k-fractional Trigonometric Functions
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1 International Journal of Contemporary Mathematical Sciences Vol. 9, 2014, no. 12, HIKARI Ltd, -Fractional Trigonometric Functions Rubén A. Cerutti Luciano L. Luque Faculty of Exact Sciences National University of the Northeast Av. Libertad 5540 (3400); Corrientes, Argentina Copyright c 2014 Rubén A. Cerutti Luciano L. Luque. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original wor is properly cited. Abstract Based on the -Mittag-Lefler function the -α-exponential Function we introduce families of functions that allows us define new fractional trigonometric functions that contain the classical trigonometric functions as particular case for some convenient election of parameters. We study some elementary properties obtain the Laplace transform of some elements of the families. Keywords: Fractional Calculus, -Calculus, Exponential Function I Introduction As is well nown the classical exponential function, from it, the classical trigonometric functions play an important role in the solution of ordinary differential equation with constant coefficients. Hence the importance of studying it its generalizations including the Mittag-Leffler function of one or two parameters, the one introduced by Praapati (c.f. [8) or for us in [5 the called -Mittag-Leffler function given by E γ,α,β (z) (γ) n, z n Γ (nα + β)n! (I.1)
2 570 Rubén A. Cerutti Luciano L. Luque where (γ) n, denote the -Pochhammer symbol given by (γ) n, γ(γ + )(γ + 2)...(γ + (n 1)) (I.2) Γ (z) is the -Gamma Function due to Diaz Pariguan (c.f.[4) Γ (z) 0 t z 1 e t dt. (I.3) Elementary calculations allow us to establish the following relationship between (γ) n, Γ (z) (γ) n, Γ (γ + n). (I.4) Γ (γ) It can be seen that if in (I.1), is taen γ α β 1, the classical exponential function is obtained E 1 1,1,1(z) e z. (I.5) By Euler equation can be established that e z e x+iy e x (cos y + i sin y), (I.6) considering the series expansion of the exponential function the sine cosine functions we have cos z eiz + e iz ( 1) n z 2n (I.7) 2 (2n)! sin z eiz e iz 2i ( 1) n z 2n+1. (I.8) (2n + 1)! The extension to fractional calculus of the exponential function carries with it the extension of trigonometric functions. It may be seen the elementary presentation (c.f.[1) the very interesting one given in [3 in which starting from a generalized exponential function e λ(x a) α (x a) α 1 λ n (x a) nα Γ [α(n + 1) (x a) α 1 E α,α [λ(x a) α, (I.9) where x > a, λ C, α R +, α R E α,β (x) is the two parametters Mittag-Leffler function. E α,β (x) x n Γ(nα + β) (I.10)
3 -Fractional trigonometric functions 571 may be extended to Fractional Calculus definitions of sine cosine function by ( 1) n λ 2n (x a) (2n+1)α 1 cos α [λ(x α) (I.11) Γ [(2n + 1)α verifying sin α [λ(x α) ( 1) n λ 2n+1 (x a) (n+1)2α 1 Γ [(n + 1)2α D α a + [sin α(λx) λ cos α (λx) D α a + [cos α(λx) λ sin α (λx) (I.12) (I.13) (I.14) where D α a denote the Riemann-Liouville fractional derivative of order α defined by + (D α a +f) (x) [ D n I n α a f (x) (I.15) + being D the usual derivative operator I α a + the fractional Riemann-Liouville integral operator of order α. Through this paper we will use frequently D γ [(x a) α Γ(α + 1) (x a)α γ (I.16) Γ(α γ + 1) (c.f. [2), (c.f.[4). Γ(x) ( ) x 1 1 Γ (x) (I.17) II Riemann-Liouville Fractional Derivative of order a of the -α-exponential Function In this paragraph, based on the -α-exponential Function, new definitions of trigonometric functions are presented some elementary properties of them are studied. To do that we will begin computing the Riemann-Liouville fractional derivative of order α/ of the -α-exponential Function. Let e az γ,α be the -α-exponential Function (c.f.[7) given by e az γ,α z α 1 E γ,α,α z α 1 ( α ) az ) n ( α (γ) n,γ az Γ [(n + 1)α n!. (II.1)
