Download Definitions of complex order integrals and complex order derivatives

1
1
Raoelina Andriambololona, 2 Ranaivoson Tokiniaina,3 Hanitriarivo Rakotoson
Theoretical Physics Department, Institut National des Sciences et Techniques Nucléaires (INSTN-Madagascar)
[email protected], [email protected], [email protected]
2
INSTN-Madagascar, [email protected]
3
INSTN-Madagascar, [email protected]
Abstract— For a complex number , the -order integral of a function fulfilling some conditions is defined as the action of an
operator, noted , on . The definition of the operator is given firstly for the case of complex number with positive real part. Then,
using the fact that the operator of one order derivative, noted , is the left hand side inverse of the operator , an -order derivative
operator, noted , is also defined for complex number with positive real part. Finally, considering the relation = , the
definition of the -order integral and -order derivative is extended for any complex number . An extension of the definition domain
of the operators is given too.
Keywords— integral, derivative, complex order, operator, fractional integral, fractional derivative, gamma function
I. INTRODUCTION
The problem of the extension of real integer-order
derivatives to fractional derivatives is an old one introduced
by Leibnitz in 1695. Several approaches have been done and
bibliography may be found in references [1], [2]
In our work [3], we have given an unified definition for both
derivatives and indefinite integrals of any power function
defined on . Notions of linear, semi-linear,
commutative and semi-commutative properties of fractional
derivatives have been introduced too.
Our investigations are going on. In another work [4], we have
utilized a more general approach. We have considered a larger
set of causal functions, which is the definition domain of order integral operators . The -order derivative operator is derived from the -order integral operators . Properties of
and for any positive real and positive real have been
studied. Remarkable relations verified by , , and for transcendental numbers and have been given too.
We have studied the case of and real numbers (case of field)[4].
In the present paper, we will give the extension to the case of
complex numbers ( field). The extension is neither trivial nor
straightforward because we have to study the existence of the
-order integral operator and the -order derivative
operator in the case of complex numbers and .
It was shown in [4] that for a derivable and integrable function
of real variable verifying the condition
0 for 0
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1.1
and for , a -order integral and -order derivative of
the function can be defined respectively as
%
1
!
" # $ #'#
Γ
&
$
!1 " ) $ )
')
Γ
&
1.2
1.2*
1.3
with , . Γ is the Euler gamma function .50, .60. Now we
extend those results for .
Remark
We would like to point out that throughout our work, we
utilize the notation for the operator product i.e
2 3 2 .3 [
00
456 stands for the writing of the usual 45, 6
II. DEFINITION OF -ORDER INTEGRAL OPERATOR FOR A COMPLEX NUMBER WITH 78 , 0
Theorem 1
For a function verifying the conditions 0 for
0, the integral defined by the relation
%
1
!
" # $ #'#
Γ
&
2.1
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Raoelina Andriambololona et al. International Journal of Latest Research in Science and Technology.
is absolutely convergent for and 9 , 0.
Proof
Let : ; <= then
%
" # $
|
" #|@$
|#|
>
#> |Γ|
Γ
%
" #@$
" #@$
|#|'#
A !>
#> '# !
|Γ|
Γ
&
&
Using an integration by parts, we have for 9 : , 0
%
!
&
" #@$
|#|'#
|Γ|
"
1
B C<D " # |#| " C<D;
" #: |#|
:|Γ| #E
"
#E0
%
" !
" #@ |#|F'#G
C<DI " #
HE%
&
@ |#|
0
As an extension of the relation1.2 we consider the following
definition.
Definition 1
Let be a function which fulfills the condition 0 for
0. For complex number with 9 , 0,we define the
-order integral of the function , considered as the action of
the operator on this function, by the relation
%
1
!
" # $ #'#
Γ
&
C<DJ
" # |#| 0
For %
" #@
" #@$
|#|'# !
|#|F'#
!
|Γ|
:|Γ|
&
&
K .0, ;∞. and for : , 0
0 # A 0 " # A 0 " #@ @
A0
%
" #@
@
:|Γ|
:|Γ|
%
" #@
@
|#|F'# A!
!|#|F'#
:|Γ|
:|Γ|
&
%
A!
&
&
" #
|#|F'# |
|
:|Γ|
:|Γ|
@
@
" #"1
A !>
#> '#
Γ
0
%
!
