Cardinal Functions and Integral Functions

INTERNATIONAL JOURNAL OF GEOMETRY Vol. 1 (2012), No. 1, 27 - 40 CARDINAL FUNCTIONS AND INTEGRAL FUNCTIONS MIRCEA E. ŞELARIU, FLORENTIN SMARANDACHE and MARIAN NIŢU Abstract. This paper presents the correspondences of the eccentric mathematics of cardinal and integral functions and centric mathematics, or ordinary mathematics. Centric functions will also be presented in the introductory section, because they are, although widely used in undulatory physics, little known. In centric mathematics, cardinal sine and cosine are de…ned as well as the integrals. Both circular and hyperbolic ones. In eccentric mathematics, all these central functions multiplies from one to in…nity, due to the in…nity of possible choices where to place a point. This point is called eccenter S(s; ") which lies in the plane of unit circle UC(O; R = 1) or of the equilateral unity hyperbola HU(O; a = 1; b = 1). Additionally, in eccentric mathematics there are series of other important special functions, as aex , bex , dex , rex , etc. If we divide them by the argument , they can also become cardinal eccentric circular functions, whose primitives automatically become integral eccentric circular functions. All supermatematics eccentric circular functions (SFM-EC) can be of variable excentric , which are continuous functions in linear numerical eccentricity domain s 2 [ 1; 1], or of centric variable , which are continuous for any value of s. This means that s 2 [ 1; +1]. ————————————– Keywords and phrases: C-Circular , CC- C centric, CE- C Eccentric, CEL-C Elevated, CEX-C Exotic, F-Function, FMC-F Centric Mathematics, M- Matemathics, MC-M Centric, ME-M Excentric, S-Super, SM- S Matematics, FSM-F Supermatematics FSM-CE- FSM Eccentric Circulars, FSM-CEL- FSM-C Elevated, FSM-CEC- FSM-CE- Cardinals, FSM-CELCFSM-CEL Cardinals (2010) Mathematics Subject Classi…cation: 32A17 28 M. Şelariu, F. Smarandache and M. Niţu 1. INTRODUCTION: CENTRIC CARDINAL SINE FUNCTION According to any standard dictionary, the word "cardinal" is synonymous with "principal", "essential", "fundamental". In centric mathematics (CM), or ordinary mathematics, cardinal is, on the one hand, a number equal to a number of …nite aggregate, called the power of the aggregate, and on the other hand, known as the sine cardinal sinc(x) or cosine cardinal cosc(x), is a special function de…ned by the centric circular function (CCF). sin(x) and cos(x) are commonly used in undulatory physics (see Figure 1) and whose graph, the graph of cardinal sine, which is called as "Mexican hat" (sombrero) because of its shape (see Figure 2). Note that sinc(x) cardinal sine function is given in the speciality literature, in three variants (1) 1; sin(x) ; x sin(x) =1 = x +1 X ( 1)n x2n = (2n + 1)! sinc(x) = ( n=0 d(sinc(x)) cos(x) = dx x for x = 0 for x 2 [ 1; +1] n 0 x2 x4 + 6 120 x76 x8 + + 0[x]11 5040 362880 ! sinc( ) = 2 sin(x) = cosc(x) x2 (2) sinc(x) = (3) sinca (x) = 2 ; sinc(x) ; x sin( x) x sin( x a x a ) It is a special function because its primitive, called sine integral and denoted Si(x) Centric circular cardinal Modi…ed centric circular sine functions cardinal sine functions Figure 1: The graphs of centric circular functions cardinal sine, in 2D, as known in literature Cardinal functions and integral functions 29 Figure 2: Cardinal sine function in 3D Mexican hat (sombrero) (4) Si(x) = x Z 0 = x = x = +1 X n=0 Z x sin(t) sinc(t) dt dt = t 0 x3 x5 x7 x9 + + + 0[x]11 18 600 35280 3265920 x8 x5 x7 + + ::: 3:3! 5:5! 