Chapter 2 Local Fractional Calculus of One-variable ...

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Chapter 2 Local Fractional Calculus of One‐variable Function In order to seek truth it is necessary once in the course of our life to doubt as far as possible all things. R. Descartes (1596-1650)

Classical calculus of function of one variable is determined on a one-dimensional continuously line domain (a Cartesian coordinate in Euclidean space). However, the one-dimensional fractal orthogonal coordinate is discontinuous, and determined on a cantor set. In this chapter we extent the idea of the classical calculus determined on the one-dimensional Cartesian coordinate in Euclidean space to a new integral determined on the determined on the one-dimensional fractal orthogonal coordinate. The calculus is called as the local fractional calculus of the function on Cantor set.

2.1 Introduction to local fractional calculus Local fractional calculus is a new branch of mathematics that deals with derivatives and integrals of the functions defined on fractal sets. It is used to explain behavior of continuous but nowhere differentiable function. Local fractional calculus (is also called Fractal calculus) was first introduced by Kolwankar and Gangal. It is explain the behavior of continuous but nowhere differentiable function. They proposed particular notation that they had used in their publication for the local fractional derivative of a function defined on fractal sets [33]:  x0 Dx f  x  

d f  x dx

x  x0

: lim

d   f  x   f  x0   d  x  x0 



x  x0

, 0    1.

Local fractional integrals of a function defined on fractal sets is defined as

I

a b

 

N 1

f  x   lim  f  x

with 1dxi  x  is the unit function defined upon

N 

i 0

* i

d  1dxi  x 

 d x

 xi , xi 1  and

i 1

 xi 

,

xi* some suitable point of the interval

33

 xi , xi 1  , where  xi , xi 1  , i  0,....., N 1 , x0  a, xN  m , is a partition of the interval  a, b .

2.2 Historical development of local fractional calculus 

Kolwankar and Gangal local fractional calculus

Kolwankar and Gangal (1996) suggested the notations of local fractional derivative by the expression [33]

d f  x x0 Dx f  x   dx 

: lim

x  x0

d   f  x   f  x0   d  x  x0 



x  x0

, 0    1.

(2.1)

Kolwankar and Gangal’s the local fractional integral is [33]

I

 

a b

N 1

f  x   lim  f  x N 

i 0

* i

d  1dxi  x 

 d x

i 1

 xi 

,

(2.2)

The notations of the local fractional derivative and integrals were broadly applying in mathematical science and engineering [33-63].  Jumarie’s fractional calculus However, Jumarie used a generalization of Taylor series for obtaining a formula for fractional order derivative. Jumarie’s notation for the fractional derivative is [64]

d  D f x f  x   x0 x dx 

 where  f  x  



  1 k 0

k

x  x0

: lim

  f  x   f  x0   h

h0

(2.3)

    f  x    k  h  for 0    1 . k 

Jumarie’s notation for the fractional integrals is [64] x



x

  f  x    f  t  dt  :   x  t  t  0 Ix 

0

 1

0

f  t dt , 0    1 .

(2.4)

The above notations for the fractional derivative and integral are taking into account the non-differentiable functions. This notation of fractional derivative and integrals was broadly applying in the field from mathematical science to engineering [64-84].  Parvate and Gangal ‘s fractal calculus Generally, to understand the fractal behaviour of functions, Parvate and Gangal introduces the local

34

fractional derivative as follows [85-89]:  x0 Dx f  x  

d f  x dx

x  x0

 F  lim

x  x0

f  x   f  x0  , S F  x   S F  x0  

(2.5)

where F  lim is the notion of the limit of f  x  through the points of fractal set F . x  x0

Successively, Parvate and Gangal introduces the local fractional integral as follows [85-89]: N 1





  f  x    f  x  d F x   f  x j  S F  x j 1   S F  x j  , 0    1 , a Ib b

a



j 0

(2.6)

Adda and Cresson’s local fractional derivative

Meanwhile, Adda and Cresson proposed a local fractional derivative [90-92]

d f  x x0 Dx f  x   dx 

: lim Dy,   f  f  x0    x  

x  x0

xx0

(2.7)

with    and Dy ,  is Riemann-Liouville derivative operator. 

Gao-Yang-Kang’s local fractional calculus

Normally, to get to the simple and strict definition, Gao,Yang and Kang went through Jumarie definition, Kolwankar and Gangal definition, and Adda and Cresson definition to obtain the notation of the local fractional derivative [10-11, 21-24, 93-96]  x0 Dx f  x  

d f  x dx

x  x0

: lim

  f  x   f  x0  

xx0

 x  x0 



,

(2.8)

where   f  x   f  x0     1     f  x   f  x0   . From the formula (2.8), we find that

x0

Dx f  x : lim xx0

f  x  f  x0 

 x  x0   1   

,

 x  x0  is fractal mass functions. 1 where H   F   x0 , x     1     1    

Successively, Gao,Yang and Kang introduces the local fractional integral as follows [10-11, 21-24, 93-96]:

35

a

Ib

 

N 1  b  1 1 f x  f t dt  f t  t lim        j j  1    a  1    t  0 j  0





where  t j  t j  1  t j and  t  max  t1 ,  t 2 ,  t j ,...

(2.9)

for 0    1 , and  t j , t j 1  ,

for j  0,..., N  1 , t0  a and t N  b , is a partition of the interval

 a, b  .

Taking into account the limit, we have the definition of the local fractional integral. Because of

 x  x0 



 x  x0 [10-11] Gao,Yang and Kang ‘s local fractional derivative changes

the form as follows[10-11, 21-24, 93-96]  x0 Dx f  x  

d f  x dx

: lim

x  x0

  f  x   f  x0   x  x0

xx0

.

(2.10)

The notations of the local fractional derivative and integrals were broadly applying in mathematical science and engineering [10-11, 21-24, 93-127].  Chen’s fractal derivative Chen had given the notion of the fractal derivative as follows [128-134]: x0

Dx f  x  

d f  x dx

x  x0

: lim xx0

f  x   f  x0  . x  x0

(2.11)

From Yang’s results [14, 17]

H   F   0, x    x , H   F   x0 , 0     x0 and

H   F   0, x    H   F   x0 , 0    H   F   x0 , x   , we have [10, 11, 17, 93]

x  x0   x  x0  , 

such that

x0

Dx f  x  

d f  x dx

where  is fractal dimension.

x  x0

: lim

x  x0

f  x   f  x0  

x  x0



 lim

x  x0

f  x   f  x0 

 x  x0 



,

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If we consider Chen fractal derivative [128-134], we get a new integral formula [17] a

F

b

I b   f  x    f  t  dt   lim 



t 0

a

j  N 1

 f  t  t 



j

j 0

j

.

