Application of Poisson Integral Formula on Solving Some Definite Integrals

Transcription

1 Jounal of Copute and Electonic Sciences Available online at jcesblue-apog 015 JCES Jounal Vol 1(), pp 4-47, 30 Apil, 015 Application of Poisson Integal Foula on Solving Soe Definite Integals Chii-Huei Yu 1*, Tsai-Jung Chen and Tsung-Ming Chen 3 1- Depatent of Infoation Technology, Nan Jeon Univesity of Science and Technology - Depatent of Vehicle Engineeing, National Pingtung Univesity of Science and Technology 3- Depatent of Electical Engineeing, Nan Jeon Univesity of Science and Technology Coesponding Autho: Chii-Huei Yu Received: 15 Mach, 015 Accepted: 0 Apil, 015 Published: 30 Apil, 015 A B S T R A C T This pape studies six types of definite integals and uses Maple fo veification The closed fos of these definite integals can be obtained ainly using Poisson integal foula On the othe hand, soe exaples ae used to deonstate the calculations Keywods: definite integals, closed fos, Poisson integal foula, Maple 015 JCES Jounal All ights eseved INTRODUCTION In calculus and engineeing atheatics, thee ae any ethods to solve the integal pobles including change of vaiables ethod, integation by pats ethod, patial factions ethod, tigonoetic substitution ethod, etc In this pape, we study the following six types of definite integals which ae not easy to obtain thei answes using the ethods entioned above exp( cos ) cos( sin ) d scos( ) s 0, (1) exp( cos ) sin( sin ) d 0 scos( ) s, () cos sin( cos ) cosh( sin ) sin cos( cos )sinh( sin ) d 0 s cos( ) s, (3) sin sin( cos ) cosh( sin ) cos cos( cos )sinh( sin ) d 0 s cos( ) s, (4) cos cos( cos ) cosh( sin ) sin sin( cos )sinh( sin ) d 0 s cos( ) s, (5) sin cos( cos ) cosh( sin ) cos sin( cos )sinh( sin ) d 0 s cos( ) s, (6)

2 J Cop & Elect Sci, 1 (): 4-47, 015, s, s 3whee ae eal nubes,, and is a non-negative intege We can obtain the closed fos of these definite integals ainly using Poisson integal foula; these ae the ajo esults of this pape (ie, Theoes 1-3) Adas et al [1], Nyblo [], and Oste [3] povided soe techniques to solve the integal pobles Yu [4-9], Yu and B -H Chen [30], and T -J Chen and Yu [31-33] used coplex powe seies ethod, integation te by te theoe, diffeentiation with espect to a paaete, Paseval s theoe, and genealized Cauchy integal foula to solve soe types of integals In this pape, soe exaples ae used to deonstate the poposed calculations, and the anual calculations ae veified using Maple 1 Main Results Soe foulas used in this pape ae intoduced below 1 Eule s foula: e ix cos x i sin x, whee i 1, and x is any eal nube DeMoive s foula: (cosx isin x) cosx isin x, whee is any intege, and x is any eal nube The following two foulas can be found in [34, p6] 3 sin( a ib) sin acosh bicos asinh b, whee a,b ae eal nubes 4 cos( a ib) cos acosh bisin asinh b, whee a,b ae eal nubes An ipotant foula used in this study is intoduced below, which can be found in [35, p 145] 5 Poisson integal foula:, s ae eal nubes, and z Suppose that analytic on the open disc z C f s, then If f is defined and continuous on the closed disc z C z s ) i f ( e ) scos( ) s i ( se 0 In the following, we deteine the closed fos of the definite integals (1) and (), s, s Theoe 1 If ae eal nubes,, and is a non-negative intege, then the definite integals exp( cos ) cos( sin ) s d exp( scos ) cos( ssin ) 0 scos( ) s ( s ), (7) and exp( cos ) sin( sin ) s d exp( s cos ) sin( ssin ) 0 scos( ) s ( s ) (8) f ( z) z e z f (z) Poof Let, then is analytic on the whole coplex plane Using Poisson integal foula yields i i i i s ( e ) exp( e ) ( se ) exp( se ) d 0 scos( ) s (9) By Eule s foula and DeMoive s foula, we have d and is 43 P a g e

