 1.4.2
,,
The proof of this relation is analogous to the preceding example. In terms of Landau symbols
(1.4.12) is
Example 1.4.3
This follows from (1.4.9). This asymptotic relation can also be written as
nx
(), ,
oe x n=→∞∀, (1.4.10)
xn
eOx x n≠→∞∀ (1.4.11)
xn
−−
ex x n
xn
−−
eox x n
−−
exx n
−−
(), ,
=→∞∀
1/
xn
nx
(), 0,
oe x n
=→∀
,0,
1/
→∞ ∀≺≺ . (1.4.12)
(1.4.13)
→∀
. (1.4.14)
. (1.4.15)
Example 1.4.4
Again, applying L’Hopital’s rule immediately leads to (1.4.16). In terms of Landau symbols,
(1.4.16) is
Example 1.4.5
log( ) ( ),
for any positive real
.
sin( ) , 0xx x→∼
. (1.4.16)
sin( ) ( ), 0xOx x=→. (1.4.17)
xox x=→∞, (1.4.18)
s
Chapter 1. Topological, metric, functional, and vector spaces
f
x
f
f
x
42
____________________________________________________________________________
1.4.3. Asymptotic series
Definition 1.4.3 (Asymptotic series)
()
Asymptotic series for a given function
() ()
xax
∼
x at
∞
∑
k
0
=
k
x→ is a functional series of the kind:
0
, (1.4.19)
where
are some functions, and symbol ~ indicates that the series in the right-hand side
ax
()
k
of (1.4.19) does not need be convergent, but it must satisfy the following property:
N
∀−= →
Remark 1.4.3 (Poincaré series)
Introduction of the asymptotic series is quite often associated with French mathematician
Poincaré, and is sometimes called Poincaré series.
Example 1.4.6
The following asymptotic series satisfies (1.4.20), while it is divergent at any x
Example 1.4.7
N
() () ( (),
xaxoaxxx
∑
kN
0
=
k
∞
exp( ) ( 1) ( 1)!
xt k
−−−
∫
t
x
dt
∞
∼
∑
k
1
=
1
k
−
k
x
. (1.4.21)
. (1.4.20)
0
Similarly to the preceding example the following asymptotic series also satisfies condition
(1.4.20)
Example 1.4.8
The following example (Bourbaki, 2003, Ch.V, §3) is interesting as it reveals that
asymptotically equivalent expressions can lead to different results at integrating
x
dt x
∫
log( )
a
∞
k
(1)!
∑
k
−
=
1
∼ . (1.4.22)
t
k
log
()
Chapter 1. Topological, metric, functional, and vector spaces
x
D
43
____________________________________________________________________________
sin sin sin
But the integral
xx x
∼ . (1.4.23)
xx x
⎛⎞
1,
+→∞
⎜⎟
⎝⎠
x
∞
converges, while the integral
is divergent.
sin 1 2
dx
∫
a
=π−
x
∞
∫
a
⎛⎞
2FrenelS
⎜⎟
⎜⎟
2
⎝⎠
sin sin
xx
⎛⎞
1
+
⎜⎟
xx
⎝⎠
⎛⎞
a
⎜⎟
⎝⎠
. (1.4.25)
dx
. (1.4.24)
π
1.5. Equation Chapter 1 Section 5 Generalized
functions
Herein we give a concise introduction to the theory of generalized functions
(distributions). The explication follows ideas proposed by Shilov and Gelfand (1964,
1968) and Schwartz (2008). Basic properties of integral kernels follow monographs by
Stein (1971) and Stein and Shakarchi (2003).