&
" #@
@
|#|F'# |
|
:|Γ|
:|Γ|
Then for : 9 , 0, we obtain
%
" # $
@
|
|2.2
!>
#> '# Γ
:|Γ|
&
The integral is absolutely convergent for : ISSN:2278-5299
$
!1 " )$ )
')
Γ
&
according to the theorem 1, the integral is well
defined.
2M3 2M3NO %
; <*
2 .PQ*CR
; <<R*CR
0
So
%
2.3
Example
@
HE&
9 , 0.
%
1
!
" # $ #'#
Γ
&
$
!1 " ) $ )
')
S
&
$
2M3
2M3 !1 " )$ )2M3 ')
Γ
&
2M3
4, ; <* ; 1
Γ
in which 4 is the extension of the Euler’s beta function for
complex numbers
$
4, ; <* ; 1 !1 " ) $ )2M3 ')
&
ΓΓ ; <* ; 1
Γ ; ; <* ; 1
in which Γ is the extension of Euler’s gamma function for
complex numbers
2M3 2M3 ΓΓ ; <* ; 1
Γ Γ ; ; <* ; 1
Γα ; 1 ; iβ
2M3
Γs ; a ; ib ; 1
If : ; <= in the relation 2.4, we have
2.4
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Raoelina Andriambololona et al. International Journal of Latest Research in Science and Technology.
2M3 %
H
1
! '# " # ^$ ! 'h # " h_ $ h
Γ$ Γc &
&
Γ ; <* ; 1
2@M3M[
Γ: ; <= ; ; <* ; 1
Γ ; <* ; 1
2@
Γ: ; ; 1 ; <= ; <*
. .PQ= ; *CR
; <<R\= ; *CR
]
From the relation 2.4, it can be easily deduced that
2.5
^ _ 2M3 ^_ 2M3 _ ^ 2M3 2.6
III. PROPERTIES OF THE OPERATOR FOR 78 , 0
Theorem 2
The set ` of function verifying the conditions 0 if
0 is a vectorial space on the field of complex number
and the operator is a linear operator on `.
Proof
The set ` of functions which verify the conditions 1.1 is a
vectorial space on the field :
a , b
A ae, f
` c : 0 and b
0 if 0
A e ; fb
c : e
; fb
0 if 0
The operator is a linear operator on `
a
`g
%
1
!
" # $ #'#
Γ
0 if 0 A A A ae, f
e
&
c , a , b
%
`
&
e
&
; f b
0 if 0
%
3.1
Theorem 3
For all complex numbers $ and c with 9$ , 0 and
9c , 0, we have the semi-group property
^ _ ^_ _ ^ Proof
^ _ ISSN:2278-5299
&
%
&
i
! 'h ! '# " #^$ # " h_$ b#, h
&
%
3.4
for b#, h h
^ _ h
1
! 'hh ! '# " #1"1 # " h2"1
Γ$ Γc 0
we perform the change of variable
)
#"h
"h
# h ; )
" h '# " h')
^ _ `c:
&
H
4$ c e
f
!
" # $ #'# ;
!
" # $ b#'#
S
S
%
! '# " # ^$ ! 'h # " h_ $ b#, h
we obtain the Euler’s beta function
1
; fb
!
" # $ .e# ; fb#0'#
Γ
%
we apply the Dirichlet’s formula given by Whittaker and
Watson [7] [8]
^ _ 1
1
! 'hh " h1 ;2 "1 ! ') 1 " )1 "1 )2 "1
Γ$ Γc 0
0
`
3.3
3.2
Γ$ Γc Γ$ ; c %
1
! 'hh " h^ _$
Γ$ ; c &
^_ ^ _ is symmetric in $ , c , then
^ _ ^_ _ ^ we obtain the semi-group property of j , which stands true for
any function satisfying the existence of j (for any
with the condition 9 , 0).
IV. DEFINITION OF -ORDER DERIVATIVE OPERATOR FOR A COMPLEX NUMBER WITH 78 , 0
By considering the fact that the operator of one order
derivative $ is a left handside inverse [9] of the operator $
and extending the relation 1.3, we may give the following
definition
Definition 2
For all function fulfilling the condition 0 for 0
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Raoelina Andriambololona et al. International Journal of Latest Research in Science and Technology.
and for a complex number with positive real part, the action
of a "order derivative operator on is defined by the
relation
4.1
A sufficient condition which determines the choice of is that
is well defined.