4 572 Rubén A. Cerutti Luciano L. Luque Taing into account formulae (I.16) we have D α [e az (γ) n, a n Γ [ α (n + 1) z α n 1 γ,α Γ n1 [(n + 1)α n! Γ [ α (n + 1). (II.2) α Reminding the relationship between the two Gamma Functions given by (I.17) we have D α [e az γ,α n1 az α 1 α (γ) n, a n 1 α (n+1) Γ [α(n + 1) z α n 1 Γ [α(n + 1) n! 1 [ α (n+1) α Γ [α(n + 1) α ( α ) (γ) n+1, az (n + 1)! Γ [α(n + 1). (II.3) If in (II.3); γ 1 is considered we obtain the formulae (23) of [2. When a iλ, λ R, x a t > 0, (II.1) can be written e iλt γ,α t α 1 E γ α,α ( α ) iλt (γ) n, i n λ n t α (n+1) 1 Γ [(n + 1)α n! (II.4) where i is the imaginary unit. By grouping summs according to the powers of the imaginary unit the parity of n, we obtain e iλt γ,α ( 1) n (γ) 2n, λ 2n t α (2n+1) 1 Γ [(2n + 1)α (2n)! + i Taing into account the above, we define cos γ,α (λt) Re ( ) e iλt γ,α ( 1) n (γ) 2n+1, λ 2n+1 t 2 α (n+1) 1. Γ [2α(n + 1) (2n + 1)! (II.5) (II.6) sin γ,α (λt) Im ( e iλt γ,α). (II.7) If γ 1, coincides with formulaes (20) (21) from [3. observed that if we adopt γ 1, we have It may be 1 cos 1,α (λt) 1 sin 1,α (λt) ( 1) n λ 2n t (2n+1) 1 Γ [(2n + 1)α ( 1) n λ 2n+1 t (2n+2) 1 Γ [(2n + 2)α
5 -Fractional trigonometric functions 573 which coincides when λ 1 with the ones in [2. Then, it may be written e iλt γ,α cos γ,α (λt) + i sin γ,α (λt). From (II.6) (II.7) when α γ 1 we have Analogously, it may be obtained From (II.8) (II.9) it result 1 cos 1,1 (t) cos(t) Re ( e it) 1 sin 1,1 (t) sin(t) Im ( e it). (II.8) e iλt γ,α cos γ,α (λt) i sin γ,α (λt). (II.9) cos γ,α (λt) e iλt γ,α + e iλt γ,α 2 (II.10) sin γ,α (λt) e iλt γ,α e iλt γ,α. (II.11) 2i Can be clearly seen that when α γ 1 results the nown expression for the classical trigonometric functions given by (I.7) (I.8). From the defining formulae (II.10) (II.11) may be derived the following properties: cos γ,α ( λt) cos γ,α (λt) (II.12) sin γ,α ( λt) sin γ,α (λt) (II.13) cos 2 γ,α(λt) + sin 2 γ,α(λt) e iλt γ,α e iλt γ,α (II.14) By replacing,α, γ λ for 1 in (II.14) is obtained the well nown relation cos 2 (t) + sin 2 (t) 1. (II.15) III The Laplace Transform of the -α-exponential Function We begin by recalling the Laplace Transform of the function t α (n+1) 1, given by L [ t α (n+1) 1 (s) Γ [ α(n + 1) s α ((n+1)) (III.1)
6 574 Rubén A. Cerutti Luciano L. Luque (c.f.[7). The series expansion of the -α-exponential Function allows us evaluate L [ [ e λt (γ) n, λ n t α (n+1) 1 γ,α L Γ [(n + 1)α n! (γ) n, λ n L [ t α (n+1) 1, Γ [(n + 1)α n! taing into account (III.1), it results L [ e λt (γ) n, λ n Γ [ α(n + 1) γ,α n!γ [(n + 1)α s α (n+1) (γ) n, 1 α n α a n n!s α n s α 1 α s α (γ) n, n! ( λ (s) α ) n. (III.2) According the notation by [8 formula (2.22) we may writte L [ ( e λt 1 α γ,α 1 λ ) γ,. (III.3) s α (s) α If in (III.3) we consider γ 1 we have L [ e λt 1 ( ) n λ α 1 s α s α s α λ (III.4) (c.f. [6). For the convergence of the series in (III.2) see [8 also [9. To obtain the Laplace Transform of cos γ,α (λt) sin γ,α (λt) taing into account (II.10) (II.11) we have L [ cos γ,α (λt) 1 { } L [e iλt γ,α + L [e iλt γ,α 2 (III.5) from (III.3) it results { ( L [ cos γ,α (λt) 1 α 1 λ 2s α (s) α ) γ, ( λ ) } γ,. (III.6) (s) α When taen α γ 1; (III.6) reduces to the classical expression s L [cos(λt) s 2 + λ 2 Analogous it may be obtained s L [sin(λt) s 2 + λ 2 (III.7) (III.8)
7 -Fractional trigonometric functions 575 IV The Function Motivated by the result expressed by the formula (II.5) looing for to be verified the classical relationship between the derivative of the sine(cosine) with the cosine(sine), by considering the variation of family of the index of the -Pochhammer symbol, we introduce the following family of Mittag-Leffler type. Definition 1 : Let α, γ C, Re(γ) > 0, Re(s) > 0, > 0 z C {0}, N 0. (λz) (γ) n+, λ n z α (n+1) 1 Γ n 0 (α(n + 1))(n + )!. (IV.1) It can be seen that if in (IV.1) is considered 0, is obtained the function -α-exponential (c.f.