1.5.1. Basic notations
Definition 1.5.1 (Inductive limit topology)
Let
functions having finite (compact) supports in
in
If
be an open subset in
Ω
(,)
SΩ denotes the set of all functions from ( )D Ω with supports containing in S . A topology
(,)DSΩ can be defined by norms
spans all the compact subsets in Ω then it is possible to introduce a special topology in
S
()D Ω , called the inductive limit topology. That is the strongest topology leaving all the
n
mp
() sup ()
Nx
ϕ≡ ∂ϕ
S
and
pmxS
be a set of all infinitely differentiable (real)
()D Ω
Ω
,
≤∈
. If
S ⊂Ω
is a proper compact subset, then
. (1.5.1)
Chapter 1. Topological, metric, functional, and vector spaces
D
p
44
____________________________________________________________________________
continuous forms on
topology is called the space of the
Remark 1.5.1
(,)DSΩ
continuous on
base functions.
. The space
()D Ω
with the corresponding
()D Ω
A sequence
() ()
n
is called convergent to zero if
Dϕ∈ Ω
lim ( ) 0,
m
NmS
ϕ= ∀ ∀
Sn
n
and
NS S n
∀∃ ϕ⊂ ∀
0
>⊂Ω >
NS nN
supp
()
In other words a sequence converges to zero, if it converges to zero in a suitable topological
space ( , )
SΩ .
Remark 1.5.2
A subset ()BD⊂Ω is bounded, if it is bounded in a suitable space (,)DSΩ .
Definition 1.5.2 (Generalized function; Distribution)
generalized function (distribution) is a linear continuous form from the dual space ()D
The
0
≥⊂Ω
mS
n
(1.5.2)
(1.5.3)
′
Ω .
Remark 1.5.3
The dual
into a topological space, in which all the forms
()D′Ω
are continuous at the fixed
can be equipped with the weakest topology
()D′Ω
,<ϕ ψ>∈
0
and variable
()Dϕ∈ Ω
0
1.5.2. Derivatives of the generalized functions
Definition 1.5.3 (Derivative of generalized function)
The derivative
of the generalized function ψ is defined by
∂ψ
′
((),())DD
σΩΩ
converting
(1.5.4)
′
ψ∈ Ω
()D
.
Chapter 1. Topological, metric, functional, and vector spaces
p
x
45
____________________________________________________________________________
()D Ω
p
: Let
∞
∫
−∞
. (1.5.5)
n
∈
p
(thus Ω is an open subset in ), if
1n =
, (1.5.6)
xdx
(1.5.7)
,(1),
Remark 1.5.4
It is easily verified that definition (1.5.5) is in accordance with the analogous definition for the
derivative of functions from
,()Dϕϕ∈ Ω and
12
then integrating by parts yields
Formula (1.5.7) is obviously extrapolated to higher derivatives and higher dimensions.
pp
<ϕ∂ψ>≡− <∂ϕψ> ∀
,()()
<ϕ ϕ >≡ ϕ ϕ
12 1 2
∞∞
,()()()(),x x dx x x dx
<ϕ∂ϕ>≡ϕ∂ϕ =−∂ϕϕ =−<∂ϕϕ>
12 1 2 1 2 12
∫∫
−∞ −∞
Example 1.5.1 (Heaviside step-function)
Let ( )
Dϕ∈ and ( )ht be Heaviside step-function:
then according to (1.5.5)
,,()(0)hhtdt
<ϕ∂ >=−<∂ϕ >=− ∂ϕ =ϕ
Thus, we got a very important relation
where
is Dirak delta-function located at zero.
δ
0
()
ht
0, 0
⎧
=
⎨
1, 0
⎩
h∂=δ
t
<
, (1.5.8)
t
≥
∞
∫
0
, (1.5.10)
0
. (1.5.9)
Chapter 1. Topological, metric, functional, and vector spaces
x
g
x
x
f
g
f
g
g
46
____________________________________________________________________________
1.5.3. Finite Part integrals and Pseudofunctions
Definition 1.5.4 (Finite Part integral)
Let a real-valued function
(),
f ∈xx
n
admits following representation in a neighborhood of
zero
⎛⎞
=+ ++
where
()
x
unit sphere
is an integrable function in a neighborhood of zero,
S ⊂ function having the zero mean value:
() ()
xx
fg
n
is a numerical coefficient, and
C
′
Ω
()
x
0
nn p
xx x
() 0
xdx
Ω=
0
∫
S
′
Ω
()
1
C
⎜⎟
⎜⎟
⎝⎠
′′
∑
pn
>
, (1.5.12)
is an arbitrary integrable function on S. Integrating
Ω
()
1
′
, (1.5.11)
′
Ω
is an integrable on the
()
0
(1.5.11) over a small ball of radius r with the origin at zero yields:
′
Ω
()
x
() () . .