According to the Theorem 1, we have to choose 9 " ,
0.
Practically we choose k and , 9. For instance
.90 ; 1, .90 is the entire part of 9, then
Γ: ; 1 ; <=
2@
Γ " : ; 1 ; <* " <=
. .PQ* " =CR
; <<R\* " =CR
]
From the relation 4.3, it can be easily shown that
^ _ 2M3 ^ _ 2M3 _ ^ 2M3 2M3 2M3 2M3 2M3 Theorem 4
^ _ ^_ _ ^ 4.2
for any $ and c with 9$ , 0 and 9c , 0
Example 4
Let us consider 2M3
According to the relation 2.4, we have
Γ ; 1 ; <*
2M3
Γ " ; ; <* ; 1
Γ ; <* ; 1
2M3 2M3 Γ" ; ; <* ; 1
Γ ; <* ; 1
2@M3[
Γ": " <= ; ; <* ; 1
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%
1
!
" # $ #'#
Γ
&
$
!1 " )$ )
')
Γ
5.1
&
5.1*
&
and
Γ " ; ; <* ; 1
.
2M3
Γ " ; ; <* ; 1 " If : ; <=
• If 9 , 0
1
m
!
" #$ #'#n
S
Γ ; 1 ; <*
Γ " ; ; <* ; 1
The result is independent on The results (4.6) suggest the following definition
%
Γ ; 1 ; <*
.
2M3 0
Γ " ; ; <* ; 1
Γ ; 1 ; <*
2M3
Γ" ; ; <* ; 1
V. DEFINITION OF -ORDER INTEGRAL OPERATOR AND ORDER DERIVATIVE OPERATOR Γ ; 1 ; <*
.
2M3 0
Γ " ; ; <* ; 1
4.6*
These relations suggest then the extension of the definition
of the operators and for any complex number .
• If 9 0
2M3 4.6
FOR ANY COMPLEX NUMBER
Proof
The proof may be easily derived from the semi-group property
of (Theorem 3)
2M3 4.5
Comparing the expressions 4.3 and 2.4, we may write the
relations
.l0$ .l0$ We have the semi-group property
4.4
4.3
5.2
The constant in the relation 5.1* is to be chosen such
is well defined. According to the Theorem 1, we
have to choose 9 ; , 0. Practically we choose k
and , 9" . For instance, one can choose .9"0 ; 1, .9"0 is the entire part of 9".
VI. EXTENSION OF THE DEFINITION DOMAIN. DEFINITION OF
-ORDER INTEGRAL OPERATOR op AND -ORDER
DERIVATIVE OPERATOR op FOR A COMPLEX NUMBER WITH 78 , 0
For a given real number & , the definition of -order integral
may be extended for more general function with the
introduction of the operator %q defined in the relation (6.1).
The action of the operator %q on a function is defined firstly,
320
Raoelina Andriambololona et al. International Journal of Latest Research in Science and Technology.
for complex number with positive real part. Then, using the
%
!
fact that the operator of one order derivative, noted , is the
$
%q
left hand side inverse of the operator %q , an -order derivative
" #@$
" & @
|#|'# C<D |#|
|Γ|
:|Γ| HE%qJ
%
;!
operator, noted %q is also defined for all complex number with positive real part. Finally, assuming the relation %q %
the definition of the -order integral and -order
q
derivative is extended for any complex number with positive
and negative real parts.
For A0
For a derivable and integrable function defined on the
by the relation
, the integral %q defined
%
1
%q !
" # $ #'#
Γ
%q
is absolutely convergent for 6.1
with 9 , 0.
Proof
Let us take : ; <=; It is to be noted that & # so
" #@ r 0, then
%
|
" #|@$
" # $
|#|
>
#> |Γ|
Γ
%
" #@$
" #@$
|#|'#
A !>
#> '# !
|Γ|
Γ
%q
%q
Using an integration by parts, we have for 9 : , 0
%
!
%q
" #@$
|#|'#
|Γ|
1
B C<D " # |#| " C<DJ
" #: |#|
:|Γ| #E
"
#E%q
"
%
" !
" #@ |#|F'#G
0
C<D " #
HE% I
@ |#|
So
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%
" #@ " 0 @
:|Γ|
:|Γ|
%
" #@
" & @
|#|F'# A !
!|#|F'#
:|Γ|
:|Γ|
%q
%
!