3 J Cop & Elect Sci, 1 (): 4-47, 015 i i i i s e exp( e ) s e exp( se ) d 0 scos( ) s (10) Theefoe, i i e exp( e ) s i i d e exp( se ) 0 scos( ) s ( s ) (11) Using the equality of eal pats of both sides of Eq (11) yields Eq (7) holds Also, by the equality of iaginay pats of both sides of Eq (11), we obtain Eq (8) qed Next, the closed fos of the definite integals (3) and (4) ae obtained below Theoe If the assuptions ae the sae as Theoe 1, then and Poof Since cos sin( cos )cosh( sin ) sin cos( cos )sinh( sin ) d s cos( ) s s [cos sin( s cos ) cosh( s sin ) sin cos( s cos ) sinh( s sin )] ( s ), (1) 0 sin sin( cos ) cosh( sin ) cos cos( cos )sinh( sin ) d s cos( ) s 0 s [sin sin( scos )cosh( ssin ) cos cos( scos )sinh( ssin )] ( s ) (13) g ( z ) z sin z is analytic on the whole coplex plane, using Poisson integal foula yields It follows that i i ( se ) sin( se ) s i i ( e ) sin( e ) 0 scos( ) s d (14) i i e sin( e ) s i i d e sin( se ) 0 scos( ) s ( s ) (15) Eq (1) can be obtained using Foula 3 and the equality of eal pats of both sides of Eq (15) On the othe hand, by Foula 3 and the equality of iaginay pats of both sides of Eq (15), we obtain Eq (13) qed Finally, we find the closed fos of the definite integals (5) and (6) Theoe 3 If the assuptions ae the sae as Theoe 1, then and cos cos( cos ) cosh( sin ) sin sin( cos )sinh( sin ) d s cos( ) s 0 s [cos cos( s cos ) cosh( s sin ) sin sin( s cos )sinh( s sin )] ( s ), (16) 44 P a g e

4 J Cop & Elect Sci, 1 (): 4-47, 015 sin cos( cos ) cosh( sin ) cos sin( cos )sinh( sin ) d s cos( ) s 0 s [sin cos( s cos ) cosh( s sin ) cos sin( s cos )sinh( s sin )] ( s ) (17) h ( z ) z cos z Poof Since is analytic on the whole coplex plane, by Poisson integal foula and Foula 4, the desied esults hold qed Exaples In the following, fo the six types of definite integals in this study, soe exaples ae poposed and we use Theoes 1-3 to deteine thei closed fos On the othe hand, Maple is used to calculate the appoxiations of soe definite integals and thei solutions fo veifying ou answes Exaple 1 In Eq (7), if 4, s,, and 3 exp(4cos ) cos(3 4sin ) d 016cos( /3, then exp( 1) 48 cos( Next, we use Maple to veify the coectness of Eq (18) >evalf(int(exp(4*cos(theta))*cos(3*theta+4*sin(theta))/(0-16*cos(theta-pi/3)),theta=0*pi),0); >evalf(pi/48*exp(1)*cos(pi+sqt(3)),0); On the othe hand, if 5, s 4, / 4, and 6 in Eq (8), then 3) (18) exp(5cos ) sin(6 5sin ) d 819 exp( 4140cos( / ) sin(3 / >evalf(int(exp(5*cos(theta))*sin(6*theta+5*sin(theta))/(41-40*cos(theta-pi/4)),theta=0*pi),0); >evalf(819*pi/14065*exp(*sqt())*sin(3*pi/+*sqt()),0); Exaple In Eq (1), let 7, s 5, / 6, and 4, then the definite integal ) (19) cos4 sin(7 cos )cosh(7sin ) sin 4 cos(7 cos )sinh(7sin ) 0 d 74 70cos( / 6 ) 65 [ 1/ sin(5 3 / )cosh(5/ ) 3 / cos(5 3 / )sinh(5/ )] 881 (0) We also use Maple to veify the coectness of Eq (0) >evalf(int((cos(4*theta)*sin(7*cos(theta))*cosh(7*sin(theta))-sin(4*theta)*cos(7*cos(theta))*sinh(7*sin(theta)))/(74-70*cos(theta-pi/6)),theta=0*pi),0); >evalf(65*pi/881*(-1/*sin(5*sqt(3)/)*cosh(5/)-sqt(3)/*cos(5*sqt(3)/)*sinh(5/)),0); Also, if 3, s,, and in Eq (13), then 45 P a g e

