Infinite Series Forms of Double Integrals

Transcription

1 Iteratoal Joural of Data Evelopmet Aalyss ad *Operatos Research*, 4, Vol., No., 6- Avalable ole at Scece ad Educato Publshg DOI:.69/jdeaor--- Ifte Seres Forms of Double Itegrals Ch-Hue Yu,*, Sh-Der Sheu Departmet of Maagemet ad Iformato, Na Jeo Uversty of Scece ad Techology, Taa Cty, Tawa Departmet of Iformato Techology, Na Jeo Uversty of Scece ad Techology, Taa Cty, Tawa *Correspodg author: Receved Jue 6, 4; Revsed Jue 6, 4; Accepted Jue 6, 4 Abstract Ths paper studes sx types of double tegrals ad uses Maple for verfcato. The fte seres forms of these double tegrals ca be obtaed usg Taylor seres expasos ad tegrato term by term theorem. I addto, some examples are used to demostrate the calculatos. Keywords: double tegrals, fte seres forms, Taylor seres expasos, tegrato term by term theorem, Maple Cte Ths Artcle: Ch-Hue Yu, ad Sh-Der Sheu, Ifte Seres Forms of Double Itegrals. Iteratoal Joural of Data Evelopmet Aalyss ad *Operatos Research*, vol., o. (4: 6-. do:.69/jdeaor---.. Itroducto I calculus ad egeerg mathematcs, there are may methods to solve the tegral problems cludg chage of varables method, tegrato by parts method, partal fractos method, trgoometrc substtuto method, ad so o. I ths paper, we study the followg sx types of double tegrals whch are ot easy to obta ther aswers usg the methods metoed above. r r r r cos r r r cos r e cos( + r s, ( e s( + r s, ( cos s( r cos cosh( r s s cos( r cos sh( r s, (3 r r s s( r cos cosh( r s + cos cos( r cos sh( r s, (4 r cos cos( r cos cosh( r s drd r + s s( r cos sh( r s, (5 r s cos( r cos cosh( r s drd r cos s( r cos sh( r s, (6 where r, r,, are ay real umbers, ad s ay postve teger. We ca obta the fte seres forms of these double tegrals usg Taylor seres expasos ad tegrato term by term theorem; these are the major results of ths paper (.e., Theorems -3. Adams et al. [], Nyblom [], ad Oster [3] provded some techques to solve the tegral problems. Yu [4-9], Yu ad B. -H. Che [3], Yu ad Sheu [3], ad T. -J. Che ad Yu [3,33,34] used complex power seres method, tegrato term by term theorem, dfferetato wth respect to a parameter, Parseval s theorem, area mea value theorem, ad geeralzed Cauchy tegral formula to solve some types of tegral problems. I ths paper, three examples are used to demostrate the proposed calculatos, ad the maual calculatos are verfed usg Maple.. Ma Results Some formulas ad theorems used ths paper are troduced below... Euler s Formula cos x + s x, where, ad x s ay real umber. e x.. DeMovre s Formula m (cos x + s x cos mx + s mx, where m s ay teger, ad x s ay real umber. The followgs are the Taylor seres expasos of some aalytc fuctos. z.3. e z, where z s ay Complex! Number ( +.4. s z z, where z s ay ( +! Complex Number (.5. cos z z, where z s ay (! Complex Number The followg two formulas ca be foud [[35], p 6]