7:7! ( 1)n x2n ; 8x 2 R (2n + 1)2 (2n)! can not be expressed exactly by elementary functions, but only by expansion of power series, as shown in equation (4). Therefore, its derivative is 0 (5) 8x 2 R; Si (x) = d(Si(x)) sin(x) = = sinc(x); dx x an integral sine function Si(x), that satis…es the di¤erential equation 000 (6) 00 0 x f (x) + 2f (x) + x f (x) = 0 ! f (x) = Si(x): The Gibbs phenomenon appears at the approximation of the square with a continuous and di¤erentiable Fourier series (Figure 3 right I). This operation could be substitute with the circular eccentric supermathematics functions (CE-SMF), because the eccentric derivative function of eccentric variable can express exactly this rectangular function (Figure 3 N top) or square (Figure 3 H below) as shown on their graphs (Figure 3 J left). 1 cos p fx; x =2 ; 1 sin(x =2)2 ; 2:01 g 1 2 P 4x Si nc[2 (2k 1)x]; fk; ngfn; 5gfx; 0; 1g 30 M. Şelariu, F. Smarandache and M. Niţu Gibbs phenomenon for a square wave with n = 5 1 2 dex[( ; 2 ); S(1; 0)] and n = 10 Figure 3: Comparison between the square function, eccentric derivative and its approximation by Fourier serial expansion Integral sine function (4) can be approximated with su¢cient accuracy. The maximum di¤erence is less than 1%, except the area near the origin. By the CE-SMF eccentric amplitude of eccentric variable (7) F ( ) = 1:57 aex[ ; S(0:6; 0)]; as shown on the graph on Figure 5. R Sin x; fx; 20; 20g R Sin (x + Iy) fx; 20; 20g; fy; 3; 3g Figure 4: The graph of integral sine function Si(x) N compared with the graph CE-SMF Eccentric amplitude 1; 57aex[ ; S(0; 6; 0)] of eccentric variable H Cardinal functions and integral functions 31 Figure 5: The di¤erence between integral sine and CE-SMF eccentric amplitude F ( ) = 1; 5aex[ ; S(0; 6; 0] of eccentric variable 2. ECCENTRIC CIRCULAR SUPERMATHEMATICS CARDINAL FUNCTIONS, CARDINAL ECCENTRIC SINE (ECC-SMF) Like all other supermathematics functions (SMF),they may be eccentric (ECC-SMF), elevated (ELC-SMF) and exotic (CEX-SMF), of eccentric variable , of centric variable 1;2 of main determination, of index 1, or secondary determination of index 2. At the passage from centric circular domain to the eccentric one, by positioning of the eccenter S(s; ") in any point in the plane of the unit circle, all supermathematics functions multiply from one to in…nity. It means that in CM there exists each unique function for a certain type. In EM there are in…nitely many such functions, and for s = 0 one will get the centric function. In other words, any supermathematics function contains both the eccentric and the centric ones. Notations sexc(x) and respectively, Sexc(x) are not standard in the literature and thus will be de…ned in three variants by the relations: (8) sexc(x) = of eccentric variable (8’) sex[ ; S(s; s)] sex(x) = x and Sexc(x) = Sex(x) Sex[ ; S(s; s)] = x of eccentric variable . sex( x) ; x of eccentric variable , noted also by sexc (x) and (9) sexc(x) = sex( x) Sex[ ; S(s; s)] = ; x of eccentric variable , noted also by Sexc (x) (9’) (10) Sexc(x) = sexca (x) = sex x a x a = sex ; 32 M. Şelariu, F. Smarandache and M. Niţu of eccentric variable , with the graphs from Figure 6 and Figure 7. Sex aa Sex ax = (10’) Sexa (x) = x a a Sin ArcSin [s Sin ( )] fs; 0; +1g; f ; ; 4 g a ArcSin [s Sin fs; 1; 0g; f ; ArcSin [s Sin fs; 0; 1g; f ; Sin ( Sin ( )] ; g )] ; g Figure 6: The ECCC-SMF graphs sexc1 [ ; S(s; ")] of eccentric variable Sin +ArcSin [s Sin ( )] fs; 0; 1g; f ; 4 ; 4 g ArcSin [0.1s Sin ( )] Sin fs; 10; 0g; f ; ; g ArcSin [0.1s Sin fs; 0:1; 0g; f ; Sin ( )] ; g Figure 7: Graphs ECCC-SMF sexc2 [ ; S(s; ")]; eccentric variable Cardinal functions and integral functions 33 3. ECCENTRIC CIRCULAR SUPERMATHEMATICS FUNCTIONS CARDINAL ELEVATED SINE AND COSINE (ECC- SMF-CEL) Supermathematical elevated circular functions (ELC-SMF), elevated sine sel( ) and elevated cosine cel( ), is the projection of the fazor/vector ~r = rex( ) rad( ) = rex[ ; S(s; ")] rad( ) on the two coordinate axis XS and YS respectively, with the origin in the eccenter S(s; "), the axis parallel with the axis x and y which originate in O(0; 0). If the eccentric cosine and sine are the coordinates of the point W (x; y), by the origin O(0; 0) of the intersection of