%q
%q
" #@
" & @
|#|F'# .|
| " C<DJ |#|0
:|Γ|
:|Γ|
HE%q
then
%
!>
%q
" # $
" & @
#> '# C<D |#|
Γ
:|Γ| HE%qJ
%
;!
%q
" & @
" #@
|#|F'# C<D |#| ;
:|Γ|
:|Γ| HE%qJ
" & @
.|
| " C<DJ |#|0
:|Γ|
HE%q
Then for : 9 , 0
%
%
%q
%q
" # $
" #@
|#|F'#
!>
#> '# !
Γ
:|Γ|
" & @
|
|
:|Γ|
The integral %q is absolutely convergent for : 9 , 0.
Theorem 6
Let $ be the operator of one –order derivative
0
C<D " #@ |#| " & @ C<DJ |#|
HE%qJ
K 0
& , ;∞. and for : , 0, we have
& # A 0 " # A 0 " #@ @
Theorem 5
interval K 0
& , ;∞., &
%q
" #@
|#|F'#
:|Γ|
HE%q
$ '
'
For any complex number with 9 , 0, we have
$ \%q s
%$
q
6.2
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Raoelina Andriambololona et al. International Journal of Latest Research in Science and Technology.
Proof
O % v
% for any R
%
1
$ %q $ .
!
" # $ #'#0
Γ
Definition 5
Let be a derivable and integrable function defined on the
interval K 0
& , ;∞.. We may define an -order derivative
operator %q for a complex number by using the fact that the
operator of one order derivative $ is a left hand side inverse
of the operator %$q
%q
" 1 %
! " # c #'#
Γ %q
%
1
! " #c #'# %$
q
Γ " 1 %q
%q %
q
From this theorem, it can be easily shown that
•
the operator is a left hand side inverse of the operator
$
%$q
• for R, k, R r $
\%q s
%$
q
6.3
These results justify the introduction of the following
definition
Definition 4
Let be a derivable and integrable function defined on the
interval K 0
& , ;∞.. For a complex number with 9 ,
0, we define an -order integral of the function , noted %q ,
by the relation
%q 1
! " # $ #'#
Γ %q
%
6.4
& . From the
%
1
! " # $ #'#
Γ &
ΓsΓp ; 1 t
Γ5 ; 1
t
Γs ; p ; 1 Γ Γs ; p ; 1
6.5
Example 6.2
Let us take & E "∞ and % . Then
% v
%
1
! " # $ H '#
Γ v
For instance, for 1, we have
%
$ % ! H '# %
v
v
6.6
A sufficient condition which determines the choice of is that
is well defined. According to the relation (5.2), we
%
q
have to choose 9 " , 0 . Practically, we choose k
and , 9, for instance .90 ; 1, in which
.90 is the entire part of .90.
%q .l0$ .l0$ 6.7
Example 6.3
Let us study the example of t and & 0,
According to (6.6), we have
& t So
Γ " Γ5 ; 1 t
Γ " ; 5 ; 1 Γ " Γ5 ; 1
t
Γ " ; 5 ; 1
& t x
According to the theorem 5, the integral %q is well
defined.
Example 6.1
Let us take & 0 and t with 5
relation (2.4), we have
k
Γ5 ; 1
t y
Γ " ; 5 ; 1
Γ5 ; 1
t
Γ" ; 5 ; 1
VII. CONCLUSIONS
According to the results that we have obtained, particularly
the relations (5.1a), (5.1b), (6.4) and (6.5), we may conclude
that the approach that we have considered allows the
extension of the definitions of -order integrals and -order
derivatives for the case of complex order . Generally
speaking, we obtain the same formal expressions as in the
case of [4]. We have considered the case of functions
which fulfil more general condition (paragraph VI) too.
Deep analysis of the above results can be done to obtain a
better understanding about the fundamental and physical
meaning of integrals and derivatives for the case of complex
order.
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[1]
[2]
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fractional differential equations”, Bertram Ross (Editor). Publisher:
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Application of differentiation and integration to arbitrary order”,
Acad.Press, N.York and London, 1974.
Using mathematical induction, it may be proven easily that
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Raoelina Andriambololona et al. International Journal of Latest Research in Science and Technology.
[3]
[4]
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10.11648/j.pamj.20130201.12
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10.11648/j.pamj.20130201.11
ISSN:2278-5299
[5]
[6]
[7]
[8]
[9]
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323