[7) e λz γ,α z α/ 1 E γ,α,α ( λz α/ ), (IV.2) When 0 1 is obtained the function α-exponencial function given by (I.9). We will demostrate that the function satisfies the following properties: 1. 0 (λz) e λz γ,α. 2. D α/ [ 0 (λz) 3. D α/ [ λ α/ 1 (λz). (λz) λ α/ +1 (λz). 4. ( D α/) ) ( (e ) λz γ,α λ α/ E,γ,α (λz). Proof. The first one results from (IV.1) when 0. Moreover, it suffices to prove 3),to see that also verified 2). Finaly we will show the property 3) by induction on the index. We will prove that 4) is verified. By using the definition the relationship (I.16) D α/ ( ) (λz) n 1 n 1 (γ) n+ λ n Γ [α(n + 1) (n + )! Dα/ ( z α/(n+1) 1) (γ) n+ λ n Γ( α Γ [α(n + 1) (n + )! (n + 1)) Γ( αn ) z α n 1
8 576 Rubén A. Cerutti Luciano L. Luque Now, taing into account the relationship between the two gamma functions, the classical -Gamma, Γ( αn αn ) 1 n Γ(αn), results D α/ ( ) (λz) n 1 α (γ) n+ λ n Γ [α(n + 1) (n + )! n 1 α Γ(α(n + 1)) z α n 1 Γ(αn) (γ) n++1 λ n+1 Γ [α(n + 1) (n + + 1)! z α n 1 λ α/ +1 (λz). (IV.3) We will prove the property 4) by inducction on. In view the properties 1) y 2) we nown that this relationship is verified to 1. Suppose it holds for n, we see that this implies that equality holds for n + 1: ( ) D α/ +1 (E,γ,α 0 ) D ( α/ D α/) [ (E,γ,α (λ 0 ) D ) α/ α/ E,γ,α ( λ α/) D α/ ( ) ( λ α/) +1 E,γ,α +1. (IV.4) that is what we wanted to prove. IV.1 Trigonometric Functions On the basis of (I.1), looing for another generalization of the classical trigonometric functions cosine sine, we put by Definition 2 : Let α, γ C, Re(γ) > 0, Re(s) > 0, > 0 z C {0} y N 0 { } cos,γ,α (λz) Re (iλz) o equivalently { sin,γ,α (λz) Im } (IV.5) (iλz) cos,γ,α (λz) n 0 sin,γ,α (λz) n 0 ( 1) n (γ) 2n+, λ 2n z α/(2n+1) 1. (IV.6) Γ [α(2n + 1) (2n + )! ( 1) n (γ) 2n+1+, λ 2n+1 z α/(2n+2) 1. (IV.7) Γ [α(2n + 2) (2n )!
9 -Fractional trigonometric functions 577 The functions introduced in (IV.5) satisfies the following properties: Lemma: Given α, γ C, Re(γ) > 0, Re(s) > 0, > 0 z C {0} y N 0, then D α/ { sin,γ,α (λz)} λ α/ +1 cos,γ,α (λz). D α/ { cos,γ,α (λz)} λ α/ +1 sin,γ,α (λz). (IV.8) (IV.9) Proof. By using the definition (IV.1) the properties demostrated in IV.1, we have D α/ [ (iλz) λ α/ +1 (iλz) λ α/ n 0 (γ) n++1, (iλ) n z α/(n+1) 1 Γ [α(n + 1) (n + + 1)!. (IV.10) Grouping of course with the powers of imaginary unit, it results D α/ [ (iλz) λ α/ n 0 ( 1) n (γ) 2n+2+,λ 2n+1 z α/(2n+2) 1 Γ [α(2n + 2) (2n )! + + i λ α/ ( 1) n (γ) 2n+1+,λ 2n z α/(2n+1) 1 Γ n 0 [α(2n + 1) (2n )!. As (iλz) cos,γ,α (λz) + i sin,γ,α (λz), results the assertion. (IV.11) References [1 Almusharrf, A. Development of Fractional Trigonometry an applications of Fractional Calculus to Pharmacoenetic Model. Maters Theses Specialist Proyects. (2011). Paper [2 Bonilla, B.; Ribero, M.; Truillo, J. Linear Differential Equations of fractional order. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments Applications in Physics Engineering. Springer. (2007).
10 578 Rubén A. Cerutti Luciano L. Luque [3 Bonilla, B.; Rivero, M. Truillo,J.; Fractional Differential Ecuations as alternative models to nonlinear differential equations. Applied Mathematic Computation. Applied Math. Computation. 187.(2007). [4 Diaz, R.; Pariguan, E. On Hypergeometric Functions -Pochhammer symbol. Divulgaciones Matematicas.Vol.15 2, (2007). [5 Dorrego, G.; Cerutti, R. The -Mittag-Leffler Function. Int. J. Contemp. Math. Sciences, no 15, Vol. 7, (2012). [6 Kilvas, H.; Srivastava, H, Truillo, J. Theory Application of Fractional Differential Equations. Elsevier.(2006). [7 Luque, L.; Cerutti, R.; The -α-exponential Function. Int. J. of Math. Analysis. Hiari. Vol.7, no 9-12, (2013). [8 Praapati J.; Shula, A. Decomposition of Generalized Mittag-Leffler function its properties. Advances in Pure Math.. No. 2. (2012).DOI: /apm [9 Shah, P.; Jana, R. Results on Generalized Mittag-Leffler Function via Laplace Transform. Applied Math. Sciences. 7(12).(2013). Received: August 1, 2014
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