Mthemtis n Sttistis (): -, 0 DOI: 0.89/ms.0.000 http://www.hpub.og A Stuy of Some Integl Poblems Ug Mple Chii-Huei Yu Deptment of Mngement n Infomtion, Nn Jeon Univesity of Siene n Tehnology, Tinn City, 776, Tiwn *Coesponing Autho: hiihuei@mil.njt.eu.tw Copyight 0 Hoizon Reseh Publishing All ights eseve. Abstt This ppe tes the mthemtil softwe Mple s the uiliy tool to stuy fou types of integls. We n obtin the Fouie seies epnsions of these fou types of integls by ug integtion tem by tem theoem. On the othe hn, we povie two emples to o lultion ptilly. The eseh methos opte in this stuy involve fining solutions though mnul lultions n veifying these solutions by ug Mple. Keywos Integls, Fouie Seies Epnsions, Integtion Tem By Tem Theoem, Mple. Intoution In lulus n engineeing mthemtis ouses, we lent mny methos to solve the integl poblems inluing hnge of vibles metho, integtion by pts metho, ptil ftions metho, tigonometi substitution metho, n so on. In this ppe, we minly stuy the following fou types of integls whih e not esy to obtin thei nswes ug the methos mentione bove. [ os( osh[ () os[ os( h[ () os[ os( t + b)osh[ () [ os( t + b)h[ () Whee,, b e el numbes, 0. We n obtin the Fouie seies epnsions of these fou types of integls by ug integtion tem by tem theoem; these e the mjo esults of this ppe (i.e., Theoems, ). As fo the stuy of elte integl poblems n efe to [-]. On the othe hn, we popose some integls to o lultion ptilly. The eseh methos opte in this stuy involve fining solutions though mnul lultions n veifying these solutions by ug Mple. This type of eseh metho not only llows the isovey of lultion eos, but lso helps moify the oiginl ietions of thining fom mnul n Mple lultions. Fo this eson, Mple povies insights n guine eging poblem-solving methos.. Min Results Fistly, we intoue nottion n some fomuls use in this stuy... Nottion Let z + ib be omple numbe, whee i,, b e el numbes. We enote the el pt of z by Re( z), n b the imginy pt of z by Im(z)... Fomuls... Eule's fomul θ e i osθ + iθ, whee θ is ny el numbe.... DeMoive's fomul n (os θ + i θ ) osnθ + i nθ, whee n is ny intege, θ is ny el numbe.... ([6, p]) ( u + iv) uoshv + i osuhv, whee u, v e el numbes.... ([6, p]) os( u + iv) osu osh v i uh v, whee u, v e el numbes.... ([6, p6]) + z ( ) z 0 ( + )!, whee z is ny omple numbe...6. ([6, p6]) os z ( ) z, whee z is ny omple 0 ()! numbe.
A Stuy of Some Integl Poblems Ug Mple Net, we intoue n impotnt theoem use in this ppe... Integtion tem by tem theoem ([7, p69]) Suppose { g ( t) } 0 is sequene of Lebesgue integble funtions efine on n intevl [, ]. If g t ( ) 0 is onvegent, then g t ( ) g ( ) t. 0 0 The following is the fist mjo esult of this stuy, we obtin the Fouie seies epnsions of the integls () n ()... Theoem Assume,, b, e el numbes, 0. Then thee eists onstnt C suh tht fo ll R, the integl 0 ( ) [ os( osh[ + [( + )( + b)] ( + )!( + ) An thee eists onstnt C suh tht fo ll the integl 0... Poof Beuse ( ) (By Fomul..) () R, os[ os( h[ Re + os[( + )( + b)] ( + )!( + ) (6) 0 [ os( + b)]osh[ + b)] ( ) (Ug Fomul..) Re{[ epi( + b)]} [ epi( + b)] ( + )! + + Re ( ) epi[( + )( + b)] 0 ( + )! (By DeMoive's fomul) 0 ( ) + (By Eule's fomul) os[( + )( + b)] ( + )! (7) Thus, fo ll R, the integl [ os( osh[ + ( ) 0 ( + + os[( + )( )! ( ) os[( + )( ( + )! 0 (By integtion tem by tem theoem) 0 ( ) + [( + )( + b)] ( + )!