Fractional Trigonometric Functions in Complexvalued Space: Applications of Complex Number to Local Fractional Calculus of Complex Function

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1 From the SelectedWorks of Xiao-Jun Yang June 4, 2 Fractional Trigonometric Functions in omplevalued Space: Applications of omple Number to Local Fractional alculus of omple Function Yang Xiao-Jun Available at:

2 Fractional Trigonometric Functions in omple-valued Space: Applications of omple Number to Local Fractional alculus of omple Function Yang Xiao-Jun Department of Mathematics Mechanics, hina University of Mining Technology, Xuhou ampus, Xuhou, Jiangsu, 228, P. R. This paper presents the fractional trigonometric functions in comple-valued space proposes a short outline of local fractional calculus of comple function in fractal spaces. Key words: Fractional trigonometric function, comple function, local fractional calculus, fractal space MS2: 3E99, 28A8, 28A99 Introduction The trigonometric functions played an important role in both mathematics engineering. Recently, the fractional trigonometric functions in real-valued space were discussed []. Recently, the fractional trigonometric functions in realvalued space were discussed in fractal space their eponent was fractal dimension [2,3]. In similar manner the fractional trigonometric functions in comple-valued space were structured [2.3]. There are many definitions of local fractional calculus [2-]. Hereby we write down Gao-Yang-Kang s local fractional derivatives [2-6] d f f f f lim, (.) d f f f f. with Gao-Yang-Kang s local fractional integrals [2-6] jn b aib f f t dt lim f tj tj a, (.2) t j where tj tj tj t ma t, t2, t j,..., tj, tj for,..., j N, t a, t N b, is a partition of the intervalab,. Based on local fractional calculus, local fractional Fourier transforms [2],denoted by,, F f : E i f d,, (.3) local fractional Laplace transforms [3], denoted by

3 , L fs s: E s f d,, (.4) as new tools to deal with local fractional differential equations local differential systems, were proposed. More recently, a new imaginary unit proposed in [2,3]. As a pursuit of the work we suggest fractional trigonometric functions in comple-valued space their application to local fractional calculus of comple function. 2 The real-valued fractional trigonometric functions In this section, we start with real-valued Mittag-Leffler function in fractal spaces. Here transforms method is proposed. 2. Mittag-Leffler function in fractal space Definition Let E :, E, denote a continuously function, which is so-called Mittag-Leffler function [2,3] E k : k,. (2.) k Remark. The parameter is fractal dimension. There always eists the relation where is constant. E E y y, for y,, We have the following relations E E y E y,, (2.2) where the function E i solution of the equation As a direct result, we have [2] E i E i y E i y, (2.3) is periodic with the period P defined as the E i P, (2.4) i 2 =-. (2.5) E E i y E i y, for y,, (2.6) 2

4 Taking into account the relation (2.6) with y, we arrive at the result E =. (2.7) Definition 2 The fractional trigonometric function is denoted by with : cos sin E i i, (2.8) cos : sin : Successively, it follows from (2.9) (2.) that 2k k k 2 k (2.9) 2 k + k. (2.) k 2k cos = (2.) sin =. (2.2) Remark 2. Taking into account the fractal dimension, the formulas (2.9) (2.) become respectively cos 2k k k 2k sin k Hence, we have following result. The function of the equation 2 k + k. 2k E i P is periodic with the period P defined as the solution E i P, then P =2. (2.3) 2.2 Transforms method Definition 3 The circle of fractional order, which is defined by the equality y R, yr,,, R,. (2.4) 3

5 Definition 4 The fractional-order circle region of order,, which is defined by the epression y R, yr,,, R,. (2.5) Definition 5 The fractional-order equation of the roundness is defined by the equality y + R, yr,,,, R,. (2.6) Definition 6 The fractional-order equation of the sphere is defined by the equality y + R, yr,,,, R,. (2.7) For (2.4) then there is a fractional-order trigonometric transform where 2 R. y R R cos, (2.7) sin For (2.4) there is a fractional-order trigonometric transform where 2. u v w R R R sin cos sin sin cos, (2.9) 3 The comple-valued fractional trigonometric functions In this section, we start with comple-valued Mittag-Leffler function in fractal spaces. Definition 7 Let E :, E the comple-valued Mittag-Leffler function, denote a continuously function, which is so-called 4

6 E k : k,, (3.) Remark 3. The parameter is fractal dimension. we always arrive at the relation where is constant. 2 2 E E, for, 2, As a direct result, we have the following formulas: Definition E E E,, E E E,, 2 A fractional-order comple number is given by, (3.2). (3.3) i y,, y,,, (3.4) its conjugate of comple number is denoted by i y,, y,,, (3.5) its fractional modulus is defined by the epression 2 2 y. (3.6) It s easy to see that if i y is purely real, that is, Re h, if is purely imaginary, then Im y.. On the other Definition 9 The fractional trigonometric function is denoted by E i : cos i sin, (3.7) with cos : sin : 2k k k 2 k (3.8) 2 k + k. (3.9) k 2k Remark 4. In special case of fractional-order comple number becomes iy,, y,, (3.) 5