We find that a

F

I b   f  x    1    a I b   f  x  . 



The notation of the local fractional derivative was broadly applying in mathematical science and engineering [17, 128-134].  He’s fractal derivative He proposed that this definition (2.12) is much simpler but lack of physical understanding and introduced a new fractal derivative for engineering application [17, 74,135]: x0

Dx f  x  

d f  x dx

x x0

: lim

xL0

f  x   f  x0  . KL0

(2.12)

For the history of fractional calculus, we see [136-138]. The fractional model [139-145], fractional derivative operators and fractional integral operator [146-153], geometric and physical interpretation of fractional integration and fractional differentiation [154-158], fractional differential equations [159-182], fractional dynamics and system [183-196], fractional kinetics [197-201], and theory of fractional order calculus and its associated results [202-229] were studied.

2.3 Local fractional derivative 2.3.1 Local fractional derivative Suppose that f  x   C  a, b  . For 1    0 ,   0 and x   x0   , x0    , the limit [10, 11, 23, 94]  x0 Dx f  x  : lim

 1    f  x  f  x0  

 x  x0 



xx0

exists and is finite, then f  x  is said to have the right-hand local fractional derivative of order at x  x0 . Local fractional derivative is noted as [10, 11, 94]

d f  x dx

x  x0

 or f    x0  .

(2.13)



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From the formula (2.13), we find that

x0

Dx f  x : lim xx0

 x  x0  is fractal mass functions of set  1    

where

f  x  f  x0 

 x  x0   1   

,

C  x, x0  .

2.3.2 Left-hand and right-hand local fractional derivative Suppose that f  x   C  a, b  . For 1    0 ,   0 and x   x0   , x0  , the limit [10, 11, 23, 94]

Dx f  x : lim 

x0

1   f  x  f  x0  

(2.14)

 xx 

  0

xx0

exists and is finite, then f  x  is said to have the left-hand local fractional derivative of order

x  x0 . Left-hand local fractional derivative is noted as

d f  x dx

x  x0

 

 or f   x0  .

Suppose that f  x   C  a, b  . For 1    0 ,   0 and x   x0 , x0    , the limit

Dx f  x : lim 

x0

[10, 11, 94]

1   f  x  f  x0 

 x  x0 



xx0

 at

(2.15)

exists and is finite, then f  x  is said to have the right-hand local fractional derivative of order  at

x  x0 . Left-hand local fractional derivative is noted as [10, 11, 23, 94] d f  x dx

x  x0

 

 or f   x0  .

As a direct result, taking into account the definition of local fractional derivative, we induce this proposition: Proposition 2.1

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If

d f  x dx

xx0

and

d f  x dx

exist and

xx0

d f  x dx

x  x0





d f  x dx

d f  x dx

x  x0

x  x0







d f  x dx



d f  x dx

x  x0

x  x0

, then [10, 11, 23, 94] .

2.3.3 The increment of a function For 0    1 , the expression

[10, 11, 94]

 f  x   f    x  x     x  



(2.16)

is called the increment of f  x  , where  x is increment of x and   0 as  x  0 .

2.3.4 The local fractional differential  For 0    1 , the expression d f  f

 

 x  dx 



is called the local fractional differential of

f  x  . Suppose that there exists any point x0   a, b  such that [10, 11, 94] d f  x x  x0  f    x0  ,  dx

D  a, b 

(2.17)

is called  - local fractional derivative set.

As direct results, we have the following propositions: Proposition 2.2 [10, 11] Suppose that f  D  a, b  , then f  C  a, b  . Proof. From (2.16) and (2.17), we arrive at the relation

f  x   f    x0  x  x0     x  x0   f  x0  . 



(2.18)

Taking the limit of formula (2.17), we have

lim f  x   f  x0  .

x  x0

For any x0 , by using the formula (2.19), we also obtain result. Proposition 2.3 [10, 11, 23, 94]

(2.19)

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If f  x   D  a, b  , then f  x  is local fractional differentiable on  a, b  . Proof. From (2.16), we arrive at the relation

 f  x   f    x  x     x  , 



(2.20)

where lim   0 . x  x0

If we replace  f  x  and  x  by d  f  x  and  dx  in (2.20), respectively, this identity implies that 



d  f  x   f    x  dx     dx  . 



(2.21)

Successively, taking into account the formula   0 in (2.21), we deduce the result. Suppose that f  x  , g  x   D  a, b  , the following differentiation rules are valid [10, 11, 94].

d   f  x   g  x  dx d   f  x  g  x  dx



 g  x

d f  x

dx d f  x dx



d g  x

;

dx d g  x  f  x ; dx

 f  x    d   g  x d f  x  f  x d g  x  g  x   dx dx if g  x   0 ; 2  dx g  x

d   Cf  x   dx

C

d f  x dx

if C is a constant;

(2.22) (2.23)

(2.24)

(2.25)

If y  x    f  u  x  where u  x   g  x  , then [10, 11, 23, 94]  d y  x  f   g  x  g1  x ,  dx



where

f



 g  x   and



(2.26)

d y  x  f 1  g  x  g   x , dx

(2.27)

g    x  exist. 1

If y  x    f  u  x  where u  x   g  x  , then [10, 11, 23, 94]

where

f    g  x   and g  1



 x  exist.

40

Local fractional differentiable rules of elementary functions are valid [10, 11, 23]: (1)

(2)

(3)

d xk (1 k)  k1 ;  x  dx (1  k 1 )

d E  x  

dx

 E  x  ;

d E  kx  dx

 kE  kx  , if k is a constant.

(4)

d sin x  cos x ; dx

(5)

d cos x  sin x . dx

(6)

d E  kx2  

dx

 2 kE  kx  .

2.3.5 The higher-order derivative The 2 - local fractional derivative of f  x  , for 0    1 , at x  x0 is defined by [10, 11, 23]

d D  f  x    Dx  Dx  f  x   dx 2 x





 d   2   dx f  x   f  x .  

(2.28)

As a direct application, we have n - local fractional derivative n times       Dx  ...  Dx  f  x 

x  x0

 Dxn f  x 

x  x0

 f

n 

 x

x  x0

 f

n 

 x0  .