( + ) Whee C is some onstnt. On the othe hn, ug Eule's fomul, DeMoive's fomul n Fomul..,.., we hve os[ os( + b)]h[ + b)] Im{[ epi( + b)]} 0 ( ) + [( + )( + b)] ( + )! Theefoe, by integtion tem by tem theoem, we n show tht thee eists onstnt C suh tht fo ll R, the integl 0 ( ).. Rem (8) os[ os( h[ + In Theoem, beuse fo eh 0 0 os[( + )( + b)] ( + )!( + ) q.e.. ( ) + R, os[( + )( ( + )! + ( + )!( + ) os[( + )( 0 + < (+ )!(+ ) It follows tht we n use integtion tem by tem theoem to show tht () hols. The sme eson tht we n pove (6) by ug integtion tem by tem theoem. Net, we etemine the Fouie seies epnsions of the integls () n ()..6. Theoem If the ssumptions e the sme s Theoem. Then thee
Mthemtis n Sttistis (): -, 0 eists onstnt C suh tht fo ll R, the integl os[ os( osh[ + ( ) [()( + b)] ()! An thee eists onstnt C suh tht fo ll the integl ( ).6.. Poof (9) R, [ os( h[ os[()( + b)] ()! (0) By Eule's fomul, DeMoive's fomul n Fomul..,..6, we hve + Thus, fo ll os[ os( + b)]osh[ + b)] R 0 ( ) ( ) Re{os[ epi( + b)]} os[()( + b)] ()! os[()( + b)] ()!, the integl () os[ os( osh[ + ( ) os[()( ()! (By integtion tem by tem theoem) + ( ) [()( + b)] ()! Whee C is some onstnt. Similly, by Eule's fomul, DeMoive's fomul n Fomul..,..6, we obtin [ os( + b)]h[ + b)] ( ) Im{os[ epi( + b)]} [()( + b)] ()! () By integtion tem by tem theoem, it follows tht fo ll R, the integl [ os( h[ ( ) Whee C is some onstnt. os[()( + b)] ()! q.e...7. Rem In Theoem, the eson tht we n use integtion tem by tem theoem to pove (9) n (0) is the sme s Rem.. Emples In the following, fo the fou types of integls in this stuy, we popose some integls n use Theoems, to etemine thei Fouie seies epnsions. In ition, we evlute some efinite integls n employ Mple to lulte the ppoimtions of these efinite integls n thei solutions fo veifying ou nswes... Emple In Theoem, ting,, b π / into (), we obtin the following integl π π os t + osh t + + ( ) ( ) 0 ( )!( ) + + + + π () Thus, we n etemine the efinite integl fom t π / to t π / 6, π π os t + osh t + π / 6 π / ( ) 0 ( ) 0 + ( + ) π ( + )!( + ) + ( + ) π () ( + )!( + ) We use Mple to veify the oetness of (). >evlf(int((*os(*t+pi/))*osh(*(*t+pi/)),t Pi/..Pi/6),); >evlf(/*sum((-)^*^(*+)/((*+)!*(*+))*s in((*+)*pi/),0..infinity)- /*sum((-)^*^(*+)/(( *+)!*(*+))*((*+)*Pi/),0..infinity),); The bove nswes obtine by Mple ppes I
A Stuy of Some Integl Poblems Ug Mple ( ), it is beuse Mple lultes by ug speil funtions built in. Both the imginy pts of the bove nswes e zeo, so n be ignoe. On the othe hn, in Theoem, ting,, b π / into (6), we hve the following integl os π os t h π t + ( ) ( ) os ( ) 0 ( )!( ) + + + Hene, we n etemine the efinite integl fom t π /0 to t π /, os π os t h π () π t π / π / 0 ( ) 0 + ( ) (6 + ) π os ( + )!( + ) + ( ) ( + ) π + ( ) os 0 ( + )!( + ) (6) Net, we use Mple to veify the oetness of (6). >evlf(int(os(sqt()*os(*t-*pi/))*h(sqt()*( *t-*pi/)),tpi/0..*pi/),); >evlf(-/*sum((-)^*sqt()^(*+)/((*+)!*(* + ))*os((6*+)*pi/),0..infinity)+/*sum((-)^*sqt ()^(*+)/((*+)!*(*+))*os((*+)*pi/),0.. infinity),); Also, both the imginy pts of the bove nswes obtine by Mple e zeo, so n be ignoe... Emple In Theoem, ting /, 7, b π / 6 into (9), we n etemine the following integl os t t os 7 + osh 7 + 6 6 (/) + ( ) () 7 7 ()! + 6 π (7) Thus, we obtin the efinite integl fom t π / to t π / 7, os os 7t + osh 7t + 6 π / 7 π / 6 π ( / ) π + ( ) 7 ()! ( / ) π ( ) (8) 7 ()! We use Mple to veify the oetness of (8) s follows: >evlf(int(os(/*os(7*t+*pi/6))*osh(/*(7*t+ *Pi/6)),tPi/..*Pi/7),); >evlf(*pi/+/7*sum((-)^*(/)^(*)/((*)!*(* ))*(**Pi/),..infinity)-/7*sum((-)^*(/)^(* )/((*)!*(*))*(**Pi/),..infinity),); Also, both the imginy pts of the bove nswes obtine by Mple e zeo, so n be ignoe. On the othe hn, in Theoem, if ting 9,, b π / into (0), we obtin the following integl π π 9os t h 9 t 9 ( ) os ()! π (9) Theefoe, we hve the efinite integl fom t π / 6 to t π /, π π 9os t h 9 t π / π / 6 9 π ( ) os ()! 9 ( ) (0) ()! Ug Mple to veify the oetness of (0) s follows: >evlf(int((9*os(*t-*pi/))*h(9*(*t-*pi/)),t Pi/6..Pi/),); >evlf(/*sum((-)^*9^(*)/((*)!*(*))*os(** Pi/),..infinity)-/*sum((-)^*9^(*)/((*)!*(*)),..infinity),);
