A new operator giving integrals and derivatives operators for any order at the same time Raoelina Andriambololona Theoretical Physics Department, Institut National des Sciences et Techniques Nucléaires (INSTN-Madagascar) [email protected], [email protected], [email protected], [email protected] Abstract - Let be the set of integrable and derivable causal functions of defined on the real interval from to infinity, being real, such is equal to zero for lower than or equal to . We give the expression of one operator that yield s the integral operator and derivative() operators of the function at any -order. For positive integer real number, we obtain the ordinary s-iterated integral of . For negative integer real number we obtain the -order ordinary derivatives of . Any positive real or positive real part of complex number corresponds to s-integral operator of .| Any| negative real number or negative real part of complex number corresponds to - order derivative operator of . The results are applied for being a monom. And remarkable relations concerning the and order|| integrals and and order derivatives are given, for and transcendental numbers. Similar results may also be obtained for anticausal functions. For particular values of and , the operator gives exactly Liouville fractional integral, Riemann fractional integral, Caputo fractional derivative, Liouville-Caputo fractional derivative. Finally, the new operator is neither integral nor derivative operator. It is integral and derivative operators at the same time. It deserves of being named : raoelinian operator is proposed. Keywords –operators, fractional integrals, fractional derivatives, gamma function, Euler’s gamma function, Euler’s beta function In geometric representation, and correspond to the I. INTRODUCTION rotations of and respectively. and correspond to # # We tackle the problem of integral and derivative using the the rotations of$ and$ respectively. operator approach. And we are looking for the unique operator # # Then, we may define one new operator such giving integrals and derivatives at the same time and for any " $ " $ order. % Our guidelines of thought are the following. I.1 Is it possible to derive the integral operator from the % ( )() = sin ( " ) 2 derivative operator or inversely ? for any . For positive (respectively negative) integer, we obtain the ordinary integral (respect. derivative) operator. We have shown that it is easier to derive derivative operator & ' from integral operator rather than the inverse [1] [2] The extension for is also possible. These relations are &utilized ( in the Fresnel representation to I.2 In classical definition, the order of the integral and study RLC circuits (passive resistor R, self-inductance L and derivative operators must be positive integer numbers . Is it capacitance C) for sinusoidal electricity in terms of phase possible to extend them to any real and complex numbers? We difference. The potential difference for R is in phase with have shown that the answer is yes [1] [2] the current . is delayed in phase of with for C ) # I.3 Our research has been based on the important remarks () = %). ) is faster of with for L$ concerning trigonometric and exponential functions. 1 # () = *+,-./ . 0 2) ) $ I.3.1 Trigonometric functions 41 For positive integer , it is easily shown by iterating times () = 3 45 ) the derivative operators and integral operator according I.3.2 Exponential function to the following results The second remark is related to exponential functions 78 78 78 78 (+ ) = 9+ (+ ) = 9 + ()( ) = + ()( ) = sin ( + ) 6 78 : 78 ! 6 78 : 78 ! 2 2 ! (+ ) = 9 + (+ ) = 9 + We may then define one new operator such as ( )( ) = + ( )( ) = ( + ) 2 2 % ()( ) = " ()( ) = sin ( " ) 78 : 78 2 2 ! % ( + ) = 9 ( + ) corresponds to the integral (respect.derivative) operator for ( )( ) = " ( )( ) = ( " ) 2 2 %positive (respect. negative) values of . 1 In algebraic representation, the derivative operator is for . U is the ordinary derivative operator for -order expressed in terms of multiplication, the integral operator by a 4 U division. This remark is utilized to solve an integro– 5b) linear & P property48 of and . differential equation with constant coefficients by postulating D D an exponential solution with constant. The integro- III. PROPERTIES OF THE OPERATOR differential equation is transformed into an algebraic equation +;(-) - with constant coefficients. Then we have to look for roots , a III.1 Linear property < well known problem. - The result is also utilized to study RLC circuits by the so- It is easy to demonstrate the relation called complex number method. D D D I.3.3 Remark : the two approaches I.3.1 and I.3.2 are for any % (X0 + Y, .)( ) = X% (0)(). + Y% (,)() equivalent by using the Euler’s relation between exponential and trigonometric functions. 