Graphs of q-exponentials and q-trigonometric functions

Transcription

1 Grahs of -otals ad -trgoomtrc fuctos Amla Carola Saravga To ct ths vrso: Amla Carola Saravga. Grahs of -otals ad -trgoomtrc fuctos. 26. <hal > HAL Id: hal htts://hal.archvs-ouvrts.fr/hal Submttd o 6 Oct 26 HAL s a mult-dsclary o accss archv for th dost ad dssmato of sctfc rsarch documts, whthr thy ar ublshd or ot. Th documts may com from tachg ad rsarch sttutos Frac or abroad, or from ublc or rvat rsarch ctrs. L archv ouvrt lurdsclar HAL, st dsté au déôt t à la dffuso d documts sctfus d vau rchrch, ublés ou o, émaat ds établssmts d sgmt t d rchrch fraças ou étragrs, ds laborators ublcs ou rvés.

2 Grahs of -otals ad -trgoomtrc fuctos Amla Carola Saravga Poltcco d Toro Abstract Hr w wll show th bhavor of som of -fuctos. I artcular w lot th - otal ad th -trgoomtrc fuctos. Sc ths fuctos ar ot grally cludd as softwar routs, a Fortra rogram was cssary to gv thm. Kywords -calculus, -otal, -trgoomtrc fuctos, Tsalls -fuctos. Itroducto Frak Hlto Jackso (87-96) was a Eglsh clrgyma ad mathmatca who systmatcally studd what today s kow as th -calculus. I artcular, h studd som -fuctos ad th -aalogs of drvatv ad tgral []. Actually, Jackso startd hs studs troducg th -dffrc orator, a orator th orgs of whch ca b trackd back to Eduard H ad Lohard Eulr [2]. For ths raso, th -dffrc orator s also kow as th Eulr-H-Jackso orator [3]. Also Carl Frdrch Gauss was volvd th -calculus sc h roosd som rlatos such as th -aalog of bomals [4]. As told [5], th -calculus has varous dalcts, ad for ths raso t s kow as uatum calculus, tm-scal calculus or calculus of arttos [5]. Hr w wll us th otato gv th book by Kac ad Chug [6]. Ths book s dscussg two modfd vrsos of a uatum calculus, dfd as th h-calculus ad th -calculus. Th lttr h dcats th Plack's costat ad th lttr stads for uatum. Lt us ot that ths two vrsos of calculus rduc to Nwto's calculus th lmt whr h or. Hr, th framwork of th -calculus of [6], w wll show th bhavors of som of th - fuctos. I artcular w wll lot th -otal ad of th -trgoomtrc fuctos. W wll s that, th lmt whr, ths fuctos bcom th fuctos commoly usd. Sc ths -fuctos ar ot grally cludd as softwar routs, w rard a Fortra rogram for gvg thm. Bfor showg th rsults, lt us dscuss shortly th -dffrc orator ad th rlatd -drvatv. Th uatum dffrc orator. I th -calculus, th -dffrc s smly gv by: d From ths dffrc, th -drvatv s: D f f f ( ) f ( ) f ( ) f ( ) Th -drvatv rducs to th Nwto s drvatv as. W hav also th h-drvatv: f ( + h) f ( ) D h f h I th lmt as h, ths rducs to th usual drvatv.

3 Takg h ( ), w may s that: f ( + h) f ( ) f ( + ( ) ) f ( ) f ( ) f ( ) Dh f D h From th two formulas for chags wh a uatty h s addd to, whras varabl s multld by a factor. Lt us cosdr th fucto D f ad D h f, w s that D h f cocrs how th fucto f () D f cosdrs how t chags wh th f ( ). If w calculat ts -drvatv, w obta: f () D ( ) Comarg th ordary calculus, gvg, to Euato (), w ca df th tgr [ ] by: ( )' Thrfor Euato () turs out to b: 2 [ ] D [ ] As a cosuc, a -th -drvatv of -drvatv, grats th -factoral: f ( ), whch s obtad by rrtg tms th [ ]! [ ][ ]...[3][2][] Form th -factorals, w ca df -bomal coffcts: [ ]! [ m]![ m]! Ths mas that w ca us th usual Taylor formula, rlacg th drvatvs by th - drvatvs ad th factorals by -factorals. Th -otals As gv [6], th -aalog of th otal fucto s dfd as [ ]! Sc w hav that [6]:

4 ( )( ) L( ) ( ) [ ]! Th -otal has th followg rsso too: ( ) ( )( ) L( ) Aothr -aalog of th otal fucto s [6]: E L A rorty of th -otal fuctos s y + y f y y Du to ths commutato rlato, ad y ar ot symmtrc ad thrfor: y Wh, th -otals bcom th usual otal. W ca s ths th followg Fgur. Th -trgoomtrc fuctos As gv [6], th -aalog of th trgoomtrc fuctos ar dfd as: s 2 ; cos y + 2 S E E 2 ; Cos E + E 2 W hav that: (2) cos Cos + s S. I th Fgur 2, w ca s th grahs of th -trgoomtrc fuctos for two dffrt valus of. Wh, ths fuctos bcom th usual os. To chck th calculato, th Fgur 2, t s show also th valu of cos Cos + s S. It s ual to, as t must b bcaus gv by (2). From th lots, w s that th bhavor of th -trgoomtrc

5 fuctos bcom mor dffrt from that of th trgoomtrc fuctos, wh th valu of s larg. Lt us also ot that wh s clos to, th -fuctos ar clos to cos ad s. Othr -fuctos It s cssary to ot that othr -fuctos st, whch ar lkd to th formulato of th troy gv by Costato Tsalls. I 948 [7], Claud Shao dfd th troy S of a dscrt radom varabl Ξ as th ctd valu of th formato cott: S I log [8]. I ths rsso, I s th formato cott of Ξ, th b robablty of -vt s ad b s th bas of th usd logarthm. Commo valus of th bas ar 2, Eulr s umbr, ad. Tsalls gralzd th Shao troy th followg mar [9]: S Lt us ot that, f w cosdr th -drvatv of th uatty, w hav: D Thrfor, wh : lm D lm S Th, th Tsalls troy s lkd to th -drvatv. But, th framwork of th Tsalls aroach to statstcs, w hav a dformato of th otal fucto, th Tsalls - otal fucto, gv by []: (3) ( ) /( ) [ + ( ) ] ( ) f f ad + ( ) > Ths s dffrt from th -otal rvously dscussd. If th fucto (3) s add Taylor srs, w hav []: (4) ( ) + Q! I (4), w hav ( ) ( 2 )( 3 2) [ ( ) ] Q L. From th rsso (4) of th Tsalls -otal, gv for coml, w ca obta th Tsalls trgoomtrc fuctos: cos + j ( ) j Q (2 j)! 2 j 2 j ; s j ( ) j Q 2 j (2 j+ )! 2 j+

6 Rfrcs. H. Jackso (98). O -fuctos ad a crta dffrc orators, Tras. Roy. Soc. Ed., T. Erst (22). A Comrhsv Tratmt of -Calculus, Srgr Scc & Busss Mda. 3. M. H. Aaby, Z. S. Masour (22). -Fractoal Calculus ad Euatos, Srgr. 4. Raja Roy (2). Sourcs th Dvlomt of Mathmatcs: Srs ad Products from th Fftth to th Twty-frst Ctury, Cambrdg Uvrsty Prss. 5. T. Erst (28). Th dffrt togus of -calculus. Procdgs of th Estoa Acadmy of Sccs, 28, 57, 2, 8 99 DOI:.376/roc V. Kac, Pokma Chug (22). Quatum Calculus, Srgr, Brl. 7. C. E. Shao (948). A Mathmatcal Thory of Commucato. Bll Systm Tchcal Joural 2 (3): DOI:.2/j tb M. Borda (2). Fudamtals Iformato Thory ad Codg. Srgr. ISBN C. Tsalls (988). Possbl Gralzato of Boltzma-Gbbs Statstcs, Joural of Statstcal Physcs, 52: DOI:.7/BF6429. S. Umarov, C. Tsalls, S. Stbrg (28). O a -Ctral Lmt Thorm Cosstt wth Notsv Statstcal Mchacs. Mla J. Math. Brkhausr Vrlag. 76: do:.7/s y. E. P. Borgs (998). O a -gralzato of crcular ad hyrbolc fuctos. J. Phys. A: Math. G

7 Fgur : Bhavor of,, E for dffrt valus of.

8 Fgur 2: -trgoomtrc fuctos for two dffrt valus of.