The Application of Parseval s Theorem to Integral Problems

Transcription

1 Applied Mathematics ad Physics, 0, Vol., No., -9 Available olie at Sciece ad Educatio Publishig DOI:0.69/amp--- The Applicatio of Paseval s Theoem to Itegal Poblems Chii-Huei Yu * Depatmet of Maagemet ad Ifomatio, Na Jeo Uivesity of Sciece ad Techology, Taia City, Taiwa *Coespodig autho: chiihuei@ju.edu.tw Received Novembe 0, 03; Revised Decembe, 03; Accepted Jauay 08, 0 Abstact This pape uses the mathematical softwae Maple as a auxiliay tool to study s types of defiite itegals. We ca obtai the ifiite seies foms of these defiite itegals by usig Paseval s theoem. O the othe had, we povide some examples to do calculatio pactically. The eseach methods adopted i this study ivolved fidig solutios though maual calculatios ad veifyig the aswes by usig Maple. Keywods: defiite itegals, ifiite seies foms, Paseval s theoem, Maple Cite This Aticle: Chii-Huei Yu, The Applicatio of Paseval s Theoem to Itegal Poblems. Applied Mathematics ad Physics, o. (0): -9. doi: 0.69/amp---.. Itoductio As the ifomatio techology advaces, whethe computes ca become compaable with huma bais to pefom abstact tasks, such as abstact at simila to the paitigs of Picasso ad musical compositios simila to those of Beethove, is a atual questio. Cuetly, this appeas uattaiable. I additio, whethe computes ca solve abstact ad difficult mathematical poblems ad develop abstact mathematical theoies such as those of mathematicias also appeas ufeasible. Nevetheless, i seekig fo alteatives, we ca study what assistace mathematical softwae ca povide. This study itoduces how to use the mathematical softwae Maple to coduct mathematical eseach. The mai easos of usig Maple i this pape ae its simple istuctios ad ease of use, which eable the begies to lea the opeatig techiques i a shot peiod. By employig the poweful computig capabilities of Maple, difficult poblems ca be easily solved. Eve whe Maple caot detemie the solutio, poblem-solvig hits ca be idetified ad ifeed fom the appoximatios calculated ad solutios to simila poblems, as detemied by Maple. Fo this easo, Maple ca povide isights ito scietific eseach. I calculus ad egieeig mathematics couses, we leat may methods to solve the itegal poblems icludig the chage of vaiables method, the itegatio by pats method, the patial factios method, the tigoometic substitutio method, ad so o. I this study, we evaluate the followig s types of defiite itegals which ae ot easy to obtai thei aswes usig the methods metioed above. sih( cos x) cosh( cos x) () si( si x)cos( si x) () 0 (3) sih( cos x) cosh( cos x) () si( si x)cos( si x) (5) 0 (6) whee is a eal umbe. We ca obtai the ifiite seies foms of these defiite itegals by usig Paseval s theoem; these ae the majo esults of this pape (i.e., Theoems ad ). The study of elated itegal poblems ca efe to [-5]. Additioally, we popose some defiite itegals to do calculatio pactically. The eseach methods adopted i this study ivolved fidig solutios though maual calculatios ad veifyig these aswes by usig Maple. This type of eseach method ot oly allows the discovey of calculatio eos, but also helps modify the oigial diectios of thikig fom maual ad Maple calculatios. Fo this easo, Maple povides isights ad guidace egadig poblem-solvig methods.. Mai Results Fistly, we itoduce a otatio, a defiitio ad some fomulas used i this aticle.