0 & =, , & = X & (, Y & ( III.2 Semi-group property of II. DEFINITION OF THE OPERATOR < < DW DV DWLDV DV DW Theorem for any and . %It is% assumed= % that= the % derivative% operator (respectively ) is the inverse of the integral operator Let be the set of integrable and derivable causal functions $ (respectively ). defined on the interval such = for . Then the operator >=?.,+A?,D . &' 0() = 0 Proof C . % The proof is given in Appendix . 8 This property is very useful because it simplifies the D 1 D: % (0)() = G ( " H) 0(H)JH demonstration of many relations in particular to define the F() I derivative operator (see section IV.2) D D: = GI (1 " K) 0(K)JK III.3 Principle of correspondence F() where is the extension of8 Euler’s gamma function for It is easily shown the following properties any F() gives at the same time the integrals and derivatives ( &', operators &() at any order . If or D , the operator gives the extensionL of the integral & ' M+() N 0 /RDS % (0)() = % (0)() = (0)() Z & P,Z0 & = operator of at any Dorder . If or , the ( & () D % operator corresponds to the extension: of derivative D :[ [ D 0 & ' M+() O 0 operator at any order . DS:[/R % (0)() = % (0)() = (0)() ZR & P, Z0 & = % where is the ordinary integral of order , and is the D || ordinary derivative operator of order . [ Proof R The proof has been given in our paper [1] for real order , in IV. OBTENTION OF THE INTEGRAL OPERATOR AND our paper [2] for complex order , in our paper [3] for real and ] DERIVATIVE OPERATOR FROM _ \ complex order for causal functions and for real and complex order for anticausal functions [4]. Then contains the IV.1 Obtention of the integral^ operator< from integral and derivative operators at any order . D a % It may be easily shown ` < Let us recall the unique strategy applied in our approach [1] [2][3] [4] ( )( ) ( )( ) L ( ) 1) Define the integral operator and the derivative Then% 0 = 0 Z & ' or M+ N 0 id & ( of a causal functions at first order . (0)() 2) Define(0)() and 0 for any . D D % = 3) Extend to (0)(, )then to (0)( ) & P IV.2 Obtention of the derivative operator from 4) Extend to & Q & ' We have two possibilities to define derivative oper< ators from 5) and finally, look & ( for in which case we have the following conditions . 5a) the principle of correspondence D %IV.2.1 The left-hand derivative operator _ ^e 8 5W 5V 5U [ :7 7:[ 7 7:[ D f : = % % = % /R (0)() = G G G T G 0(* ) J* J* T J* for any and any DS I I I I no sommation on 9 & P R & 'Lor M+() N 0 id R &( for or 9 & P D J [ 7 7:[ /RDS (0)() = (0)() f (0)() = % (0)() J 2 8 8 7 1 7:[: D 1 D () = G ( " H) 0(H)JH % (0)() = G ( " H) 0 (H)JH F(g " R) I Let us change to F ( + 1) I By successive application of derivative operator on the integral, it is easily found that is independent on l R [ 8 and we obtain f (0)() 9 :[ 1 :[ () % (0)() = G ( " H) 0 (H)JH I F(1 " R) 8 [ [ 1 :[ then = j (0)() f (0)() = G ( " H) 0(H)JH F(1 " R) I If where is a constant :[ [ The Liouville-Caputo fractional% = derivative j definition is [5] 0() = h h :[ [ ( " .) 8 f (h) = h i 0 0 h i 0 [ 1 :[ () F(1 " R) f2 L (0)() = G ( " H) 0 (H)JH The Riemann fractional derivative is defined by [5] hich is exactly F(1 " Rwith) :m the particular value ∞. [ nThe Caputo fractionalj ( 0derivative)() is defined by [5] . = " 8 [ J 1 :[ j1 (0)() = G ( " H) 0(H)JH 8 k J F(1 " R) [ 1 :[ J which is exactly for . 2L (0)() = G ( " H) ? 0(H)oJH [ which is withF(1 "the R particular) k value JH . f (0)() . = 0 [ IV.2.2 The right-hand derivative operator j (0)() . = 0 _ IV.2.3 Remark < ^ [ 7:[ :7 7:[ 7 If the derivative operator (respectively ) is the inverse of j = % % = % for any and any the integral operator (respectively ) then we have the no sommation on . semi-group property for the operator and we have one L M+() N 0 &( 9 & P R & ' or id R derivative operator . If it is notD the case we don’t 9 [ [ % [ 7:[ 7 have the semi-group propertyf = j for the operator but the semi- j (0)() = % ( 0)() group property stands for and . D % 8 D D 1 7:[: (7) = G ( " H) 0 (H)JH V. APPLICATIONS OF THE RESULTS [1] By successiveF integration(9 " R) I by part of the function under the integration sign, we obtain that the second member is Let us apply now our results to the function for p independent on . any positive real . 0() = ; 9 8 [ 1 :[ () j (0)() = G ( " H) 0 (H)JH q p F(; + 1) pLD ;r pLD If where F is(1 a "constant R) I % ( ) = = F(; + + 1) (; + )r ( ) for any positive real numbers and . 0 = h h [ j ; (h) = 0 even if the constant s t s F(+ + 1) #Ls +r #Ls ( ) = % ( ) = = h i 0 F(+ + + 1) (+ + )r is then the good choice instead of if you require [ [ # # F( + 1) sL# r #Ls thej (principle0) of correspondence that the derivativef of a ( ) = % ( ) = = constant is null.