Application of Parseval s Theorem on Evaluating Some Definite Integrals

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1 Tukish Jounal of Analysis and Numbe Theoy, 4, Vol., No., -5 Available online at Science and Education Publishing DOI:.69/tjant--- Application of Paseval s Theoem on Evaluating Some Definite Integals Chii-Huei Yu * Depatment of Management and Infomation, Nan Jeon Univesity of Science and Technology, Tainan City, Taiwan *Coesponding autho: chiihuei@nju.edu.tw Abstact This pape uses the mathematical softwae Maple fo the auxiliay tool to study s types of definite integals. We can detemine the infinite seies foms of these definite integals by using Paseval s theoem. On the othe hand, we povide some definite integals to do calculation pactically. The eseach methods adopted in this study involved finding solutions though manual calculations and veifying these solutions using Maple. Keywods: definite integals, infinite seies foms, Paseval s theoem, Maple Cite This Aticle: Chii-Huei Yu, Application of Paseval s Theoem on Evaluating Some Definite Integals. Tukish Jounal of Analysis and Numbe Theoy, no. (4): -5. doi:.69/tjant---.. Intoduction As infomation technology advances, whethe computes can become compaable with human bains to pefom abstact tasks, such as abstact at simila to the paintings of Picasso and musical compositions simila to those of Mozat, is a natual uestion. Cuently, this appeas unattainable. In addition, whethe computes can solve abstact and difficult mathematical poblems and develop abstact mathematical theoies such as those of mathematicians also appeas unfeasible. Nevetheless, in seeking fo altenatives, we can study what assistance mathematical softwae can povide. This study intoduces how to conduct mathematical eseach using the mathematical softwae Maple. The main easons of using Maple in this study ae its simple instuctions and ease of use, which enable beginnes to lean the opeating techniues in a shot peiod. By employing the poweful computing capabilities of Maple, difficult poblems can be easily solved. Even when Maple cannot detemine the solution, poblem-solving hints can be identified and infeed fom the appoximate values calculated and solutions to simila poblems, as detemined by Maple. Fo this eason, Maple can povide insights into scientific eseach. In calculus and engineeing mathematics couses, we leant many methods to solve the integal poblems including change of vaiables method, integation by pats method, patial factions method, tigonometic substitution method, and so on. In this pape, we study the following s types of definite integals which ae not easy to obtain thei answes using the methods mentioned above. sinh ( cos x ) cos ( sin x ) () cosh ( cos x ) sin ( sin x ) () [sinh ( cos x ) + sin ( sin x )] (3) cosh ( cos x ) cos ( sin x ) (4) sinh ( cos x ) sin ( sin x ) (5) [cosh ( cos x ) + cos ( sin x )] (6) whee is any eal numbe. We can obtain the infinite seies foms of these definite integals by using Paseval s theoem; these ae the majo esults of this pape (i.e., Theoems and ). As fo the study of elated integal poblems can efe to [-7]. On the othe hand, we popose some definite integals to do calculation pactically. The eseach methods adopted in this study involved finding solutions though manual calculations and veifying these solutions by using Maple. This type of eseach method not only allows the discovey of calculation eos, but also helps modify the oiginal diections of thinking fom manual and Maple calculations. Fo this eason, Maple povides insights and guidance egading poblem-solving methods.. Main Results Fistly, we intoduce a notation and a definition and some fomulas used in this aticle... Notation Let z = a + ib be a complex numbe, whee i =, ab, ae eal numbes. We denote a the eal pat of z by Re( z ), and b the imaginay pat of z by Im( z )... Definition

2 Tukish Jounal of Analysis and Numbe Theoy Suppose f( x ) is a continuous function defined on [, ], then the Fouie seies expansion of f( x ) is a + ( ak cos kx + bk sin kx), whee a = f ( x), and ak = f ( x)cos kx, bk = f ( x)sin kx fo all positive integes k..3. Fomulas.3.. Eule s Fomula e = cos x+ isin x, whee x is any eal numbe..3.. DeMoive s Fomula n (cos x + i sin x) = cos nx + i sin nx, whee n is any intege, and x is any eal numbe Taylo Seies Expansion of Hypebolic Sine Function ([8]) k+ z sinh( z) =, whee z is any complex numbe. (k + )!.3.4. Taylo Seies Expansion of Hypebolic Cosine Function ([8]) k z cosh( z) =, whee z is any complex numbe. ( k)! Next, we intoduce an impotant theoem used in this study..4. Paseval s Theoem ([9]) If f( x ) is a continuous function defined on [, ], and f() = f( ). If the Fouie seies expansion of f( x ) is a + ( ak cos kx + bk sin kx), then a f ( x) ( ak bk ) = + +. Befoe deiving the fist majo esult of this pape, we need a lemma..5. Lemma Suppose p, ae any eal numbes. Then sinh( p+ i) = sinh p cos + icosh p sin (7) sinh p cos + cosh p sin = sinh p+ sin (8) Poof sinh( p + i) [ p+ i ( p+ i) = e e ] [ p (cos sin ) p = e + i e (cos i sin )] p p p p = ( e e )cos + i ( e + e )sin = sinh p cos + icosh p sin And sinh p cos + cosh p sin = sinh p ( sin ) + cosh p sin = sinh p+ sin Next, we detemine the infinite seies foms of the definite integals (), () and (3)..6. Theoem Suppose is any eal numbe. Then the definite integals sinh ( cos x) cos ( sin x) + [(k + )!] cosh ( cos x) sin ( sin x) + [(k + )!] [sinh ( cos x) + sin ( sin x)] + = [(k + )!] Poof Because (9) () () sinh( cos x) cos( sin x) = Re[sinh( e )] ( By (7)) k+ ( e ) = Re ( By Fomula.3.3.) = (k + )! k+ i(k+ ) x e = Re ( By DeMoive ' s fomula) = (k + )! k+ = cos(k + ) x ( By Eule ' s fomula) (k + )! () sinh ( cos x) cos ( sin x) + [(k + )!] (Using () and Paseval s theoem) Similaly, because cosh( cos x) sin( sin x) = Im[sinh( e )] ( By (7)) k+ ( e ) = Im = (k + )!

