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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
x
xxx
xxx
21
____________________________________________________________________________
Proposition 1.2.2
Exponential function is a real analytic function everywhere in (1.2.33).
Proof
Proof of this proposition follows from Corollary to Proposition 1.1, where series (1.2.6) have
coefficients
Definition 1.2.5 (Hyperbolic sine function; Hyperbolic cosine function)
Hyperbolic sine and cosine functions can be defined by the following equivalent equations.
1) These are functions defined by the following series:
a =
k
.
1
357 21
sinh( ) ...
cosh( ) 1 ...
xx x x x
≡++++=
x
1! 3! 5! 7! (2 1)!
246 2
≡++++=
x
xx x
∑
n
2! 4! 6! (2 )!
∞
=
∑
n
n
−
(1.2.35)
−
n
1
n
∞
(1.2.36)
n
0
=
2) These functions are solutions of the following differential equation [1]:
2
⎛⎞
d
1()0
−=
⎜⎟
2
⎜⎟
dx
⎝⎠
fx
(1.2.37)
3) These functions relate to the exponential function by
x
−
ee
sinh( )
x
−
= (1.2.38)
2
x
−
ee
x
cosh( )
+
=
(1.2.39)
2
Definition 1.2.6 (Hyperbolic tangent function; Hyperbolic cotangent function)
Hyperbolic tangent and cotangent functions can be defined by the following equivalent
equations.
1) These functions are expressed in terms of ratios of hyperbolic sine and cosine functions:
tanh( )
sinh( )
= , (1.2.40)
cosh( )
coth( )
cosh( )
= . (1.2.41)
sinh( )
2) These are functions expressed in terms of exponential function:
Chapter 1. Topological, metric, functional, and vector spaces
xxx
xxx
x
B
x
x
xxx
22
____________________________________________________________________________
−
ee
−
tanh( )
=
x
ee
coth( )
=
x
ee
ee
3) These are functions expressed in terms of the power series:
22
kk
∞
xx
tanh( )
coth( )
2(2 1)
=
∑
=
1
k
12
=
∑
xk
k
(2 )!
∞
=
1
(2 )!
, (1.2.42)
x
−
+
−
+
. (1.2.43)
x
−
−
B
−
2
k
−
21
k
, (1.2.44)
k
2
k
B
2
k
2
k
. (1.2.45)
x
Bernulli numbers
appearing in expressions (1.2.44), (1.2.45) can be computed by
2k
Laplace formula
1
⎛⎞
⎜⎟
2!
⎜⎟
11
⎜⎟
⎜⎟
3! 2!
Bn
=−
n
n
(1) !det
⎜⎟
111
⎜⎟
4! 3! 2!
⎜⎟
⎜⎟
⎜⎟
111 1
⎜⎟
⎜⎟
(1)! !(1)! 2!
nnn
+−
⎝⎠
1 0 ... 0
1 ... 0
... 0
....
...
(1.2.46)
4) These functions are solutions of the following (nonlinear) differential equation:
2
⎛⎞
dd
⎜⎟
⎜⎟
dx
⎝⎠
() () 0
fx fx
+=
2
dx
. (1.2.47)
Some basic properties of exponential and hyperbolic functions:
22
cosh ( ) sinh ( ) 1xx−=
, (1.2.48)
sinh(2 ) 2sinh( )cosh( )
cosh(2) cosh() sinh()
tanh(2 )
=
xx= , (1.2.49)
22
xx=+, (1.2.50)
2 tanh( )
1tanh()
+
, (1.2.51)
2
Chapter 1. Topological, metric, functional, and vector spaces
x
A
A
A
A
A
B
A
A
A
A
A
B
x
x
x
x
23
____________________________________________________________________________
2
coth(2 )
coth ( ) 1
x
= , (1.2.52)
AB
sinh sinh 2sinh cosh
cosh cosh 2cosh cosh
cosh cosh 2sinh sinh
±=
AB
+=
AB
−=
x
+
2coth( )
BAB
±
∓
22
BAB
+−
22
BAB
+−
22
, (1.2.53)
, (1.2.54)
, (1.2.55)
tanh tanh
±=
AB
sinh( )
cosh cosh
coth coth
±=
AB
sinh( )
sinh sinh
±
B
, (1.2.56)
B
±
A
, (1.2.57)
B
sinh( ) sinh cosh cosh sinh
cosh( ) cosh cosh sinh sinh
BABAB±= ± , (1.2.58)
BABAB±= ± , (1.2.59)
tanh( )
coth( )
±=
AB
±=
AB
tanh tanh
1tanhtanh
coth coth 1
coth coth
±
±
AB
±
B
, (1.2.60)
B
±
, (1.2.61)
A
d
sinh( ) cosh( )
=
, (1.2.62)
x
dx
d
cosh() sinh()
=
, (1.2.63)
x
dx
d
tanh( )
x
dx
d
coth( )
x
dx
=
cosh ( )
=
sinh ( )
1
, (1.2.64)
2
1
, (1.2.65)
2
Chapter 1. Topological, metric, functional, and vector spaces
x
∑
x
∑
x
x
24
____________________________________________________________________________
1.2.3. Logarithmic and related other functions
and series
In this section we present useful power series defining some other functions of a real
variable.