2 7 Iteratoal Joural of Data Evelopmet Aalyss ad *Operatos Research*.6. s( a + b s a cosh b + cos ash b, where ab, are ay real Numbers.7. cos( a + b cos a cosh b s ash b, where ab, are ay real Numbers A mportat theorem used ths study s troduced below, whch ca be foud [[36], p 69]..8. Itegrato Term by Term Theorem Suppose that { } g s a sequece of Lebesgue tegrable fuctos defed o I. If I g s coverget, the g I g I. Frstly, we determe the fte seres forms of the double tegrals ( ad (. Theorem. Suppose that r, r,, are real umbers, ad s a postve teger. The the double tegrals: ad r r cos r e cos( + r s + + ( r r!( + ( + [s( + s( + ] r r cos r e s( + r s + + ( r r!( + ( + [cos( + cos( + ] Proof By Formula.3, we have: re z z e, (7 (8 + z. (9! z, where r, are ay real umbers, the: re + ( re e ( re. (! By Euler s formula ad DeMovre s formula, we obta: It follows that: e re ( + e e r e. (! rcos ( + rs e ( + r e. (! Usg the equalty of real parts of both sdes of Eq. ( yelds: r cos e cos( + rs r cos[( + ](3! Therefore, r r cos r r ( by tegrato term by term theorem e cos( + r s r cos[( ] drd r +! r r cos[( ] drd! + r + + ( r r!( + ( + [s( + s( + ] O the other had, by the equalty of magary parts of both sdes of Eq. ( yelds: r cos e s( + rs r s[( + ]. (4! Usg tegrato term by term theorem, we ca easly obta: r r cos r e s( + r s + + ( r r!( + ( + [cos( + cos( + ] Next, the fte seres forms of the double tegrals (3 ad (4 ca be obtaed below. Theorem. If the assumptos are the same as Theorem, the the double tegrals: ad r cos s( r cos cosh( r s drd r s cos( r cos sh( r s ( (5 + + ( r r ( +!( + ( + + [s( + + s( + + ] r s s( r cos cosh( r s drd r cos cos( r cos sh( r s + + ( (6 + + ( r r ( +!( + ( + + [cos( + + cos( + + ] Proof Usg Formula.4 yelds: re ( + +. (7 ( +! z s z z z, where r, are ay real umbers, the: ( + + ( re s( re ( re. (8 ( +! By Euler s formula, DeMovre s formula, ad Formula.6, we obta:

3 Iteratoal Joural of Data Evelopmet Aalyss ad *Operatos Research* 8 s( r cos cosh( r s (cos + s cos( r cos sh( r s +. (9 ( + (+ + r e ( +! Usg the equalty of real parts of both sdes of Eq. (9 yelds: Thus, r r cos s( r cos cosh( r s s cos( r cos sh( r s ( + r cos[(+ + ] ( +! r ( + r ( +! ( r + (! r + cos s( r cos cosh( r s s cos( r cos sh( r s. ( r cos[( + + ] r cos[( + + ] ( by tegrato term by term theorem ( + + ( r r (+!(+ (+ + [s(+ + s( + + ] O the other had, usg the equalty of magary parts of both sdes of Eq. (9 yelds: s s( r cos cosh( r s + cos cos( r cos sh( r s ( + r s[(+ + ] ( +!. ( By tegrato term by term theorem, we obta Eq. (6. Fally, we determe the fte seres forms of the double tegrals (5 ad (6. Theorem 3 If the assumptos are the same as Theorem, the the double tegrals: ad r r cos cos( r cos cosh( r s + s s( r cos sh( r s ( + + ( r r (!(+ ( + r r [s( + s( + ] + s cos( r cos cosh( r s cos s( r cos sh( r s ( + + ( r r (!(+ ( + [cos( + cos( + ] Proof By Formula.5, we have: ( (3 ( + z cos z z. (4 (! re z, where r, are ay real umbers, the: ( + ( re cos( re ( re. (5 (! Usg Formula.7 yelds: cos( r cos cosh( r s (cos + s s( r cos sh( r s. (6 ( ( + r e (! By the equalty of real parts of both sdes of Eq. (6, we have: Hece, r r cos cos( rcos cosh( rs + s s( rcos sh( rs ( r cos[( + ] (! cos cos( r cos cosh( r s + s s( r cos sh( r s r ( r (! ( r (! r ( r cos[( + ] r cos[( + ] ( r (!(+ ( r [s( + s( + ]. (7 O the other had, the equalty of magary parts of both sdes of Eq. (6 mples that: s cos( r cos cosh( r s cos s( r cos sh( r s. (8 ( r s[( + ] (! Thus, usg tegrato term by term theorem yelds Eq. (3 holds. 3. Examples I the followg, for the sx types of double tegrals ths study, we provde three examples ad use Theorems -3 to determe ther fte seres forms. I addto, Maple s used to calculate the approxmatos of some double tegrals ad ther solutos for verfyg our aswers. Example. By Eq. (7, we obta: π /3 3 r cos π /6 e cos(4 + r s + + (3!( + ( + 4 ( + 4 π ( + 4 π s s 3 6. (9