2.3.6 Theorems of local fractional derivatives Theorem 2.4 (Local fractional Rolle’s theorem) [10, 11, 23] Suppose that f  C  a, b  and f  D  a, b  . If f  a   f  b  then there exists a point

(2.29)

41

c   a, b  with f



c   0 ,

(2.30)

where   (0,1] .  Proof. Case 1: f  x   0 in  a, b  . Then for all x in (a, b) we have f    x   0 .

Case 2: f  x   0 in

 a, b . Since f  x 

is continuous there are points at which f  x  attains its

maximum and minimum values, denoted by M and m respectively. Because f  x   0 , at least one of the values M , m is not zero. Suppose, for instance, M  0 and that f  c   M . For this case, f  c  x   f  c  is taken into account. If  x  0 , then we have the relations

1   f  c x  f  c 

 x



0

and

lim

1   f  c x  f  c 

x0

 x



0;

Similarly, if  x  0 , then we have

1   f  c x  f  c 

 x



0

and

lim

x0

1  f  c x  f  c 

 x



 0.

Since f  x   D  a, b  , applying Proposition 2.1, we find it happen only if the right-hand and left-hand derivatives are both equal to zero, in which case argument can be used in case M  0 and m  0 .

f



c   0

as required. As similar

42

Hence we arrive at the formula (2.29). A generalized local fractional Rolle’s theorem was discussed [44]. Theorem 2.5[10, 11, 23, 112] Suppose that f  x   C  a, b  and f  x   D  a, b  . Then there exists a point    a, b  with

f     b  a f  b  f  a  , 1  

(2.31)

where   (0,1] . Proof. Define   x  a   F  x 1  f  x  f  a    f  b  f  a   b a  

with

(2.32)

  (0,1] .

Then we have the following relations

F  a   0 and F  b   0 .

(2.33)

In this case, applying Theorem 2.4 to the function F  x  , we have the following identity

F      f     

 1     f  b   f  a    0 , a  b  b  a 

(2.34)

Successively, taking into account the formula (2.33), we obtain the result. Hence this completed the proof of theorem. Theorem 2.6 [10, 11, 23, 112] Suppose that f  x  , g  x   C  a, b  and f  x  , g  x   D  a, b  . If g  b   g  a  then there exists a point c   a, b  with

f b  f  a  f   c  .  g  b   g  a  g    c  

Proof.

(2.35)

Let us define

G  x    1     f  x   f  a   

 1     f  b   f  a    g  x   f  a   .  g  b   g  a  

(2.36)

43

Then we have the formulas

G  a   0 and G  b   0 . Taking the above formulas into account and applying Theorem 2.4 in formula (2.35), we have the relation

G



 x  0 .

(2.37)

In this case, for a  c  b we have the following identity

f



 f  b   f  a  

 x  

 g  b   g  a  

g



 x  0 .

(2.38)

Successively, that is to say, we arrive at the formula (2.35). Therefore this completed the proof of theorem. Theorem 2.7[10, 11, 23, 112] Suppose that f  x   C  a, b  and f  x   D  a, b  . If lim f  x  0 and lim g  x   0 xx0

xx0

as x tends to x0 and A denotes either a real number or one of the symbols  ,  . Suppose that

lim

x  x0

f g



 

 x   A . Then it is also true that  x lim

x  x0

f  x  A. g  x

(2.39)

Proof. Let f  x   C  a, b  and f  x   D  a, b  . We have a  x0  b such that

f  x0   0 and g  x0   0 . Taking the above formulas into account and applying Theorem 2.5, we have x0  c  x such that

f  x  f  x   f  x0  f    c    . g  x  g  x   g  x0  g    c  

(2.40)

When x tends to x0 in (2.40), we obtain the following identity

f  x f    x  lim    A. g  x  x x0 g  x  

lim

x  x0

(2.41)

44

similarly，when x tends to x0 ,we have

f  x f    x lim  lim    A. x  x0 g  x  x  x0 g  x 

(2.42)

Combing (2.41) and (2.42), we complete the proof of the theorem.

2.4 Application of local fractional derivative 2.4.1 Extreme values A function has an absolute maximum at c if f  c   f  x  for x all in D , where D is domain of

f  x  . The number f  c  is called the maximum value of f  x  on D .Similarly, f  x  has an absolute minimum at c if f  c   f  x  for x all in D and the number f  c  is called the minimum value of f  x  on D . The maximum and minimum values of f  x  are called the extreme values of

f  x .

2.4.2 Local extreme value A function has a local maximum at c if f  c   f  x  when x is near to c .Similarly, f  x  has a local minimum at c if f  c   f  x  when x is near to c . Theorem 2.8 (Local fractional Fermat’s Theorem) [10, 11, 23, 112] If f  x  has a local maximum or minimum at x  c , and if

f



c   0 .

f



 c  exists, then (2.43)

45

2.4.3 Critical number A critical number of a function f  x  is a number in the domain of

f



 c   0 or

f



f



c

such that either

 c  does not exist.

As a direct application of local fractional Fermat’s Theorem, we have the following theorem: Theorem 2.9[10, 11, 23, 112] If f  x  has a local maximum or minimum at x  c , then c is a critical number of f  x  . Theorem 2.10 (Increasing/Decreasing test) [10, 11, 23, 112] (a) If

f



(b) If

f



 x  0

on an interval, then f  x  is increasing on that interval.

 x  0

on an interval, then f  x  is decreasing on that interval.

Proof. (a) Let x1 and x2 be any two numbers in the interval with x1  x2 . According to the definition of an increasing function we have to show that

f  x1   f  x2  .  Because we give that f    x   0 , we know that is local fractional differentiable on

 x1 , x2  . So, by

the local fractional mean value theorem there is a number c between x1 and x2 such that

f  x2   f  x1   f 



 c  x2  x1 



.

Since x2  x1 , by assumption, we have

f



 c   0 for x2  x1  0 .

Thus, we have f  x2   f  x1   0 . This shows that f  x  is increasing. Part (b) is proved similarly. Hence the proof of this theorem is completed. As direct application, we have this following theorem: Theorem 2.11 (The  -derivative test) [10, 11, 23, 112] Suppose that c is a critical number of a local fractional function f  x  .