the straight line d = d + [ d^ ae\, revolving around the point S(s; "), the elevated cosine and sine are the same coordinates to the eccenter S(s; "); ie, considering the origin of the coordinate straight rectangular axes XSY /as landmark in S(s; "). Therefore, the relations between these functions are as follows: (11) x = cex( ) = X + s cos(") = cel( ) + s cos(") y = Y + s sin(") = sex( ) = sel( ) + s sin(") Thus, for " = 0, ie S eccenter S located on the axis x > 0; sex( ) = sel( ), and for " = 2 ; cex( ) = cel( ), as shown on Figure 8. On Figure 8 were represented simultaneously the elevated cel( ) and the sel( ) graphics functions, but also graphs of cex( ) functions, respectively, for comparison and revealing sex( ) elevation Eccentricity of the functions is the same, of s = 0:4, with the attached drawing and sel( ) are " = 2 , and cel( ) has " = 0. Figure 8: Comparison between elevated supermathematics function and eccentric functions 34 M. Şelariu, F. Smarandache and M. Niţu Figure 9: Elevated supermathematics function and cardinal eccentric functions celc(x) J and selc(x) I of s = 0:4 Figure 10: Cardinal eccentric elevated supermathematics function celc(x) J and selc(x) I Elevate functions (11) divided by become cosine functions and cardinal elevated sine, denoted celc( ) = [ ; S] and selc( ) = [ ; S], given by the equations 8 > < X = celc( ) = celc[ ; S(s; ")] = cexc( ) s cos(s) (12) > : Y = selc( ) = selc[ ; S(s; ")] = sexc( ) s sin(s) with the graphs on Figure 9 and Figure 10. Cardinal functions and integral functions 35 4. NEW SUPERMATHEMATICS CARDINAL ECCENTRIC CIRCULAR FUNCTIONS (ECCC-SMF) The functions that will be introduced in this section are unknown in mathematics literature. These functions are centrics and cardinal functions or integrals. They are supermathematics eccentric functions amplitude, beta, radial, eccentric derivative of eccentric variable [1], [2], [3], [4], [6], [7] cardinals and cardinal cvadrilobe functions [5]. Eccentric amplitude function cardinal aex( ), denoted as (x) = aex[ ; S(s; ")]; x ; is expressed in (13) aexc( ) = aex( = aex[ ; S(s; s)] = arcsin[s sin( s] and the graphs from Figure 11. Sin ( ) ; fs; 0; 1g; f ; 4 ; +4 g Figure 11: The graph of cardinal eccentric circular supermathematics function aexc( ) The beta cardinal eccentric function will be bex[ S(s; s)] arcsin[s sin( bex( ) = = (14) bexc( ) = s)] ; with the graphs from Figure 12. Figure 12: The graph of cardinal eccentric circular supermathematics function bexc( ) (fs; 1; 1g; f ; 4 ; 4 g) 36 M. Şelariu, F. Smarandache and M. Niţu The cardinal eccentric function of eccentric variable (15) rexc1;2 ( ) = = rex[ ; S(s; s)] rex( ) s cos( = is expressed by p s) 1 s2 sin( s) and the graphs from Figure 13, and the same function, but of centric variable is expressed by (16) Rexc( = Rex[ 1;2 S(s; s)] 1;2 sCos( )+ p 1 [s Si n( )]2 fs; 0; 1g; f ; 4 ; 4 g ; 1;2 ) = = p Rex( 1;2 ) 1;2 1 + s2 sCos( ) 2s cos( 1;2 s) 1;2 p 1 [s Si n( )]2 ; fs; 1; 0g; f ; 4 ; 4 g sCos( ) p 1 [s Si n( )]2 fs; 0; 1g; f ; 4 ; 4 g Figure 13: The graph of cardinal eccentric circular supermathematical function rexc1;2 ( ) And the graphs for Rexc( 1), from Figure 14. Figure 14: The graph of cardinal eccentric radial circular supermathematics function Rex c( ) An eccentric circular supermathematics function with large applications, representing the function of transmitting speeds and/or the turning speeds of all known planar mechanisms is the derived eccentric dex1;2 ( ) and Dex( 1;2 ), functions that by dividing/reporting with arguments and, respectively, ; Cardinal functions and integral functions 37 lead to corresponding cardinal functions, denoted dexc1;2 ( ), respectively Dexc( 1;2 ) and expressions (17) dexc1;2 ( ) = dex1;2 ( ) = dex1;2 [ ; S(s; s)] = 1 s cos( ") 1 s2 sin2 ( ") (18) Dexc( 1;2 ) = Dex( 1;2 ) = Dexf [ 1;2 1;2 S(s; s)]g = 1;2 p 1 + s2 2s cos( 1;2 1;2 