∫∫ ∫ ∫
BB B B
rr r r
=+ ++
dx
xx
dx P V dx dx
0
nnp
xxx
⎛⎞
C
⎜⎟
⎜⎟
⎝⎠
∑
pn
>
Ω
()
x
1
′
(1.5.13)
The first term in (1.5.13) is finite, the second term evaluated as the Principle Value integral
vanishes, and two remaining terms in brackets are infinite at any
r . Now, we define the Finite
Part integral, as
F.P. () () P.V. ()
∫∫ ∫ ∫
BB B B
dx
xx x
rr r r
Remark 1.5.5
This concept of the Finite Part integrals is due to Hadamard.
Definition 1.5.5 (Pseudofunction)
Now we consider a very special, but often occurring case of the generalized functions of the
kind
′
Ω
()
x
≡+ =
dx dx
,F.P.dx<ϕ ψ>≡ ϕψ
0
n
x
∫
, (1.5.15)
dx
. (1.5.14)
Chapter 1. Topological, metric, functional, and vector spaces
x
x
x
47
____________________________________________________________________________
where
which the finite part integral in the right-hand side of (1.5.15) is well defined for any
()Dϕ∈ Ω
Example 1.5.2
).
()Dϕ∈ Ω
and
′
ψ∈ Ω
()D
is a special generalized function called a pseudofunction, at
Let
, pseudofunction ψ be
1n =
()
ψ=
x
m
−
, (1.5.16)
and
()Dϕ∈ . Expanding ()
ϕ into Taylor’s series yields:
m
() , 0
ϕ= + →
xxoxx
∑
k
0
=
ϕ
()
k
k
(0)
!
(
m
)
. (1.5.17)
k
Substituting (1.5.16), (1.5.17) into (1.5.15) and taking into account (1.5.14), yields
()
k
m
⎛⎞
ϕ
mkm
<ϕ >≡ + =
−−
xx x ox dx
(), F.P.
=+=
lim P.V.
0
δ→
(1)
m
ϕ
==
(1)!
m
⎜⎟
∑
∫
⎜⎟
k
=
⎝⎠
δ
⎛⎞
ϕ
⎜⎟
∫
⎜⎟
⎝⎠
−δ
−
(0)
lim P.V.
δ→
−
(0)
k
!
0
(1)
m
−
(0)
−
(1)!
m
δ
−
xdx
0
∫
−δ
()
1
mm
−−
1`
m
Ox x dx
(
)
(1.5.18)
where P.V stands for the Principle Value.
1.6. Equation Chapter 1 Section 6 Elements
of vector algebra
Herein we remind some basic properties of vector algebra in the n-dimensional Euclidean
and Hermitian spaces. For references see Bourbaki (1998, 2002), Humphreys (2008), and
Korn and Korn (2000).
Chapter 1. Topological, metric, functional, and vector spaces
48
____________________________________________________________________________
1.6.1. Basic notations
Definition 1.6.1 (Euclidian space; Hermitian space)
A.
Let
following formula:
The finite dimensional space
Euclidian space. Quite often the sign
known as Einstein summation convention over repeated indices:
n
be real n -dimensional vector space with the scalar product defined by the
n
. (1.6.1)
⋅= ∀
ab ab
ab
∑
kk
=
1
k
n
equipped with the scalar product (1.6.1) is called
,
n
∈
in expressions like (1.6.1) is omitted, this is
Σ
We shall use this convention in the sequel. If in a particular expression no summation
over repeated indices is needed, we shall cross the summation sign:
If the finite dimensional space
B.