46

 x

changes from positive to negative at x  c , then has a local maximum at x  c .



 x

changes from negative to positive at x  c , then has a local minimum at x  c .



 x

does not change sign at x  c (for example, if

(a) If

f



(b) If

f

(c) If

f

f



 x

is positive on both sides

of x  c or negative on both sides), then f  x  has no local maximum or minimum at c . Theorem 2.12 (The 2 -derivative test) [10, 11, 23, 112] Suppose that (a) If

f



(b) If

f



f

2 

 x

is local fractional continuous near x  c .

c   0

2 and f    c   0 , then f  x  has a local minimum at x  c .

c   0

2 and f    c   0 , then f  x  has a local maximum at x  c .

2 Proof. (a) Applying the definition of f    c  , we have inequalities



 1+  f    x   f    c 

 x-c 



and

for c  x because

f

2 



 0

for x  c

(2.44)

 0

(2.45)

 1+  f    x   f    c

 x-c



 x  0 .

Thus, taking into account the inequalities (2.44) and (2.45), we show that

f



 x  0 , x  c

 and f    x   0 , c  x .

The function f  x  has a local minimum at x  c . Part (b) is proved similarly. Hence the proof of this theorem is completed.

47

2.5 Local fractional integral and its existence 2.5.1 Local fractional integral Let f  x   C  a, b  , local fractional integral of the function f  x  is given by [10, 11]

a

Ib

 

N 1  b  1 1 f x  f t dt  f t  t lim        j j  1    a  1    t  0 j  0



(2.46)



with 0    1 ,  t j  t j  1  t j and  t  max  t1 ,  t 2 ,  t j ,... , where  t j , t j 1  ,

j  0,..., N  1 and x0  a  x1  ...  xi  ...  xN 1  xN  b , is a partition of the interval

 a, b  .

For convenience, we assume that

I

a a

 

f  x   0 if a  b

and   I   f  x  b Ia  f  x if a  b .

a b

Proposition 2.13 [10, 11, 23, 112] Suppose that f  x  defined on a fractal set of fractal dimension  and is bounded on

 a, b 

(or

f  C  a, b ), then a necessary and sufficient condition for the existence of  b 1 f t dt     1    a

(2.47)

is that the fractal set of discontinuities of f  t  have generalized Lebesgue measure zero. Proof. Here we shortly make the proof of this proposition. A necessary condition is proved. Because there exists  b 1 , f t dt     1    a

(2.48)

applying the notation of local fractional integral to formula (2.55), we arrive at the relation N 1  b  1 1 f t dt  f  t j  t j  . lim       1    a  1    t 0 j 0

(2.49)

48

Let the partition of

 a, b 

ti , ti 1 

is

for i  0,....., N  1 , where

t0  a  t1  ...  ti  ...  t N 1  t N  b .

 

Hence f t j

is discontinuous.

In accordance with Theorem 1.3, taking the formula (2.49) into account, we show that

 t    t 

i 1

j

N 1

 ti     and   ti 1  ti     . 



i 0

N 1

Hence  ti , ti 1  has generalized Lebesgue measure zero. i 0

A sufficient condition is proved. Let the partition of

 a, b 

is

ti , ti 1 

for i  0,....., N  1 , where

t0  a  t1  ...  ti  ...  t N 1  t N  b . N 1

And further,  ti , ti 1  has generalized Lebesgue measure zero. i 0

Hence there exist   0 and 0    1 such that N 1

 t i 0

 ti     . 

i 1

When ti  ti 1  ti , we have

 ti 



  ti 1  ti 



one-to-one correspondence with ti . Hence we have the following relation N 1  1 lim  f  t j  t j   1    t 0 j 0

as ti  0 and

N 1

 t i 0

i 1

 ti     . 

So, we derive the result. Theorem 2.14 [10, 11, 23, 112] Suppose that f  x   C  a, b  , then f  x  is local fractional integral on [a, b]. Proof. See the proof of Theorem 2.13.

49

2.5.2 Properties of local fractional integral Property 2.15[10, 11, 23, 112] Suppose that f  x  , g  x   C  a, b  , then

I

a b

 

   f  x   g  x    a I b   f  x   a I b   g  x  .

(2.50)

Proof. Taking into account the definition of local fractional integral, the proof of the property is completed. Property 2.16[10, 11, 23, 112] Suppose that f  x   C  a, b  and C is a constant, then

I

a b

 

 Cf  x    C a I b   f  x  .

(2.51)

Proof. Taking into account the notation of local fractional integral, the proof of the property is completed. Property 2.17[10, 11, 23, 112] Suppose that f  x   1 , then   1 a Ib 

(b  a ) .    1

(2.52)

Proof. This property is obtained by the mean value theorem for local fractional integrals. Property 2.18[10, 11, 23, 112] Suppose that f  x   C  a, b  and f  x   0 . Then we have

I   f  x  0 

a b

with b  a  0 . Proof.

Let f  x   C  a, b  and f  x   0 . We have

f  xi   0 ， i  0,....., N  1 .

(2.53)

50

Let the partition of

 a, b 

 xi , xi 1 

is

for i  0,....., N  1 , where

x0  a  x1  ...  xi  ...  xN 1  xN  b . Because  xi   0 , we have 

  f  x  a Ib 

j  N 1 1  lim  f  xi  xi   0 .  1    t 0 j 0

Hence the proof of the property is completed. Property 2.19[10, 11, 23, 112] Suppose that f  x  , g  x   C  a, b  and f  x   g  x  , Then we have

I

 

a b

f  x   a Ib  g  x  

(2.54)

with b  a  0 . Proof.

Let f  x   g  x   0 , then we have the relation

f  x   g  x   C  a, b  . Taking into account Property 2.18 and using the definition of local fractional integral, we take the integration

I

a b

 

 f  x   g  x   0 .

Applying Property 2.15, the proof of the property is completed. Property 2.20 [10, 11, 23, 112] Suppose that f  x   C  a, b  . Let M and m are the maximum and minimum values of f  x  over the interval  a, b  , respectively. Then we have

M

(b  a ) (b  a )   a Ib  f  x   m    1    1

with b  a  0 . Proof.

Let f  x   C  a, b  , we have

(2.55)

51

m  f  x  M . Taking the integration, for b  a  0 we have  

I

a b

M  a I b  g  x   a I b  m . 