Mthemtis n Sttistis (): -, 0 The imginy pts of the bove nswes obtine by Mple e eithe zeo o vey smll, so n be ignoe.. Conlusion Fom the bove isussion, we now the integtion tem by tem theoem plys signifint ole in the theoetil infeenes of this stuy. In ft, the pplition of this theoem is etensive, n n be use to esily solve mny iffiult poblems; we enevo to onut futhe stuies on elte pplitions. On the othe hn, Mple lso plys vitl ssistive ole in poblem-solving. In the futue, we will eten the eseh topi to othe lulus n engineeing mthemtis poblems n solve these poblems by ug Mple. These esults will be use s tehing mteils fo Mple on eution n eseh to enhne the onnottions of lulus n engineeing mthemtis. REFERENCES [] A. A. Ams, H. Gottliebsen, S. A. Linton, n U. Mtin, Automte theoem poving in suppot of ompute lgeb: symboli efinite integtion s se stuy, Poeeings of the 999 Intentionl Symposium on Symboli n Algebi Computtion, pp. -60, Vnouve, Cn, 999. [] C. Oste, Limit of efinite integl, SIAM Review, Vol., No., pp. -6, 99. [] M. A. Nyblom, On the evlution of efinite integl involving neste sque oot funtions, Roy Mountin Jounl of Mthemtis, Vol. 7, No., pp. 0-0, 007. [] C. -H. Yu, A stuy on integl poblems by ug Mple, Intentionl Jounl of Avne Reseh in Compute Siene n Softwe Engineeing, Vol., Issue. 7, pp. -6, 0. [] C. -H. Yu, Evluting some integls with Mple, Intentionl Jounl of Compute Siene n Mobile Computing, Vol., Issue. 7, pp. 66-7, 0. [6] C.-H. Yu, Applition of Mple on evluting the lose foms of two types of integls, Poeeings of the 7th Mobile Computing Woshop, ID6, 0. [7] C.-H. Yu, Applition of Mple on some integl poblems, Poeeings of the Intentionl Confeene on Sfety & Seuity Mngement n Engineeing Tehnology 0, pp. 90-9, 0. [8] C.-H. Yu, Applition of Mple on the integl poblem of some type of tionl funtions, Poeeings of the Annul Meeting n Aemi Confeene fo Assoition of IE, D7-D6, 0. [9] C. -H. Yu, Ug Mple to stuy two types of integls, Intentionl Jounl of Reseh in Compute Applitions n Robotis, Vol., Issue., pp. -, 0. [0] C.-H. Yu, Applition of Mple on evlution of efinite integls, Applie Mehnis n Mteils, in pess. [] C. -H. Yu, Solving some integls with Mple, Intentionl Jounl of Reseh in Aeonutil n Mehnil Engineeing, Vol., Issue., pp. 9-, 0. [] C. -H. Yu, Ug Mple to stuy the integls of tigonometi funtions, Poeeings of the 6th IEEE/Intentionl Confeene on Avne Infoomm Tehnology, No. 009, 0. [] C. -H. Yu, A stuy of the integls of tigonometi funtions with Mple, Poeeings of the Institute of Inustil Enginees A Confeene 0, Spinge, Vol., pp. 60-, 0. [] C.-H. Yu, Applition of Mple on some type of integl poblem, Poeeings of Ubiquitous-Home Confeene 0, pp.06-0, 0. [] C.-H. Yu, Applition of Mple: ting two speil integl poblems s emples, Poeeings of the 8th Intentionl Confeene on Knowlege Community, pp.80-8, 0. [6] W. R. Dei, Intoutoy Comple Anlysis n Applitions, New Yo: Aemi Pess, 97. [7] T. M. Apostol, Mthemtil Anlysis, n e., Boston: Aison-Wesley, 97.