Proposition 1.2.2 (Neumann series)
Proof
Let
then
1x <
A.The following power series, known as the Neumann series, is absolutely convergent
∞
k
k
(1.2.66)
0
=
and
∞
1
−
(1 )
−=
k
. (1.2.67)
x
k
0
=
The following power series is also absolutely convergent
∞
∑
k
=
k
−
()
0
(1.2.68)
and
∞
−
(1 )
1
+=−
∑
k
=
k
x
()
0
(1.2.69)
Proof A. The series (1.2.66) is absolutely convergent due to assumption
(1.2.67) we consider the following product:
∞∞∞
(1 ) 1
⎛⎞⎛⎞⎛ ⎞
xx x x
−⋅ = − =
⎜⎟⎜⎟⎜ ⎟
∑∑∑
⎜⎟⎜⎟⎜ ⎟
000
kkk
===
⎝⎠⎝⎠⎝ ⎠
The latter equation reveals that Neumann series (1.2.66) defines the inverse to
is analogous.
Proposition 1.2.3 (Logarithmic function)
Let
then
1x <
kkk
1
+
. (1.2.70)
. To prove
1x <
(1 )x− . Proof B
Chapter 1. Topological, metric, functional, and vector spaces
x
x
H
x
25
____________________________________________________________________________
k
1
k
+
(1.2.71)
k
Proof
∞
ln(1 ) ( 1)
x
+= −
∑
k
1
=
The proof flows out from constructing Taylor’s expansions of ln(1 )
series in the right-hand side of (1.2.71).
Proposition 1.2.4
Let
x <∞
Proof
The proof relies on Taylor’s expansions and Stirling’s estimate (1.2.4).
The role of hypergeometrical functions is underlined by the fact that a lot of functions of
mathematical physics are expressed or can be expressed as solutions of linear hypergeometrical
differential equations. However, we should note that there remain plenty of other functions that
do not obey hypergeometrical equations.
then
+ at
2
∞
2
1
xk
−
ee
⎛⎞
=× −
(1)
⎜⎟
∑
⎜⎟
0
k
=
⎝⎠
k
x
. (1.2.72)
!
k
1.2.4. Hypergeometric functions
. This gives the
1x <
For further reading see Abramowitz and Stegun (1972) and Slater (2008).
Definition 1.2.7 (Hypergeometric function; Gauss’s hypergeometric function)
The hypergeometric function (also called Gauss’s hypergeometric function) is denoted as
(,;;)
21
abcx
; it is a regular solution of the linear homogeneous differential equation called
the hypergeometrical differential equation:
2
(1) 1 0
x x a b x c aby
dy dy
−+++−+=
dx
⎡⎤
()
⎣⎦
2
dx
. (1.2.73)
The hypergeometric function can also be defined by convergent at
∞
ab
()()
(,;;)
Habcx
21
=
∑
n
nn
cn
()
0
=
n
, (1.2.74)
n
!
series
1x <
Chapter 1. Topological, metric, functional, and vector spaces
H
x
H
26
____________________________________________________________________________
where
Proposition 1.2.5 (Gauss Hypergeometric theorem)
Remark 1.2.2
Differential equation (1.2.73) has three (regular) singular points at 0; 1; and infinity. It can be
shown (Abramowitz and Stegun, 1972) that any linear ODE of the second order with at most
three regular singular points can be transformed into hypergeometric equation.
Definition 1.2.8 (Generalized hypergeometric function)
are Pochhammer symbols
a
()
n
1; ( 1)( 2)...( 1), 1
aaaaaann==+++−>. (1.2.75)
() ()
0
Habc
21
n
(,;;1)
()( )
ccab
ΓΓ−−
=
()()
Γ−Γ−
ca cb
, (1.2.76)
The
generalized hypergeometric function is denoted as
following
where
linear ordinary differential equation
⎡⎤
( 1)...( 1) ( )...( ) ( ) 0
DDb Db xDa Da yx
+− +−− + + =
⎣⎦
11
qp
/Dddx= .
The generalized hypergeometric function can also be defined by the following convergent at
series
1x <
where
ab
()()
kk
Ha ab bx
( ,..., ; ,..., ; )
pq p q
;
nn
11
are Pochhammer symbols; see (1.2.75).
=
∑
n
Definition 1.2.9 (Confluent hypergeometric function)
A function
denoted as
(;;)
abx
1111
.
()Mx
is called a confluent hypergeometric function. This is sometimes
( ,..., ; ,..., ; )
aabbx
pq p q
∞
aa
()
bb
()
0
=
1
1
...
××
n
...
××
n
11
()
p
()
q
satisfies the
(1.2.77)
n
n
, (1.2.78)
!
n
n
Chapter 1. Topological, metric, functional, and vector spaces
{
f
{
27
____________________________________________________________________________
1.2.5. Other functions
Herein we give a brief outline of some other functions that can be frequently found in
different mathematical and physical applications.