2 Applied Mathematics ad Physics 5.. Notatio Let z a + ib be a complex umbe, whee i, a, b ae eal umbes. We deote a the eal pat of z by Re(z), ad b the imagiay pat of z by Im(z)... Defiitio Suppose f (x) is a cotiuous fuctio defied o [0, ], the Fouie seies expasio of f (x) is a0 + ( ak cos kx + bk si kx), whee k a0 0 f ( x), ak 0 f ( x)cos kx, bk 0 f ( x)si kx fo all positive iteges k..3. Fomulas.3.. Eule s Fomula e cos x+ isi x, whee x is ay eal umbe..3.. DeMoive s Fomula (cos x + i si x) cos x + i si x, whee is ay itege, ad x is ay eal umbe ([6]) sih( α + i) sihαcos + icoshαsi, whee α, ae eal umbes..3.. ([6]) cosh( α + i) coshαcos + isihαsi, whee α, ae eal umbes Taylo Seies Expasio of Hypebolic Secat Fuctio ([7]) E sech( z) z, whee z is a complex umbe, 0 ( )! z <, E ae the Eule umbes fo all o-egative iteges Taylo Seies Expasio of Hypebolic Cosecat Fuctio ([7]) ( ) B csch( z) + z, whee z is a z ( )! complex umbe, 0 < z <, ad B ae the Beoulli umbes fo all positive iteges. Next, we itoduce a impotat theoem used i this study... Paseval s Theoem ([8]) Suppose f (x) is a cotiuous fuctio defied o [0, ], f (0) f (). If the Fouie seies expasio of f (x) is a0 + ( acos x + bsi x), the a 0 0 f ( x) ( a b ) + +. Befoe deivig the fist majo esult i this study, we eed a lemma..5. Lemma Suppose α, ae eal umbes with sih α + cos 0. The cosh cos i sih si sech( α+ i) sih α + cos cosh αcos + sih αsi (sih + cos ) sih + cos.5.. Poof (7) (8) sech( α+ i) cosh( α + i) (By Fomula.3..) Ad coshαcos + i sihαsi cosh cos i sih si cosh cos + sih si coshαcos i sihαsi sih α + cos cosh αcos + sih αsi (sih α + cos ) cosh αcos + sih α( cos ) (sih α + cos ) sih α + cos I the followig, we detemie the ifiite seies foms of the defiite itegals (), () ad (3)..6. Theoem If is a eal umbe with <. The the defiite itegals cosh( cos x) cos( si x) ( E ) + [( )!] (9)

3 6 Applied Mathematics ad Physics sih( cos x) si( si x) ( E ) [( )!] 0 + ( E ) [( )!].6.. Poof Because cosh( cos x) cos( si x) Re[sec h( e )] (By (7)) E Re ( e ) 0 ( )! (By Fomula.3.5.) (0) () sih( cos x) si( si x) ( E ) [( )!] O the othe had, fom the summatio of (9) ad (0) ad usig (8), 0 + ( E ) [( )!] Befoe deivig the secod majo esult of this pape, we also eed a lemma..7. Lemma Suppose α, ae eal umbes with sih α + si 0. The sih cos i cosh si csc h( α + i) sih α + si () E ix Re e 0 ( )! (By DeMoive s fomula) E cos x 0 ( )! (By Eule s fomula) E + cos x () ( )! Thus, usig Paseval s theoem, we obtai cosh( cos x) cos( si x) ( E ) + [( )!] Similaly, because sih( cos x) si( si x) Im[sec h( e )] E (3) Im ( e ) 0 ( )! E si x ( )! By Paseval s theoem, we have sih αcos + cosh αsi (sih + si ) sih + si.7.. Poof Ad (5) csc h( α + i) sih( α + i) (By Fomula.3.3.) sihαcos + i coshαsi sih cos i cosh si sih cos + cosh si sih cos i cosh si sih α + si sih α cos + cosh α si (sih α + si ) sih α(-si )+cosh αsi (sih α + si ) sih α + si Fially, we fid the ifiite seies foms of the defiite itegals (), (5) ad (6)..8. Theoem If is a eal umbe with 0 < <. The the defiite itegals