Taking Property 2.17 into account, we have

(b  a ) (b  a )   . M  a Ib f  x   m    1    1 Hence the proof of this property is completed. Property 2.21[10, 11, 23, 112] Suppose that f  x   C  a, b  .Then we get

I

 

a b

f  x   a I b



f  x

(2.56)

with b  a  0 . Proof. By using Property 2.19 we have the relation

 a I b   f  x   a I b   f  x   a I b   f  x  when  f  x   f  x   f  x  . And therefore, we obtain

I

a b

 

f  x   a Ib



f  x .

Thereby we have the result. Property 2.22[10, 11, 23, 112] Suppose that f  x   C  a, b  and a  c  b . Then

I

a b

 

f  x   a I c  f  x   c Ib  f  x  . 



(2.57)

Proof. Let f  x   C  a, b  and a  c  b , then f  x  is local fractional integral on C  a, b  ,

C  a, c  and C  c, b . Let the partition of

 a, b 

is

 xi , xi 1  , where

52

x0  a  x1  ...  xi  ...  xN 1  xN  b and i  0,....., N  1 . We arrive at the following relation  

I

a b

Let the partition of

 a, c 

N 1 1  f  x  lim  f  xi  xi  . 0  t   1    i 0

 xi , xi 1 

is

for i  0,....., j , where

x0  a  x1  ...  xi  ....  x j  c . We get the integration  

I

a b

Let the partition of

 c, b 

j 1  f  x  lim  f  xi  xi  .   0 t  1    i 0

 xi , xi 1 

is

for i  j,....., N  1 , where

x j  c  x j 1  ...  xN 1  xN  b . We have the relation

I

 

a b

N 1 1  f  x  lim  f  xi  xi  . 0  t   1    i j

Hence we show j N 1 N 1 1 1 1    lim  f  xi  xi   lim  f  xi  xi   lim  f  xi  xi  0 0 0       t t t 1  1  1  i 0 i 0 i j

and therefore we obtain the result.

2.5.3 Theorems of local fractional integral Theorem 2.23 (The mean value theorem for local fractional integrals) [10, 11, 23, 112] Suppose that f  x   C  a, b  , there is a point

I

 

a b

 in  a, b  such that

(b a) . f  x  f    1

(2.58)

53

Proof.

Let f  x   C  a, b  , we have the relation

(b  a ) (b  a )   M  a Ib f  x   m    1    1 and therefore

M

   1 

(b  a )

I

a b

 

f  x  m .

(2.59)

From (2.59), taking Proposition 1.20 into account, we arrive at the following formula

   1 

(b  a )

 

I

a b

f  x   f    ,    a, b 

and therefore the result. Theorem 2.24 [10, 11, 23, 112]  Suppose that f  x   C  a, b  , then there is a function   x  a I x  f  x , the function has its local

fractional derivative

d   x  f  x , a  x  b . dx Proof.

(2.60)

Let x   a, b  there exists x  x   a, b  such that

  x   a I x x   f  x  . 

(2.61)

In this case, we get the increment of formula (2.62), that is to say,

  x     x  x     x  

1  x x f  t  dt   x f  t  dt   . a   1     a

(2.62)

Hence, we have that

  x  

x x 1  f  t  dt  .   1    x

Applying Theorem 2.23 to formula (2.63), we arrive at the formula   f  x   f   x I x x 

and therefore

(x)    1

(2.63)

54

   1 x I x x   f  x   f   . (x)

(2.64)

Since

   x    1      x  x     x   , we arrive at the following identity

   x  f    . (x)

Taking the limit of

(2.65)

   x  as  x  0 , we show that (x)    x   f  x , lim x 0 ( x)

(2.66)

Here there exists x  a and  x  0 such that

d   x dx

xa

 f  a  .

(2.67)

Similarly, there exists  x  0 and x  b such that

d   x dx

x b 

 f b  .

(2.68)

Combing the formulas (2.67) and (2.68), the proof of this theorem is completed. As a direct result, we derive the following theorem: Theorem 2.25 [10, 11, 23, 112] Suppose that f  x   C  a, b  , then there exists   C  a, b  such that

  x   a I x  f  x  . 

(2.69)

Proof. Taking Theorem 2.24 into account we obtain the result. Theorem 2.26 (local fractional integration is anti-differentiation) [10, 11, 23, 112]  Suppose that f  x   g    x   C  a, b  , then we have

I

a b

 

f  x   g b   g  a  .

(2.70)

55

Proof. Let

R  x   a I x  f  x  . 

(2.71)

Applying Theorem 2.24 to the formula (2.71), we have the relations

d   R  x   g  x  dx

d R  x



dx



d g  x dx

 f  x  f  x  0

and

R  x  g  x  k . Hence we arrive at the following identity

I

 

a b

f  x   R b   R  a   g b   g  a  .

The proof of this theorem is completed. For fractional order case, we see [154]. Theorem 2.27 [10, 11, 23, 112] Suppose that g  x   C1  a, b  and

 f  g  s   C  g  a  , g  a   . Then we have

I

g  a  g b

Proof.

 

f  x   a Ib



 f  g  s   g '  s  



.

(2.72)

 Let F  x   a I x   f  x  .

We arrive at the formula

I

g c g d 

 

f  x   F  g  c    F  g  d   .

By using Theorem 2.26 in (2.73), we have the following relation

F  g  c    F  g  d    c I d    D  F  g    s  and therefore

I

 

c d

Hence we have the result. As a direct application, we have

 D  F  g    s   c I d   F    g '  s   . 

(2.73)

56

d y

b y    1  f x , y dx         1    a y   b y 



d f  x, y  d b y d a y 1   dx   f  b  y  , y    f  a  y  , y      1    a y  y y y 





.

(2.74)

Theorem 2.28 (local fractional integration by parts) [10, 11, 23, 112]   Suppose that f  x  , g  x   D  a, b  and f    x  , g    x   C  a, b  . Then we have

I

 

a b

f t  g 



 t    f  t  g  t   a  a Ib  f    t  g  t  . b

(2.75)

Proof. Using the equality

d   f  t  g  t    f    t  g  t   f  t  g    t  ,  dt We arrive at the relation

 f  t  g  t   a  a I b b

 

 d   f  t  g  t      dt    .

(2.76)

Applying Theorem 2.26 to (2.76), we deduce to the following relation

I

 

a b

f t  g 



 t    f  t  g  t   a  a Ib  f    t  g  t  . b

So, we obtain the result. As a direct result of Theorem 2.24, we have this following propostion: Proposition 2.29 [10, 11, 23, 112] Suppose that for 0    1

f

k

 x  C  a, b  , then we have

 where

x0

Ix

 k 

I

x0 x

k 

f  x



 k 

 f  x ,

k times k times          k f  x   x0 I x   ... x0 I x   f  x  and f    x   Dx   ...Dx   f  x  .

