CARDINAL FUNCTIONS AND INTEGRAL FUNCTIONS. MIRCEA E. ŞELARIU, FLORENTIN SMARANDACHE and MARIAN NIŢU

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1 INTERNATIONAL JOURNAL OF GEOMETRY Vol. 1 (2012), No. 1, CARDINAL FUNCTIONS AND INTEGRAL FUNCTIONS MIRCEA E. ŞELARIU, FLORENTIN SMARANDACHE nd MARIAN NIŢU Abstrct. This pper presents the correspondences of the eccentric mthemtics of crdinl nd integrl functions nd centric mthemtics, or ordinry mthemtics. Centric functions will lso be presented in the introductory section, becuse they re, lthough widely used in undultory physics, little known. In centric mthemtics, crdinl sine nd cosine re de ned s well s the integrls. Both circulr nd hyperbolic ones. In eccentric mthemtics, ll these centrl functions multiplies from one to in nity, due to the in nity of possible choices where to plce point. This point is clled eccenter S(s; ") which lies in the plne of unit circle UC(O; R 1) or of the equilterl unity hyperbol HU(O; 1; b 1). Additionlly, in eccentric mthemtics there re series of other importnt specil functions, s e, be, de, re, etc. If we divide them by the rgument, they cn lso become crdinl eccentric circulr functions, whose primitives utomticlly become integrl eccentric circulr functions. All supermtemtics eccentric circulr functions (SFM-EC) cn be of vrible ecentric, which re continuous functions in liner numericl eccentricity domin s 2 [ 1; 1], or of centric vrible, which re continuous for ny vlue of s. This mens tht s 2 [ 1; +1]. Keywords nd phrses: C-Circulr, CC- C centric, CE- C Eccentric, CEL-C Elevted, CEX-C Eotic, F-Function, FMC-F Centric Mthemtics, M- Mtemthics, MC-M Centric, ME-M Ecentric, S-Super, SM- S Mtemtics, FSM-F Supermtemtics FSM-CE- FSM Eccentric Circulrs, FSM-CEL- FSM-C Elevted, FSM-CEC- FSM-CE- Crdinls, FSM-CELC- FSM-CEL Crdinls (2010) Mthemtics Subject Clssi ction: 32A17

2 62 M. Selriu nd F. Smrndche nd M. Nitu 1. INTRODUCTION: CENTRIC CARDINAL SINE FUNCTION According to ny stndrd dictionry, the word "crdinl" is synonymous with "principl", "essentil", "fundmentl". In centric mthemtics (CM), or ordinry mthemtics, crdinl is, on the one hnd, number equl to number of nite ggregte, clled the power of the ggregte, nd on the other hnd, known s the sine crdinl sinc() or cosine crdinl cosc(), is specil function de ned by the centric circulr function (CCF). sin() nd cos() re commonly used in undultory physics (see Figure 1) nd whose grph, the grph of crdinl sine, which is clled s "Meicn ht" (sombrero) becuse of its shpe (see Figure 2). Note tht sinc() crdinl sine function is given in the specility literture, in three vrints (1) sinc() d(sinc()) d ( 1; for 0 sin() ; for 2 [ 1; +1] n 0 sin() 1 +1X n0 cos() ( 1) n 2n (2n + 1)!! sinc( 2 ) 2 ; sin() 2 cosc() []11 sinc() ; (2) sinc() sin() (3) sinc () sin( ) It is specil function becuse its primitive, clled sine integrl nd denoted Si() Centric circulr crdinl Modi ed centric circulr sine functions crdinl sine functions Figure 1: The grphs of centric circulr functions crdinl sine, in 2D, s known in literture

3 Crdinl functions nd integrl functions 63 Figure 2: Crdinl sine function in 3D Meicn ht (sombrero) (4) Si() Z 0 +1X n0 sin(t) dt t 3 Z 0 7 sinc(t) dt []11 8 3:3! :5! 7:7! + : : : ( 1) n 2n (2n + 1) 2 (2n)! ; 8 2 R cn not be epressed ectly by elementry functions, but only by epnsion of power series, s shown in eqution (4). Therefore, its derivtive is (5) 8 2 R; Si 0 () d(si()) d sin() sinc(); n integrl sine function Si(), tht stis es the di erentil eqution (6) f 000 () + 2f 00 () + f 0 () 0! f() Si(): The Gibbs phenomenon ppers t the pproimtion of the squre with continuous nd di erentible Fourier series (Figure 3 right I). This opertion could be substitute with the circulr eccentric supermthemtics functions (CE-SMF), becuse the eccentric derivtive function of eccentric vrible cn epress ectly this rectngulr function (Figure 3 N top) or squre (Figure 3 H below) s shown on their grphs (Figure 3 J left). 1 cos f; 2 p 1 sin( 2) 2 ; ; 2:01g P Si nc[2(2k 1)]; fk; ngfn; 5gf; 0; 1g