the graphs on Figure 15. Figure 15: The graph of supermathematical cardinal eccentric radial circular function dex c1 ( ) 1 1 Because Dex( 1;2 ) = dex1;2 ( ) results that Dexc( 1;2 ) = dexc1;2 ( ) siq( ) and coq( ) are also cvadrilobe functions, dividing by their arguments lead to cardinal cvadrilobe functions siqc( ) and coqc( ) obtaining with the expressions coq[ S(s; s)] cos( s) coq( ) = = p (19) coqc( ) = 2 2 1 s sin ( s) (20) siqc( ) = siq( ) the graphs on Figure 16. = siq[ S(s; s)] sin( = p 1 s) s2 cos2 ( s) Figure 16: The graph of supermathematics cardinal cvadrilobe function ceqc( ) J and siqc( ) I s) 38 M. Şelariu, F. Smarandache and M. Niţu It is known that, by de…nite integrating of cardinal centric and eccentric functions in the …eld of supermathematics, we obtain the corresponding integral functions. Such integral supermathematics functions are presented below. For zero eccentricity, they degenerate into the centric integral functions. Otherwise they belong to the new eccentric mathematics. 5. ECCENTRIC SINE INTEGRAL FUNCTIONS Are obtained by integrating eccentric cardinal sine functions (13) and are Z x sexc( ) d (21) sie(x) = 0 with the graphs on Figure 17 for the ones with the eccentric variable x . Figure 17: The graph of eccentric integral sine function sie 1 (x)N and sie 2 (x)H Unlike the corresponding centric functions, which is denoted Sie(x), the eccentric integral sine of eccentric variable was noted sie(x), without the capital S, which will be assigned according to the convention only for the ECCC-SMF of centric variable. The eccentric integral sine function of centric variable, noted Sie(x) is obtained by integrating the cardinal eccentric sine of the eccentric circular supermathematics function, with centric variable (22) Sexc(x) = Sexc[ ; S(s; ")]; Cardinal functions and integral functions 39 thus (23) Sie(x) = Z x Sex[ ; S(s; "] ; 0 with the graphs from Figure 18. Figure 18: The graph of eccentric integral sine function sie 2 (x) 6. C O N C L U S I O N The paper highlighted the possibility of inde…nite multiplication of cardinal and integral functions from the centric mathematics domain in the eccentric mathematics’s or of supermathematics’s which is a reunion of the two mathematics. Supermathematics, cardinal and integral functions were also introduced with correspondences in centric mathematics, a series new cardinal functions that have no corresponding centric mathematics. The applications of the new supermathematics cardinal and eccentric functions certainly will not leave themselves too much expected. References [1] Şelariu, M. E., Eccentric circular functions, Com. I Conferinţa Naţional¼ a de Vibraţii în Construcţia de Maşini, Timişoara , 1978, 101-108. [2] Şelariu, M. E., Eccentric circular functions and their extension, Bul .Şt. Tehn. al I.P.T., Seria Mecanic¼ a, 25(1980), 189-196. [3] Şelariu, M. E., Supermathematica, Com.VII Conf. Internaţional¼ a. De Ing. Manag. şi Tehn.,TEHNO’95 Timişoara, 9(1995), 41-64. [4] Şelariu, M. E., Eccentric circular supermathematic functions of centric variable, Com.VII Conf. Internaţional¼ a. De Ing. Manag. şi Tehn.,TEHNO’98 Timişoara, 531548. [5] Şelariu, M. E., Quadrilobic vibration systems, The 11th International Conference on Vibration Engineering, Timişoara, Sept. 27-30, 2005, 77-82. [6] Şelariu, M. E., Supermathematica. Fundaments, Vol.II, Ed. Politehnica, Timişoara, 2007. [7] Şelariu, M. E., Supermathematica. Fundaments, Vol.II, Ed. Politehnica, Timişoara, 2011 (forthcoming). 40 M. Şelariu, F. Smarandache and M. Niţu Received: December, 2011 POLYTEHNIC UNIVERSITY OF TIMIŞOARA, ROMANIA E-mail address: mselariu@gmail.com UNIVERSITY OF NEW MEXICO-GALLUP, USA E-mail address: smarand@unm.edu INSTITUTUL NAŢIONAL DE CERCETARE - DEZVOLTARE ¼ PENTRU ELECTROCHIMIE ŞI MATERIE CONDENSATA TIMIŞOARA, ROMANIA E-mail address: nitumarian13@gmail.com View publication stats