The space
Proposition 1.6.1
A. Let ,n∈ab , then
⋅=⋅
B. Let
Proof
. (1.6.2)
ab⋅=ab
kk
.
Σ
n
is complex, then the scalar product is defined by
n
∑
=
k
ab
1
kk
⋅= ∀
ab ab
n
equipped with the scalar product (1.6.3) is called Hermitian space.
,
. (1.6.3)
n
∈
ab ba (1.6.4)
n
∈ab
, then
⋅=⋅ab ba
(1.6.5)
,
The proof follows immediately from Definitions 1.6.1.
Definition 1.6.2 (Euclidian norm)
The (Euclidean) norm of a vector is
. (1.6.6)
aa=⋅=aaa
kk
Chapter 1. Topological, metric, functional, and vector spaces
49
____________________________________________________________________________
Definition 1.6.3 (Hermitian norm)
The (Hermitian) norm of a vector is
Proposition 1.6.2
If 0≠a , then
regardless of Euclidean or Hermitian space is considered.
0>a
Proof
The proof flows out from Definition 1.6.1.
Definition 1.6.4 (Orthogonality)
Two non-zero vectors are called mutually orthogonal, if
Proposition 1.6.3
Let
,, or
nn
∈abc
then
aa=⋅=aaa
kk
. (1.6.8)
0⋅=ab
+⋅=⋅+⋅abcacbc
()
. (1.6.7)
(1.6.9)
If
,orλμ∈
and
Proof
The proof flows out from Definitions 1.6.1.A or 1.6.1.B.
1.6.2. Vector product (Gibb’s vector product)
Definition 1.6.5 (Vector product)
The binary anticommutative operation in
for arbitrary vectors
following properties:
Anticommutative property
(1)
, ∈ab
nn
∈a
λ+μ =λ +μaaa
()
or
then
3
defined by
×=abc
3
and
3
is called the vector product, if it possesses the
∈c
(1.6.10)
(1.6.11)
Chapter 1. Topological, metric, functional, and vector spaces
⎡⎤⎢
⎢⎥⎢⎥⎣
50
____________________________________________________________________________
(2)
Distributive property over addition and scalar multiplication
×=−×ab ba; (1.6.12)
(3) Jacobi identity
Remarks 1.6.1
A. It can be shown that conditions (1.6.12) - (1.6.14) are equivalent to the following single
condition for definition of the vector product
where
Expression (1.6.15) yields
×λ +γ =λ× +γ×abcabac
()
() () ()
eee
123
ab
e are orts in
k
×= + +
ab e e e
3
and ,
aa aa aa
⎛⎞ ⎛⎞ ⎛⎞
23 13 12
det det det
⎜⎟ ⎜⎟
bb bb bb
23 13 12
det aa a
×=
ab are the corresponding components of vectors a and b .
kk
123
123
bbb
123
; (1.6.13)
. (1.6.14)
0××+××+××=abccabbca
⎥
, (1.6.15)
⎦
⎜⎟
⎝⎠⎝⎠ ⎝⎠
(1.6.16)
or
The vector product can also be defined by the following geometrical definition:
B.
where
0;θ∈ π
[]
orthogonal to the plane containing vectors
ab a b ab ab ab a b×=−+−+−ab e e e
()()()
23 32 1 13 31 2 12 21 3
sin×= θab ab n
is the smaller angle between vectors
, (1.6.18)
()
and b and n is a unit vector
a
a and b . Direction of vector
rule of thumb: the forefinger of the right hand points to vector
direction of
One of the must unusual properties of the vector product is its anticommutative property:
C.
Indeed, taking in (1.6.19)
, then the thumb (orthogonal to the palm) will indicate direction of vector n.
b
××≠××ab c a bc
() ()
yields for the left-hand side
=ba
()
. (1.6.19)
(1.6.20)
0××=aa c
(1.6.17)
is defined by the
n
, the middle finger to the
a
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