Proof. Taking the theorem 2.24 into account, we have the result. Proposition 2.30 [10, 11, 23, 112] k  k1 Suppose that f    x , f  x  C  a, b  , for 0    1 , then we have

(2.77)

57

I

 k 

x0 x

where

x0

Ix

  k 1 

[f

 k 

  k 1 

 x]  x I x 0

[f

 k 1 

 x ]  f

 k 

(x  x0 )k ,  x0   k 1

(2.78)

k 1 times k 1 times      k  1      f  x   x0 I x   ... x0 I x   f  x  and f   x   Dx  ...Dx  f  x  .

Proof. Applying Theorem 2.29, we arrive at the formula

I x

x0

 k 1 

[f

 n 1 

 x ]

  x 1   n 1  f x dt      x   1    0 



 x0 I x

k 



I x

k 

x0



x0

I x  k  f  k   x   x0 I x  k  f  k   x0  .

f

k 

(2.79)

 x   f  k   x0  

Hence, using the formula (2.79) and considering the formula x0

I x  k  f  k   x0 

 f  k   x0  x0 I x  k 1  1   x  x0      1     1 2  k  2    1     f  k   a  x0 I x        x  x0     1  2   1     k ( x  x0 )  f  k   x0    k  1  f  k   x0  x0 I x  

k 1 

we have the following relation    [f x0 I x k

k 

I 0 x

 x ]  x

 k 1 

[f

 n 1 

 x ] 

f

k 

 x0 

Remark. When we consider the formula (2.80) with k  0 , we show that x0

I x0[ f 

0

 x ]  f  x   f  x0   f  x0 

and we have

I 0[ f 

a x

Hence the proof of this theorem is completed.

0

 x ]  f  x  .

( x  x0 ) k .   k  1

(2.80)

58

2.5.4 Local fractional integral of trigonometric functions For any positive integers m and n we have [10, 11] (1)

 1   sin  nx   dt   0 ;   1    

(2)

 1   cos  nx   dt   0 ;     1   

(3)

 1    sin  mx  cos  nx   dt   0 ;   1    

0, m  n  1     cos  mx cos  nx  dt     (4) ;    1 1 , m  n  0, m  n  1     (5) ； sin  mx sin  nx  dt       1   1  , m  n  

  2n  1 x  sin    2 1   (6)      1     x 2 sin   2

 dt 







 1   

.

2.5.5 Local fractional definite integral of elementary functions For a constant C these formulas are valid: b

(1)

1  E  x   dx  E  b   E  a  ;  1 a

b 1 k 1  k 1  k1  k (2) x dx  b   a  ;    1 a 1 k 1 





59

b

1  (3) sin x  dx  cos a cos b ;  1 a b

(4)

1  cos x  dx  sin b sin a .  1 a

2 .6 Local fractional Taylor’s theorem 2.6.1 Local fractional Taylor’s theorem Theorem 2.31 (Local fractional Taylor’ theorem) [10, 11, 23, 111, 112] Suppose that f

 k1

 x  C  a, b  , for k  0,1,..., n

and 0    1 , then we have

n f  k  x0  f     k n1    f  x   x x  0  x  x0   1  n 1   k 0 1 k 

 n1

with a  x0    x  b , x   a, b  , where

f

  k 1 

k 1 times     x   Dx ...Dx  f  x  .

Proof. From Theorem 2.30, we arrive at the formula

I

a x

 k 

[f

 k 

  k 1 

 x ] 

I  k [ f  k  x]  a Ix

[ f 

 x ]  a I x

  k 1 

[f

f

 k 

( x  a ) k , a   k  1

that is

 n

k0

k1

a x

 f  x  a Ix n

f

 k

k0

Applying Theorem 2.23, we show that

 n1

[f 

 n1

(x  a)k .  a  k 1

 x

k1

 x]

(2.81)

60  n 1    n 1  I [f  x

a x



  

 x 1  n    n 1  I f x dt    a x  1    a

I

 n 

a x

 f  n 1    x  a      1   

 x  a f    1    n 1   n 1  f   x  a    1   n  1   I

  n 1 

 n 



a x

with a    x , x [a, b] . Hence the proof of this theorem is completed. For fractional order Taylor’ theorem, we see [218-221]. Theorem 2.32 [10, 11, 23, 111, 112] Suppose that f

 k1

 x  C  a, b  , for k  0,1,..., n

and 0    1 , then we have

f  k   x0  k f  x    x  x0   Rn  x  x0  k 0  1  k  n

with ax0  xb, x a,b , where f

  k1

(2.82)

k 1 times   n    x0   Dx ...Dx f  x and Rn  xx0  O  xx0  .





Proof. Applying Theorem 2.31, we have the following formula

Rn  x  x0 

 x  x0 

n



f

 n 1 

  x  x0   n  1   n  1    x  x0  n 1 



f

 n 1 

  x  x   0  1   n  1  

that is

lim

x  x0

Rn  x  x0 

 x  x0 

n

 lim

x  x0

f

 n 1 

  x  x   0  1   n  1  

 0.

Hence we derive the result. Remark, this result was discussed when k  1 [12,14]. Theorem 2.33 [10, 11, 23, 111, 112] Suppose that f

 k1

 x  C  a, b  , for k  0,1,..., n

and 0    1 , then we have

,

61  n1  f  k   0 k f   x x n1  f  x   x  1  n 1   k 0  1 k  n

with 0    1 , x  (a, b) , where f

(2.83)

k1 times      x  Dx ...Dx f  x .

 k1

Proof. Applying Theorem 2.31, for x0  0 and a  x0    x  b we arrive at  f  k   0  k f    x n 1  f  x   x  .  1   n  1   k  0  1  k   n 1

n

If

(2.84)

   x in (2.90), then we obtain that f

 n 1 

  x n1  1   n  1  



f

 n 1 

 x  x n1  1   n  1  

with 0    1 . Hence, we have the result.

2.6.2 Local fractional Taylor’s series Theorem 2.34 [10, 11, 23, 111, 112] Suppose that f

 k1

 x C  a,b , for k  0,1,..., n,...

and 0    1 , then we have

f  k  x0  k f  x    x  x0  k 0 1 k  

with a  x0  x  b , x   a, b  , where

f

  k 1 

(2.85)

k 1 times     x   Dx ...Dx  f  x  .