4 Applied Mathematics ad Physics 7 sih( cos x) cos( si x) (6) ( ) ( B ) + 6 [( )!] cosh( cos x) si( si x) (7) ( ) ( B ) [( )!] 0 (8) ( ) ( B ) [( )!].8.. Poof Because sih( cos x) cos( si x) Re[csc h( e )] (By ()) ( ) B Re + ( e ) e ( )! (By Fomula.3.6.) ( ) B cos x+ cos( ) x (9) 6 ( )! Usig Paseval s theoem, we have sih( cos x) cos( si x) ( ) ( B ) + 6 [( )!] Similaly, cosh( cos x) si( si x) Im[csc h( e )] (By ()) ( ) B Im + ( e ) e ( )! (By Fomula.3.6.) ( ) B + si x si( ) x(0) 6 ( )! Theefoe, usig Paseval s theoem, we obtai cosh( cos x) si( si x) ( ) ( B ) [( )!] I additio, fom the summatio of (6) ad (7) ad by (5), we have 0 ( ) ( B ) [( )!] 3. Examples I the followig, fo the s types of defiite itegals i this study, we povide some defiite itegals ad employ Theoems ad to detemie thei ifiite seies foms. I additio, we use Maple to calculate the appoximatios of these defiite itegals ad thei solutios fo veifyig ou aswes. 3.. Example Takig ito (9), we obtai the defiite itegal cosh( cos x)cos( si x) sih ( cos x) + cos ( si x) ( E ) + [( )!] () Next, we use Maple to veify the coectess of (). >evalf(it((cosh(/*cos(x))*cos(/*si(x)))^/((sih( /*cos(x)))^+(cos(/*si(x)))^)^,x0..*pi),8); >evalf(*pi+pi*sum((eule(*))^/((*)!)^*(/)^( *),..ifiity),8); Similaly, if 5 i (0), the defiite itegal sih( 5 cos x) si( 5 si x) sih ( 5 cos x) + cos ( 5 si x) () ( E ) [( )!] 5 >evalf(it((sih(/sqt(5)*cos(x))*si(/sqt(5)*si(x)))^ /((sih(/sqt(5)*cos(x)))^+(cos(/sqt(5)*si(x)))^)^,x0..*pi),8); >evalf(pi*sum((eule(*))^/((*)!)^*(/sqt(5))^( *),..ifiity),8);

5 8 Applied Mathematics ad Physics Fially, takig 5ito (), we have sih ( 5 cos x) + cos ( 5 si x) 0 ( E ) + [( )!] 5 >evalf(it(/((sih(/5*cos(x)))^+(cos(/5*si(x)))^), x0..*pi),8); >evalf(*pi+*pi*sum((eule(*))^/((*)!)^*(/5)^ (*),..ifiity),8); 3.. Example If i (6), the sih( cos x) cos(si x) sih (cos x) + si (si x) ( ) ( B ) + 36 [( )!] >evalf(it((sih(*cos(x))*cos(*si(x)))^/((sih(*co s(x)))^+(si(*si(x)))^)^,x0..*pi),8); >evalf(pi*(/36+*sum((-^(*- ))^*(beoulli(*)) ^/((*)!)^*^(*-),..ifiity)),8); I additio, if 7 i (7), the (3) () cosh( 7 cos x)si( 7 si x) sih ( 7 cos x) + si ( 7 si x) (5) 83 ( ) ( B ) ( 7) + 76 [( )!] >evalf(it((cosh(sqt(7)*cos(x))*si(sqt(7)*si(x)))^/( (sih(sqt(7)*cos(x)))^+(si(sqt(7)*si(x)))^)^,x0.. *Pi),8); >evalf(pi*(83/76+*sum((-^(*- ))^*(beoulli (*))^/((*)!)^*(sqt(7))^(*-),..ifiity)),8); O the othe had, if 95i (8), we have 0 sih (9 5 cos x) + si (9 5 si x) (6) 39 ( ) ( B ) [( )!] 5 >evalf(it(/((sih(9/5*cos(x)))^+(si(9/5*si(x)))^), x0..*pi),8); >evalf(pi*(39/050+8*sum((-^(*- ))^*(beoulli (*))^/((*)!)^*(9/5)^(*-),..ifiity)),8);. Coclusio I this aticle, we povide a ew techique to detemie some defiite itegals. We hope this techique ca be applied to solve aothe defiite itegal poblems. O the othe had, the Paseval s theoem plays a sigificat ole i the theoetical ifeeces of this study. I fact, the applicatios of this theoem ae extesive, ad ca be used to easily solve may difficult poblems; we edeavo to coduct futhe studies o elated applicatios. I additio, Maple also plays a vital assistive ole i poblem-solvig. I the futue, we will exted the eseach topic to othe calculus ad egieeig mathematics poblems ad solve these poblems by usig Maple. These esults will be used as teachig mateials fo Maple o educatio ad eseach to ehace the cootatios of calculus ad egieeig mathematics. 5. Apped I the followig, we itoduce some Maple's commads. 5.. evalf( ); calculatig the appoximatio. Example. Evaluatig the appoximatio of. > evalf(sqt()); sum( ); Detemiig the summatio. Example. Evaluatig the ifiite seies. > sum(/^,..ifiity); it( ); Detemiig the itegal. Example. Fidig the itegal cos x. > it(cos(x),x); si( x ) 5.. beoulli(); Evaluatig the -th Beoulli umbe. Example. Detemiig the -th Beoulli umbe. > beoulli(); eule(); Calculatig the -th Eule umbe. Example. Evaluatig the 8-th Eule umbe. > eule(8); laplace(f(t),t,s); Evaluatig the Laplace tasfom of f() t. Example. Fidig the Laplace tasfom of cos at. > laplace(cos(a*t),t,s);