4 64 M. Selriu nd F. Smrndche nd M. Nitu 1 2 de[(; 2 ); S(1; 0)] Gibbs phenomenon for squre wve with n 5 nd n 10 Figure 3: Comprison between the squre function, eccentric derivtive nd its pproimtion by Fourier seril epnsion Integrl sine function (4) cn be pproimted with su cient ccurcy. The mimum di erence is less thn 1%, ecept the re ner the origin. By the CE-SMF eccentric mplitude of eccentric vrible (7) F () 1:57 e[; S(0:6; 0)]; s shown on the grph on Figure 5. Sin R ; f; 20; 20g Sin R ( + Iy) f; 20; 20g; fy; 3; 3g Figure 4: The grph of integrl sine function Si() N compred with the grph CE-SMF Eccentric mplitude 1; 57e[; S(0; 6; 0)] of eccentric vrible H

5 Crdinl functions nd integrl functions 65 Figure 5: The di erence between integrl sine nd CE-SMF eccentric mplitude F () 1; 5e[; S(0; 6; 0] of eccentric vrible 2. ECCENTRIC CIRCULAR SUPERMATHEMATICS CARDINAL FUNCTIONS, CARDINAL ECCENTRIC SINE (ECC-SMF) Like ll other supermthemtics functions (SMF),they my be eccentric (ECC-SMF), elevted (ELC-SMF) nd eotic (CEX-SMF), of eccentric vrible, of centric vrible 1;2 of min determintion, of inde 1, or secondry determintion of inde 2. At the pssge from centric circulr domin to the eccentric one, by positioning of the eccenter S(s; ") in ny point in the plne of the unit circle, ll supermthemtics functions multiply from one to in nity. It mens tht in CM there eists ech unique function for certin type. In EM there re in nitely mny such functions, nd for s 0 one will get the centric function. In other words, ny supermthemtics function contins both the eccentric nd the centric ones. Nottions sec() nd respectively, Sec() re not stndrd in the literture nd thus will be de ned in three vrints by the reltions: se[; S(s; s)] (8) sec() se() of eccentric vrible nd (8 ) Sec() Se() of eccentric vrible. (9) sec() se() ; of eccentric vrible, noted lso by sec () nd Se[; S(s; s)] (9 ) Sec() se() Se[; S(s; s)] ; of eccentric vrible, noted lso by Sec () (10) sec () se se ;

6 66 M. Selriu nd F. Smrndche nd M. Nitu of eccentric vrible, with the grphs from Figure 6 nd Figure 7. (10 ) Se () Se Se Sin ArcSin[s Sin()] fs; 0; +1g; f; ; 4g ArcSin[s Sin()] Sin fs; 1; 0g; f; ; g Sin fs; 0; 1g; f; ArcSin[s Sin()] ; g Figure 6: The ECCC-SMF grphs sec 1 [; S(s; ")] of eccentric vrible +ArcSin[s Sin()] Sin fs; 0; 1g; f; 4; 4g ArcSin[0.1s Sin()] Sin fs; 10; 0g; f; ; g ArcSin[0.1s Sin()] Sin fs; 0:1; 0g; f; ; g Figure 7: Grphs ECCC-SMF sec 2 [; S(s; ")]; eccentric vrible