Proof. In accordance with local fractional Taylor theorem we obtain that

f  x f  k   x0  f      k n 1   lim   x  x0    x  x0   x  x0  1   n  1   k  0  1  k  n

f  k   x0  k  x  x0  x  x0 k  0  1  k  n

 lim 

f  k   x0  k   x  x0  k  0  1  k  

n 1 

(2.86)

62

with a  x0    x  b , x   a, b  , where

f

  k 1 

k 1 times     x   Dx ...Dx  f  x  .

As a direct application, we derive the following theorem: Theorem 2.35 [10, 11, 23, 111, 112] Suppose that f

 k1

 x  C  a, b  , for k  0,1,..., n,...

and 0    1 , then we have

f  k  0 k f  x   x k 0 1 k  

with a  0  x  b , x   a, b  , where

f

  k 1 

(2.87)

k 1 times   x   Dx  ...Dx  f  x  .

Proof. In this case of Theorem 2.34, the proof the theorem is completed when x0  0 .

2.6.3 Local fractional Mc-Laurin’s series to elementary functions We have the following series [10, 11]:

x k (1) E  x    , x  , 0    1 ; k  0  1  k  





(2) cos x   1k k 0



(3) sin x   1

k

k0

x2k , 1 2k 

x  , 0    1 ;

x(2k1) , 1  2k 1

x  , 0    1.

2.7 Local fractional indefinite integral 2.7.1 Local fractional anti-differentiation Let f  x and g  x be two local fractional continuous functions defined on an interval

g



 x  f  x

for each x in

f  x  on  a,b .

 a,b ,

 a,b . If

then g  x is called an local fractional anti-derivative of

63

Theorem 2.36 [10, 11] If g1  x and g2  x are any two local fractional anti-derivatives of f  x on

 a,b , then there

exists some constant C such that

g1  x  g2  x C . Proof. If h x  g1  x  g2  x , then

h



 x  g1  x  g2  x  f  x  f  x  0

for all x in

 a,b .

By Theorem 2.25 there exists some constant C such that for all x in

 a,b

C  h x  g1  x  g2  x .

(2.88)

g1  x  g2  x C .

(2.89)

Hence we have the following relation

2.7.2 Local fractional indefinite integral If g  x is an local fractional anti-derivative of f  x  on

 a, b  ,

then the set

g  x  C :

C is const is called a one-parameter family of local fractional anti-derivatives of f  x  . We call this one-parameter family of local fractional anti-derivatives the local fractional indefinite integral of

f  x  on  a, b  and write 1 f  x dx  g  x C . 1 

2.7.3 Local fractional indefinite integral of elementary functions For a constant C these formulas are valid [10, 11]: (1)

1  E  x   dx  E  x  C ;  1

(2.90)

64

1 k x 1  (2) xk  dx  C ;  1 1  k 1   k1

(3)

1  sin x  dx cos x C ;  1

(4)

1  cos x  dx  sin x C .  1

2.8 Local fractional differential equations If p  x  and q  x  are defined on some open interval

 a, b  , then an equation of the form [10, 11]

d f  x  p  x f  x  q  x , 0    1 , dx is called

(2.91)

 local fractional differential equation in the variable f  x  .

Theorem 3.37 [10, 11] A model for Mittag-Leffler growth is local fractional ordinary differential equation

d y  ky  0 , k  0 , y  0   y0 . dx

(2.92)

Solution of the local fractional differential equation is

y  x   y0 E   kx  . Proof. To solve this equation, we take integration both sides with respect to x , replacing

 d y        dx  by d y as follows:  dx  1 1  d y   1  k  dx .    dx    1 y  dx  1

(2.93)

Because

d  E  kx  dx



 kE  kx  ,

(2.94)

65

we have the following relation

d  E  kx  dx



Let y  x   cE  kx



 kE  kx   0 .

(2.95)

for any constant c , we also show that

d y  ky  0 . dx

(2.96)

Because y  0   c we arrive at

y  x   y0 E   kx  .

(2.97)

Similarly, a model for Mittag-Leffler growth is local fractional differential equation

d y  ky , k  0 , y  0   y0 . dx

(2.98)

Solution of the local fractional differential equation is

y  x   y0 E  kx  ,

(2.99)

where k   . Theorem 3.38 [10, 11] If k  0 and q  x  is local fractional continuous on

 a, b  , then local fractional equation

d y  ky  q  x  dx

(2.100)

 1   y  x  E  kx   q  x E  kx   dx  c .    1   

(2.101)

has the one-parameter family of solutions

 

Proof. We multiply the given local fractional differential equation in Theorem 3.37 by E kx , which is called the integrating factor.

E  kx  Here we have the following relation

d y  p  x  E  kx  y  q  x  E  kx  .  dx

(2.102)

66

d  E  kx  y   q  x  E  kx  .  dx 

(2.103)

By using the notation of the indefinite integral, we arrive at

E  kx  y 

1  q  x  E  kx   dx  .   1   

(2.104)

From (2.104) we arrive at

y  x 

 1  1    q x E kx dx c        ,  E  kx   1   

(2.105)

where k   . This completes the proof. If p  x  and q  x  are defined on some open interval  a, b  , then an equation of the form

d 2 f  x  d f  x  p  x  q  x f  x  r  x , 0    1 , dx 2 dx

(2.106)

is called 2 local fractional differential equation in the variable f  x  . Theorem 3.39 [10, 11] If k and h are constant coefficients, then local fractional equation with constant coefficients

d 2 y d y  k  hy  0 dx 2 dx

(2.107)

has the two-parameter family of solutions

 k  k 2  4h    k  k 2  4h   y  x  AE  x   BE  x  , k 2  4h  0 ,     2 2    

(2.108)

with two constants A and B . Proof.



If E mx



is a solution of 2 local fractional ordinary differential equation, then

m2  km  h  0 .

(2.109)

From (2.109), we have the following relation

m1 



 k  k 2  4h  k  k 2  4h and m2  . 2 2

Because for any constant C CE mx differential equation we show that



is a solution of the 2 local fractional ordinary

(2.110)

67

 k  k 2  4h    k  k 2  4h   y  x  AE  x   BE  x      2 2    

(2.111)

with two constants A and B . Hence this theorem holds. Theorem 3.40 [10, 11] If k and h are constant coefficients, then local fractional equation with constant coefficients

d 2 y d y  k  hy  0 dx 2 dx

(2.112)

has the two-parameter family of solutions

 k  i k 2  4h    k  i k 2  4h   y  x  AE  x   BE  x  , k 2  4h  0 , (2.113)     2 2     with two constants A and B . Proof.