6 Applied Mathematics ad Physics 9 s s + a 5.7. ivlaplace(f(s),s,t); Evaluatig the ivese Laplace tasfom of Fs. () Example. Detemiig the ivese Laplace tasfom of a. > ivlaplace(a/(s^-a^),s,t); s a Refeeces sih( at ) [] M. A. Nyblom, O the evaluatio of a defiite itegal ivolvig ested squae oot fuctios, Rocky Moutai Joual of Mathematics, Vol. 37, No., pp , 007. [] A. A. Adams, H. Gottliebse, S. A. Lito, ad U. Mati, Automated theoem povig i suppot of compute algeba: symbolic defiite itegatio as a case study, Poceedigs of the 999 Iteatioal Symposium o Symbolic ad Algebaic Computatio, pp , Vacouve, Caada, 999. [3] C. Oste, Limit of a defiite itegal, SIAM Review, Vol. 33, No., pp. 5-6, 99. [] C. -H. Yu, Usig Maple to study two types of itegals, Iteatioal Joual of Reseach i Compute Applicatios ad Robotics, Vol., pp. -, 03. [5] C. -H. Yu, Solvig some itegals with Maple, Iteatioal Joual of Reseach i Aeoautical ad Mechaical Egieeig, Vol., pp. 9-35, 03. [6] C. -H. Yu, A study o itegal poblems by usig Maple, Iteatioal Joual of Advaced Reseach i Compute Sciece ad Softwae Egieeig, Vol. 3, pp. -6, 03. [7] C. -H. Yu, Evaluatig some itegals with Maple, Iteatioal Joual of Compute Sciece ad Mobile Computig, Vol., pp. 66-7, 03. [8] C. -H. Yu, Applicatio of Maple o evaluatio of defiite itegals, Applied Mechaics ad Mateials, Vols (0), pp , 03. [9] C. -H. Yu, Usig Maple to study the itegals of tigoometic fuctios, Poceedigs of the 6th IEEE/Iteatioal Cofeece o Advaced Ifocomm Techology, No. 009, 03. [0] C. -H. Yu, Applicatio of Maple o some type of itegal poblem, Poceedigs of the Ubiquitous-Home Cofeece 0, pp.06-0, 0. [] C. -H. Yu, Applicatio of Maple o the itegal poblem of some type of atioal fuctios, Poceedigs of the Aual Meetig ad Academic Cofeece fo Associatio of IE, D357- D36, 0. [] C. -H. Yu, A study of the itegals of tigoometic fuctios with Maple, Poceedigs of the Istitute of Idustial Egiees Asia Cofeece 03, Spige, Vol., pp , 03. [3] C. -H. Yu, Applicatio of Maple o some itegal poblems, Poceedigs of the Iteatioal Cofeece o Safety & Secuity Maagemet ad Egieeig Techology 0, pp. 90-9, 0. [] C. -H. Yu, Applicatio of Maple o evaluatig the closed foms of two types of itegals, Poceedigs of the 7th Mobile Computig Wokshop, ID6, 0. [5] C. -H. Yu, Applicatio of Maple: takig two special itegal poblems as examples, Poceedigs of the 8th Iteatioal Cofeece o Kowledge Commuity, pp.803-8, 0. [6] R. V. Chuchill ad J. W. Bow, Complex vaiables ad applicatios, th ed., McGaw-Hill, New Yok, p 65, 98. [7] Hypebolic fuctios, olie available fom og/wiki/hypebolic_fuctio. [8] D. V. Widde, Advaced calculus, d ed., Petice-Hall, New Jesey, p 8, 96.