7 Crdinl functions nd integrl functions ECCENTRIC CIRCULAR SUPERMATHEMATICS FUNCTIONS CARDINAL ELEVATED SINE AND COSINE (ECC- SMF-CEL) Supermthemticl elevted circulr functions (ELC-SMF), elevted sine sel() nd elevted cosine cel(), is the projection of the fzor/vector ~r re() rd() re[; S(s; ")] rd() on the two coordinte is X S nd Y S respectively, with the origin in the eccenter S(s; "), the is prllel with the is nd y which originte in O(0; 0). If the eccentric cosine nd sine re the coordintes of the point W (; y), by the origin O(0; 0) of the intersection of the stright line d d + [ d^e\, revolving round the point S(s; "), the elevted cosine nd sine re the sme coordintes to the eccenter S(s; "); ie, considering the origin of the coordinte stright rectngulr es XSY /s lndmrk in S(s; "). Therefore, the reltions between these functions re s follows: (11) ce() X + s cos(") cel() + s cos(") y Y + s sin(") se() sel() + s sin(") Thus, for " 0, ie S eccenter S locted on the is > 0; se() sel(), nd for " 2 ; ce() cel(), s shown on Figure 8. On Figure 8 were represented simultneously the elevted cel() nd the sel() grphics functions, but lso grphs of ce() functions, respectively, for comprison nd reveling se() elevtion Eccentricity of the functions is the sme, of s 0:4, with the ttched drwing nd sel() re " 2, nd cel() hs " 0. Figure 8: Comprison between elevted supermthemtics function nd eccentric functions

8 68 M. Selriu nd F. Smrndche nd M. Nitu Figure 9: Elevted supermthemtics function nd crdinl eccentric functions celc() J nd selc() I of s 0:4 Figure 10: Crdinl eccentric elevted supermthemtics function celc() J nd selc() I Elevte functions (11) divided by become cosine functions nd crdinl elevted sine, denoted celc() [; S] nd selc() [; S], given by the equtions 8 >< s cos(s) X celc() celc[; S(s; ")] cec() (12) >: s sin(s) Y selc() selc[; S(s; ")] sec() with the grphs on Figure 9 nd Figure 10.

9 Crdinl functions nd integrl functions NEW SUPERMATHEMATICS CARDINAL ECCENTRIC CIRCULAR FUNCTIONS (ECCC-SMF) The functions tht will be introduced in this section re unknown in mthemtics literture. These functions re centrics nd crdinl functions or integrls. They re supermthemtics eccentric functions mplitude, bet, rdil, eccentric derivtive of eccentric vrible [1], [2], [3], [4], [6], [7] crdinls nd crdinl cvdrilobe functions [5]. Eccentric mplitude function crdinl e(), denoted s is epressed in () e[; S(s; ")]; ; (13) ec() e( e[; S(s; s)] nd the grphs from Figure 11. rcsin[s sin( s] Sin() ; fs; 0; 1g; f; 4; +4g Figure 11: The grph of crdinl eccentric circulr supermthemtics function ec() The bet crdinl eccentric function will be (14) bec() be() be[s(s; s)] rcsin[s sin( s)] ; with the grphs from Figure 12. Figure 12: The grph of crdinl eccentric circulr supermthemtics function bec() (fs; 1; 1g; f; 4; 4g)

10 70 M. Selriu nd F. Smrndche nd M. Nitu The crdinl eccentric function of eccentric vrible is epressed by (15) rec 1;2 () re() re[; S(s; s)] s cos( s) p 1 s 2 sin( s) nd the grphs from Figure 13, nd the sme function, but of centric vrible is epressed by (16) Rec( 1;2 ) Re( 1;2) Re[ 1;2S(s; s)] 1;2 1;2 p 1 + s 2 2s cos( 1;2 s) 1;2 scos()+ p 1 [s Si n()] 2 ; fs; 0; 1g; f; 4; 4g p scos() 1 [s Si n()] 2 ; fs; 1; 0g; f; 4; 4g scos() p 1 [s Si n()] 2 ; fs; 0; 1g; f; 4; 4g Figure 13: The grph of crdinl eccentric circulr supermthemticl function rec 1;2 () And the grphs for Rec(1), from Figure 14. Figure 14: The grph of crdinl eccentric rdil circulr supermthemtics function Re c() An eccentric circulr supermthemtics function with lrge pplictions, representing the function of trnsmitting speeds nd/or the turning speeds of ll known plnr mechnisms is the derived eccentric de 1;2 () nd De( 1;2 ), functions tht by dividing/reporting with rguments nd, respectively,