If E mx



is a solution of 2 local fractional ordinary differential equation, then we

have

m2  km  h  0 .

(2.114)

From (2.114) we arrive at the following relations

 k  i k 2  4h  k  i k 2  4h m1  and m2  . 2 2



(2.115)



Since, for any constant C , CE mx is a solution of the 2 local fractional ordinary differential equation we have

 k  i k 2  4h    k  i k 2  4h   y  x  AE  x   BE  x      2 2     with two constants A and B . Thereby we have the result.

2.9 The extended mean value theorem Theorem 3.41 (Second Mean Value Theorem I) [125] Let f ( x) be a bounded function that is integrable on [a, b] . Let further mF and M F be the

68

infimum and supremum, respectively, of the function F ( x) 

x 1 f (t )( dt )  (1   ) a

(2.116)

on [a, b] . Then: (i) If a function g ( x ) is non-increasing with g ( x)  0 on [a, b] , then there is some point  in ( a, b) such that

mF  f ( )  M F and ( ) a b

I

f ( x) g ( x)  g (a) F ( ) .

(2.117)

(ii) If g ( x ) is any monotone function on [a, b] , then there is some point  in ( a, b) such that

mF  F ( )  M F and ( ) a b

I

f ( x) g ( x)  [ g (a)  g (b)]F ( )  g (b) a I b( ) f ( x)

(2.118)

Theorem 3.42 (Second Mean Value Theorem II). [125]

Let f ( x) be a bounded function that is integrable on [a, b] . Let further m and M  be the infimum and supremum, respectively, of the function  ( x) 

b 1 f (t )( dt )  x (1   )

(2.119)

on [a, b] . Then: (i) If a function g ( x ) is non-increasing with g ( x)  0 on [a, b] , then there is some point  in ( a, b) such that

m  ( )  M  and ( ) a b

I

f ( x) g ( x)  g (b)( ) .

(2.120)

(ii) If g ( x ) is any monotone function on [a, b] , then there is some point  in ( a, b) such that

m  ( )  M  and ( ) a b

I

f ( x) g ( x)  [ g (b)  g (a)]( )  g (a) a I b( ) f ( x) .

(2.121)

2.10 Local fractional improper integrals of first kind Local fractional improper integrals of first kind are evaluated as follows [126]: Suppose

( ) a t

I

f ( x )  F (t ) 

( ) a 

I

f ( x) 

t 1 f ( x)( dx) exists for all t  a , then we define  a (1   )

 t 1 1 f ( x)( dx)  lim f ( x)( dx)  lim a I t( ) f ( x)   a a t  t  (1   ) (1   )

provided the limit exists as a finite number. In this case， F (t ) is said to be convergent (or to converge). Otherwise, F (t ) is said to be divergent (or to diverge).

(2.122)

69

Proposition 3.43 [126]

If

both a I ( ) f ( x) and

( ) b 

I

f ( x) are convergent , k1 ， k2 are constant , then ( ) a 

I

[k1 f ( x)  k2 g ( x)] ,

(2.123)

is convergent, and ( ) a 

I

[k1 f ( x)  k2 g ( x)]  k1 a I ( ) f ( x)  k2 a I ( ) g ( x) .

(2.124)

Proposition 3.44 [126]

If f ( x) is local fractional integrable for any infinite interval [a, t ] , a  b ， then (1)

( ) a 

I

f ( x) and b I ( ) f ( x) both converge or both diverge.

（2） a I ( ) f ( x)  a I b( ) f ( x)  b I ( ) f ( x) ,

where

( ) a 

I

f ( x) and b I ( ) f ( x) both converge.

Theorem3.45 [126]

For the existence of the integral

( ) a 

I

f ( x) it is necessary and sufficient that for any given  there

exists t0  a such that ( ) t1 t2

I

f ( x)   

(2.125)

for any t1 , t2 satisfying the inequalities t1  t0 and t2  t0 . Theorem 3.46 [126]

If f ( x) is local fractional integrable for any infinite interval and ( ) a 

I

( ) a 

I

| f ( x) | is convergent, then

f ( x) ,

(2.126)

is convergent and ( ) a 

I

f ( x)  a I ( ) f ( x) .

Theorem 3.47 [126]

An integral

( ) a 

I

f ( x) with f ( x)  0 for all x  a is convergent if and only

if there exists a constant M  0 such that F (t ) 

t 1 f ( x )( dx )  M , for t  a .  a (1   )

(2.127)

The value of the improper integral is then not greater than M . Theorem 3.48 [126]

Let the inequalities 0  f ( x)  g ( x) be satisfied for all x  [a, ) . Then the convergence of the improper integral ( ) a 

I

g ( x) ,

(2.128)

( ) a 

f ( x) ,

(2.129)

implies the convergence of the improper integral

I

and the inequality

70 ( ) a 

I

while the divergence of integral

( ) a 

I

f ( x)  a I ( ) g ( x) .

(2.130)

g ( x) implies the divergence of integral

( ) a 

I

f ( x) .

Theorem 3.49 (Limit Comparison Test) [126]

Suppose

( ) a 

I

f ( x)

and

( ) a 

I

g ( x) , are improper integrals of the first kind with positive

integrands, and suppose that the limit lim

x 

f ( x)  L , g ( x)

(2.131)

exists (finite) and is not zero. Then the integrals are simultaneously convergent or divergent. Corollary 3.50 [126]

Let | f ( x) | g ( x) for all x  [a, ) . Then the convergence of the integral

 1 g ( x )(dx )  a (1   )

implies the convergence of the integral  1 f ( x )( dx ) .   (1   ) a

Theorem 3.51 [126]

Let the following conditions be satisfied. (1) f ( x) is local fractional integrable from a to any point t  [a, ) , and the integral F (t ) 

t 1 f ( x )( dx) ,  a  (1   )

(2.132)

is bounded for all t  a . (2)

g ( x ) is monotone on [ a, ) and lim g ( x)  0 x 

Then the improper integral of first kind of the form  1 f ( x ) g ( x )( dx ) ,  a  (1   )

is convergent.

(2.133)