4 9 Iteratoal Joural of Data Evelopmet Aalyss ad *Operatos Research* Next, we use Maple to verfy the correctess of Eq. (9. >evalf(doublet(exp(r*cos(theta*cos(4*theta+r*s(t heta,r..3,thetap/6..p/3,8; >evalf(sum(/(!*(+*(+4*(3^(+-^(+* (s((+4*p/3- s((+4*p/6,..fty, 8; Also usg Eq. (8 yelds: π /4 5 r cos π /8 3 e s(6 + r s + + (5 3!( + ( + 6 ( + 6 π ( + 6 π cos cos 4 8 (3 >evalf(doublet(exp(r*cos(theta*s(6*theta+r*s(t heta, r3..5,thetap/8..p/4,8; >evalf(sum(-/(!*(+*(+6*(5^(+-3^(+* (cos((+6*p/4-cos((+6*p/8,..fty, 8; Example. Usg Eq. (5 yelds: π /4 5 π /6 cos s( r cos cosh( r s s cos( r cos sh( r s ( + + (5 ( +!( + ( + 3 ( + 3 π ( + 3 π s s 4 6 (3 We also use Maple to verfy the correctess of Eq. (3. >evalf(doublet(cos(*theta*s(r*cos(theta*cosh(r *s(theta- s(*theta* cos(r*cos(theta* sh(r*s (theta, r..5, thetap/6..p/4, 8; >evalf(sum((-^/((*+!* (*+* (*+3* (5^ (*+-^(*+* (s((*+3*p/4-s ((*+3*P/6,.fty, 8; O the other had, Eq. (6 mples that: π /6 7 π / 4 s 3 s( r cos cosh( r s + cos 3 cos( r cos sh( r s + ( + + (7 4 ( +!( + ( + 4 ( + π ( + π cos cos 3 6 (3 >evalf(doublet(s(3*theta*s(r*cos(theta*cosh(r *s(theta+cos(3*theta*cos(r*cos(theta*sh(r*s(the ta, r4..7,thetap/..p/6,8; >evalf(sum((-^(+/((*+!*(*+*(*+4* (7^(*+-4^(*+*(cos((+*P/3- cos((+*p/6,. fty, 8; Example 3. By Eq. ( we have: 5 π /6 8 π /3 3 cos 5 cos( r cos cosh( r s + s 5 s( r cos sh( r s ( + + (8 3 (!( + ( + 5 ( + 5 π (4 + π s s 6 3 (33 Usg Maple to verfy the correctess of Eq. (33 as follows: >evalf(doublet(cos(5*theta*cos(r*cos(theta*cosh(r *s(theta+s(5*theta*s(r*cos(theta*sh(r*s(thet a,r3..8,theta*p/3..5*p/6,8; >evalf(sum((-^/((*!* (*+*(*+5* (8^(*+- 3^(*+*(s((*+5*P/6- s((4*+*p/3,. fty, 8; I addto, usg Eq. (3 yelds: 5 π /8 6 3 π /8 s 3 cos( r cos cosh( r s cos3 s( r cos sh( r s + ( + + (6 (!(+ (+ 3 (+ 5 π (6+ 9 π cos cos 8 8 (34 >evalf(doublet(s(3*theta*cos(r*cos(theta*cosh(r *s(theta-cos(3*theta* s(r*cos(theta* sh(r*s (theta, r..6,theta3*p/8..5*p/8,8; >evalf(sum((-^(+/((*!* (*+*(*+3* (6^(*+-^(*+*(cos((*+5*P/8-cos ((6*+9*P/8,. fty, 8; Cocluso I ths paper, we use Taylor seres expasos ad tegrato term by term theorem to solve some types of double tegrals. I fact, the applcatos of the two methods are extesve, ad ca be used to easly solve may dffcult problems; we edeavor to coduct further studes o related applcatos. I addto, Maple also plays a vtal assstve role problem-solvg. I the future, we wll exted the research topc to other calculus ad egeerg mathematcs problems ad use Maple to verfy our aswers. Refereces [] A. A. Adams, H. Gottlebse, S. A. Lto, ad U. Mart, Automated theorem provg support of computer algebra: symbolc defte tegrato as a case study, Proceedgs of the 999 Iteratoal Symposum o Symbolc ad Algebrac Computato, Caada, pp. 53-6, 999. [] M. A. Nyblom, O the evaluato of a defte tegral volvg ested square root fuctos, Rocy Mouta Joural of Mathematcs, Vol. 37, No. 4, pp. 3-34, 7. [3] C. Oster, Lmt of a defte tegral, SIAM Revew, Vol. 33, No., pp. 5-6, 99. [4] C. -H. Yu, A study of two types of defte tegrals wth Maple, Jöull Joural, Vol. 64, No., pp , 4.

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