11 Crdinl functions nd integrl functions 71 led to corresponding crdinl functions, denoted dec 1;2 (), respectively Dec( 1;2 ) nd epressions (17) dec 1;2 () de 1;2() (18) Dec( 1;2 ) De( 1;2) 1;2 the grphs on Figure 15. de 1;2[; S(s; s)] 1 scos( ") 1 s 2 sin 2 ( ") Def[ p 1;2S(s; s)]g 1 + s 2 2s cos( 1;2 s) 1;2 1;2 Figure 15: The grph of supermthemticl crdinl eccentric rdil circulr function dec 1 () 1 Becuse De( 1;2 ) de 1;2 () results tht Dec( 1 1;2) dec 1;2 () siq() nd coq() re lso cvdrilobe functions, dividing by their rguments led to crdinl cvdrilobe functions siqc() nd coqc() obtining with the epressions (19) coqc() coq() coq[s(s; s)] cos( s) p 1 s 2 sin 2 ( s) (20) siqc() siq() the grphs on Figure 16. siq[s(s; s)] sin( s) p 1 s 2 cos 2 ( s) Figure 16: The grph of supermthemtics crdinl cvdrilobe function ceqc() J nd siqc() I

12 72 M. Selriu nd F. Smrndche nd M. Nitu It is known tht, by de nite integrting of crdinl centric nd eccentric functions in the eld of supermthemtics, we obtin the corresponding integrl functions. Such integrl supermthemtics functions re presented below. For zero eccentricity, they degenerte into the centric integrl functions. Otherwise they belong to the new eccentric mthemtics. 5. ECCENTRIC SINE INTEGRAL FUNCTIONS Are obtined by integrting eccentric crdinl sine functions (13) nd re (21) sie() Z 0 sec() d with the grphs on Figure 17 for the ones with the eccentric vrible. Figure 17: The grph of eccentric integrl sine function sie 1 ()N nd sie 2 ()H Unlike the corresponding centric functions, which is denoted Sie(), the eccentric integrl sine of eccentric vrible ws noted sie(), without the cpitl S, which will be ssigned ccording to the convention only for the ECCC-SMF of centric vrible. The eccentric integrl sine function of centric vrible, noted Sie() is obtined by integrting the crdinl eccentric sine of the eccentric circulr supermthemtics function, with centric vrible (22) Sec() Sec[; S(s; ")];

13 thus (23) Sie() with the grphs from Figure 18. Crdinl functions nd integrl functions 73 Z 0 Se[; S(s; "] ; Figure 18: The grph of eccentric integrl sine function sie 2 () 6. C O N C L U S I O N The pper highlighted the possibility of inde nite multipliction of crdinl nd integrl functions from the centric mthemtics domin in the eccentric mthemtics s or of supermthemtics s which is reunion of the two mthemtics. Supermthemtics, crdinl nd integrl functions were lso introduced with correspondences in centric mthemtics, series new crdinl functions tht hve no corresponding centric mthemtics. The pplictions of the new supermthemtics crdinl nd eccentric functions certinly will not leve themselves too much epected. References [1] Şelriu, M. E., Funcţii circulre ecentrice, Com. I Conferinţ Nţionl¼ de Vibrţii în Construcţi de Mşini, Timişor, 1978, [2] Şelriu, M. E., Funcţii circulre ecentrice şi etensi lor, Bul.Şt. Tehn. l I.P.T., Seri Mecnic¼, 25(1980), [3] Şelriu, M. E., Supermtemtic, Com.VII Conf. Internţionl¼. De Ing. Mng. şi Tehn.,TEHNO 95 Timişor, 9(1995), [4] Şelriu, M. E., Funcţii supermtemtice circulre ecentrice de vribil¼ centric¼, Com.VII Conf. Internţionl¼. De Ing. Mng. şi Tehn.,TEHNO 98 Timişor, [5] Şelriu, Mirce Eugen, Qudrilobic vibrtion systems, The 11 th Interntionl Conference on Vibrtion Engineering, Timişor, Sept , 2005, [6] Şelriu, Mirce Eugen, Supermtemtic. Fundmente, Vol.II, Ed. Politehnic, Timişor, [7] Şelriu, Mirce Eugen, Supermtemtic. Fundmente, Vol.II, Ed. Politehnic, Timişor, 2011 (forthcoming).

14 74 M. Selriu nd F. Smrndche nd M. Nitu Received: December, 2011 POLYTEHNIC UNIVERSITY OF TIMIŞOARA, ROMANIA E-mil ddress: UNIVERSITY OF NEW MEXICO-GALLUP, USA E-mil ddress: INSTITUTUL NAŢIONAL DE CERCETARE - DEZVOLTARE PENTRU ELECTROCHIMIE ŞI MATERIE CONDENSAT ¼A TIMIŞOARA, ROMANIA E-mil ddress: nitumrin13@gmil.com