5 J Cop & Elect Sci, 1 (): 4-47, 015 sin sin( 3cos )cosh( 3sin ) cos cos( 3cos )sinh( 3sin ) d 131cos( 8 [ 3 /sin(1)cosh( 3) 1/ cos(1)sinh( 3)] 45 (1) >evalf(int((sin(*theta)*sin(-3*cos(theta))*cosh(-3*sin(theta))+cos(*theta)*cos(-3*cos( theta))*sinh(-3*sin(theta)))/(13+1*cos(theta-pi/3)),theta=0*pi),0); >evalf(8*pi/45*(sqt(3)/*sin(1)*cosh(sqt(3))-1/*cos(1)*sinh(sqt(3))),0); Exaple 3 In Eq (16), if 4, s,, and 5, then the definite integal cos5 cos(4 cos )cosh(4sin ) sin 5 sin(4 cos )sinh(4sin ) d 016cos( [ 1/ cos(1)cosh( 3) 3 / sin(1)sinh( 3)] 19 () Maple is used to veify the coectness of Eq () as follows: >evalf(int((cos(5*theta)*cos(4*cos(theta))*cosh(4*sin(theta))+sin(5*theta)*sin(4*cos(theta))*sinh(4*sin(theta)))/(0+16*cos(t heta-*pi/3)),theta=0*pi),15); >evalf(-pi/19*(-1/*cos(1)*cosh(-sqt(3))-sqt(3)/*sin(1)*sinh(-sqt(3))),15); In addition, let 5, s 4, 3 / 4, and 7 in Eq (17), we obtain sin 7 cos( 5cos )cosh( 5sin ) cos7 sin( 5cos )sinh( 5sin ) 0 d 41 40cos( 3 / 4 ) 3768 [ / cos( ) cosh( ) / sin( ) sinh( )] >evalf(int((sin(7*theta)*cos(-5*cos(theta))*cosh(-5*sin(theta))-cos(7*theta)*sin(-5*cos( theta))*sinh(-5*sin(theta)))/(41+40*cos(theta+3*pi/4)),theta=0*pi),18); >evalf(3768*pi/(-70315)*(sqt()/*cos(-*sqt())*cosh(-*sqt())+sqt()/*sin(-* sqt())*sinh(-*sqt())),18); (3) 4 Conclusion In this aticle, we use Poisson integal foula to solve soe types of definite integals In fact, the applications of this foula ae extensive, and can be used to easily solve any difficult pobles; we endeavo to conduct futhe studies on elated applications In addition, Maple also plays a vital assistive ole in poble-solving In the futue, we will extend the eseach topic to othe calculus and engineeing atheatics pobles and use Maple to veify ou answes REFERENCES Adas AA, Gottliebsen H, Linton SA and Matin U 1999 Autoated theoe poving in suppot of copute algeba: sybolic definite integation as a case study, Poceedings of the 1999 Intenational Syposiu on Sybolic and Algebaic Coputation, Canada, pp Chen TJ and Yu CH 014 A study on the integal pobles of tigonoetic functions using two ethods, Wulfenia Jounal, Vol 1, No 4, pp Chen TJ and Yu CH 014 Fouie seies expansions of soe definite integals, Sylwan Jounal, Vol 158, Issue 5, pp P a g e

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