7 its conjugate of comple number is denoted yields the fractional modulus defined by the epression iy,, y,, (3.) 2 2 y. (3.2) It follows the definition of classical comple number in special case of. Theorem For a fractional-order comple number i y,, y,,, (3.3) There eists an equivalent formula in the form of the trigonometric function, denoted by the epression Then 2 2 i y y cos i sin. (3.4) cos y 2 2 (3.5) sin y y 2 2. (3.6) Proof. Dividing by y 2 2 in (3.4), we get Hence we deduce the result. y 2 2 y i y y cos i sin. (3.7) 4 Application: Local fractional calculus of complevariable function In this section we give a short outline of local fractional calculus. It is a useful tool to deal with non-differentiable function in comple space. 6

8 4. Local fractional continuity of comple functions Take into account the relation E E, (4.) 2 2 with any, 2,, is constant, which is called comple Hölder inequality of E. Definition with any, 2, f f, (4.2) 2 2, is constant, f is comple Hölder function. Definition 2 Given, then for any we have f f. (4.3) Here f is called local fractional continuous at, denoted by lim f f. (4.4) Setting for any f is called local fractional continuous at, f called local fractional continuous on, denoted by f. As a direct result, we have the following result: Suppose that lim f f lim g g lim f g f g, then we have that is, (4.5) lim the last only if g. f g f g, (4.6) lim / / f g f g, (4.7) 4.2 Local fractional derivatives of comple functions Setting F, the local fractional derivative of D F F F :lim F at is,. (4.8) 7

9 If this limit eists, then the function F is said to be local fractional analytic at, d denoted by D F, F or F d If this limit eists for all in a region, then the function f local fractional analytic in a region.. is said to be As a direct result for definition of local fractional derivatives, we have the following result: Suppose that rules are valid: f g are local fractional analytic functions, the following d f g d f d g ; (4.9) d d d d f g d f d g g f ; (4.) d d d d f d g g f d f d d d g g 2 if g ; (4.) d f d f, where is a constant; (4.2) d d If y f u whereu g, then f g g d y. (4.3) d d d ( k) ( k ) k k ; (4.4) d de E ; (4.5) d d sin cos ; (4.6) d cos sin. (4.7) d 4.3 Local fractional integrals of comple functions Setting f letting f be defined, single-valued in. The local fractional integral of f along the contour in from point p to point q, is defined as 8

10 lim n i i, (4.8) I f f f d where fori,,..., n i i For convenience, we assume that, p n q. I f if. (4.9) Taking into account the definition of local fractional integrals, we have the following result: Suppose that f, g, the following rules are valid: f gd f d gd ; (4.8) k kf d f d, for a constant k ; (4.9) f d f d f d where 2;, (4.2) 2 Theorem 2 If the contour has end points function p q with orientation p to q, if f has the primitive F on, then we have f d FqFp. (4.2) Proof. The proof of the theorem is similar to that of real function is omitted. For more detail for real function, see[4,6]. Theorem 3 If is a simple closed contour, if function f has a primitive on then f d. (4.22) Proof. The definition of a closed contour is that q p. So This proof of the theorem is completed.. f d FqFp. (4.23) 9

11 orollary 4 If the contours 2 have same end points if analytic on, 2 between them, then we have Proof. If 2, then we have f 2 is local fractional f d f d. (4.24) f d f d f d. 2 This proof of the corollary is completed. orollary 5 If the closed contours, 2 is such that2 lies inside, if fractional analytic on, 2 between them, then we have f d f d 2 f is local (4.25). (4.26) Proof. Taking new same end points path using orollary 4, we deduce the result. References [] G. Jumarie. Laplace's transform of fractional order via the Mittag-Leffler function modified Riemann-Liouville derivative. Appl. Math. Lett..22, 29, [2] X. Yang, Z. Kang,. Liu. Local Fractional Fourier s Transform Based on the Local Fractional alculus. In: The 2 International onference on Electrical ontrol Engineering, pp IEEE omputer Society, 2. [3] X.Yang. Local Fractional Laplace s Transform Based on the Local Fractional alculus. In: Proc. Of The 2 International onference on omputer Science Information Engineering, pp , Springer, 2. [4] X.Yang, L.Li, R.Yang. Problems of Local Fractional Definite Integral of the One-variable Non-differentiable Function, World Sci-Tech R&D.. 3(4), 29, [5] F. Gao, X.Yang, Z. Kang. Local Fractional Newton s Method Derived from Modified Local Fractional alculus. In: Proc. of the second Scientific Engineering omputing Symposium on omputational Sciences Optimiation, pp IEEE omputer Society, 29, [6] X.Yang, Research on fractal mathematics some applications in mechanics. M.S.thesis, hina University of Mining Technology, [7] K.M.Kolwankar, A.D.Gangal, Local Fractional Fokker Planck Equation. Phys. Rev. Lett.. 8, 998,

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