Definition 1.2.10 (Floor function; Entire function)
Floor (or Entire) function is a function maps a real number to the largest integer that is not
greater than the given number. Such a function is defined by
Definition 1.2.11 (Ceiling function)
Ceiling function is a function maps a real number to the smallest integer that is not less than the
given number. Such a function is defined by
Proposition 1.2.6
Proof
The proof flows out from definitions (1.2.79) and (1.2.80).
Definition 1.2.12 (Saw function)
Saw function is a periodic function defined as
( ) ( ) integer , but 1
loor x Ent x m x m x≡= ≤ +>
( ) integer , but 1ceiling x m x m x=≥−<
ceiling x floor x
() ()
1, if is real
−=
⎧
⎨
0, if is integer
⎩
. (1.2.79)
. (1.2.80)
x
. (1.2.81)
x
Remarks 1.2.3
A. Both floor and ceiling functions are used in analyzing and constructing prime numbers.
It follows from definition (1.2.82) that
B.
() ()saw x xfloor x=−
. (1.2.82)
is periodic with the lowest period 1.
()saw x
Chapter 1. Topological, metric, functional, and vector spaces
x
∑
x
∑
28
____________________________________________________________________________
Definition 1.2.13 (Divisor function)
Divisor function
positive integer
where
often omitted, and the corresponding function
Example 1.2.1
Let
Let
is defined as the sum of x-th powers of all positive divisors of a
()
nσ
x
n :
()
ndσ=
x
is a divisor of n; the power x can be arbitrary real all complex. If
d
, (1.2.83)
dn
|
σ is called as the sigma function.
, then
15n =
000 0
111 1
−−− −
(15) 1 3 5 15
111 1
. (1.2.84)
24
15
00000 0
11111 1
11 1 1 1 1
−−−−−
28
12
. (1.2.85)
12n =
, then
(15) 1 3 5 15 4
σ=+++=
0
(15) 1 3 5 15 24
σ =+++ =
1
σ=+++=
−
1
(12) 1 2 3 4 6 12 6
σ=+++++=
0
(12) 1 2 3 4 6 12 28
σ=+++++=
1
(12) 1 2 3 4 6 12
σ=+++++=
1
−
the index is
1
=
Definition 1.2.14 (Aliquot divisor function)
Aliquot divisor function
(the divisors should be less than
where
is a divisor of n.
d
is the sum of all positive aliquot divisors of a positive integer n
()sn
n ):
sn d
Example 1.2.2
Let
(15) 1 3 5 9
Let
15n =
12n =
, then
s =++=. (1.2.87)
, then
()
=
|,
dn d n
, (1.2.86)
<
Chapter 1. Topological, metric, functional, and vector spaces
x
x
x
∑
s
29
____________________________________________________________________________
(12) 1 2 3 4 6 16s =+ + + + =
. (1.2.88)
Let
, then
8n =
Definition 1.2.15 (Prime counting
prime counting function
The
than the given positive real
Example 1.2.3
Let 5n = , then
n = , then
Let 10
(8) 1 2 4 7s =+ + =. (1.2.89)
-function)
π
is the sum of all positive prime numbers that are not greater
()
π
:
()
π=
,isprime
nxn
≤
(5) 2 3 5 3π=++=. (1.2.91)
(10 ) 2 3 5 7 4π = +++ =
000
0000
0
n
. (1.2.90)
. (1.2.92)
Conjecture 1.2.1 (Gauss and Legendre conjecture)
The prime counting function admits the following asymptotic estimate:
() ,
π→∞∼ . (1.2.93)
xx
x
ln( )
x
Proof
The proof relies on Riemann's ζ -function, which will be discussed later on in this subsection.
Definition 1.2.16 (Riemann zeta function)
The
Riemann zeta function is defined for any real (and complex) numbers
series:
∞
−− − −
( ) 1 2 ... ...
sn n
ζ≡ = + ++ +
ss s s
∑
=
n
1
by the following
(1.2.94)
Chapter 1. Topological, metric, functional, and vector spaces
s
s
x
x
30
____________________________________________________________________________
Proposition 1.2.7
The series in the right-hand side of (1.2.94) is convergent at any
complex variables the series is absolutely convergent at any complex
Proof
The proof relies on the convergence condition for the Dirichlet series, as the series in the right
hand of (1.2.94) represents with one of the special cases of the Dirichlet series.
Example 1.2.4
(1)ζ=∞, (1.2.95)
Definition 1.2.17 (Dirichlet function)
The
Dirichlet function is defined for any real
precisely, discontinuous at rational
(or in terms of the
1
>
with Re( ) 1s > ).
2
π
(2)
ζ= , (1.2.96)
(4)
ζ=
, and continuous at irrational x):
6
4
π
. (1.2.97)
90
, as the following discontinuous function (more
where
a and
Dx
are arbitrary, and
b
()
⎧
=
⎨
bx
, if is rational
⎩
.
ab≠
, (1.2.98)
ax
, if is irrational
1.3. Equation Chapter 1 Section 3 Functions of
complex variables
This chapter is devoted to exposition of basic properties of complex numbers and functions
of complex variables. For references see Brown and Churchill (2008), Titchmarsh (1976),
and Wunsch (2004).
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