3 Tukish Jounal of Analysis and Numbe Theoy 3 k+ = sin(k+ ) x (3) (k + )! cosh ( cos x) sin ( sin x) + [(k + )!] (Using (3) and Paseval s theoem) On the othe hand, fom the summation of (9) and () and using (8), we obtain [sinh ( cos x) + sin ( sin x)] + = [(k + )!] Befoe deiving the second majo esult of this study, we also need a lemma..7. Lemma Suppose p, ae any eal numbes. Then cosh( p+ i) = cosh p cos + isinh p sin (4) cosh p cos + sinh p sin = sinh p+ cos (5) Poof cosh( p + i) [ p+ i ( p+ i) ] = e + e [ p (cos sin ) p = e + i + e (cos i sin )] p p p p = ( e + e )cos + i ( e e )sin = cosh p cos + isinh p sin And cosh p cos + sinh p sin = ( + sinh p) cos + sinh p sin = sinh p+ cos Finally, we find the infinite seies foms of the definite integals (4), (5) and (6)..8. Theoem Suppose is any eal numbe. Then the definite integals cosh ( cos x) cos ( sin x) = + [( k)!] sinh ( cos x) sin ( sin x) = [( k)!] (6) (7) Poof Because [sinh ( cos x) + cos ( sin x)] = + [( k)!] cosh( cos x) cos( sin x) = Re[cosh( e )] ( By (4)) k ( e ) = Re ( By Fomula.3.4.) = ( k)! k = + cos kx ( k)! ( By DeMoive ' s fomula and Eule ' s fomula) Using (9) and Paseval s theoem, we have Similaly, because k cosh ( cos x) cos ( sin x) = + [( k)!] sinh( cos x) sin( sin x) = Im[cosh( e )] ( By (4)) k ( e ) = Im = ( k)! = sin kx ( k)! sinh ( cos x) sin ( sin x) = [( k)!] (8) (9) () (By () and Paseval s theoem) Fom the summation of (6) and (7) and using (5), we have [sinh ( cos x) + cos ( sin x)] = + [( k)!] 3. Examples In the following, fo the s types of definite integals in this study, we povide some definite integals and use Theoems and to detemine thei infinite seies foms. On the othe hand, we employ Maple to calculate the appoximations of these definite integals and thei solutions fo veifying ou answes. 3.. Example Taking = 7 into (9), we obtain the definite integal

4 4 Tukish Jounal of Analysis and Numbe Theoy sinh (7 cos x) cos (7 sin x) + 7 [(k + )!] () Next, we use Maple to veify the coectness of (). >evalf(int((sinh(7*cos(x)))^*(cos(7*sin(x)))^,x=..*pi ),8); >evalf(pi*sum(7^(4*k+)/((*k+)!)^,..infinity),8); Also, let = 3 in (), we have cosh ( 3 cos x) sin ( 3 sin x) + ( 3) [(k + )!] () >evalf(int((cosh(st(3)*cos(x)))^*(sin(st(3)*sin(x)))^,x=..*pi),8); >evalf(pi*sum((st(3))^(4*k+)/((*k+)!)^,.. infinity),8); Finally, if = 53in (), then = 5 5 sinh cos x sin sin x (5 3) 4k+ [(k + )!] (3) >evalf(int((sinh(5/3*cos(x)))^+(sin(5/3*sin(x)))^,x=.. *Pi),8); >evalf(*pi*sum((5/3)^(4*k+)/((*k+)!)^,.. infinity),8); 3.. Example Taking = 9 into (6), then the definite integal cosh (9 cos x) cos (9sin x) 9 = + [( k)!] (4) >evalf(int((cosh(9*cos(x)))^*(cos(9*sin(x)))^,x=..*pi ),8); >evalf(*pi+pi*sum(9^(4*k)/((*k)!)^,..infinity),8); In addition, let = in (7), then sinh ( cos x) sin ( sin x) ( ) = [( k)!] (5) >evalf(int((sinh(st()*cos(x)))^*(sin(st()*sin(x))) ^,x=..*pi),8); >evalf(pi*sum((st())^(4*k)/((*k)!)^,..infinity), 8); Finally, if = 7 in (8), we have sinh cos x cos sin x = + ( 7) 4k [( k)!] (6) >evalf(int((sinh(/7*cos(x)))^+(cos(/7*sin(x)))^,x=..*pi),8); >evalf(*pi+*pi*sum((/7)^(4*k)/((*k)!)^,.. infinity),8); 4. Conclusion In this pape, we povide a new techniue to detemine some definite integals. We hope this techniue can be applied to solve anothe definite integal poblems. On the othe hand, the Paseval s theoem plays a significant ole in the theoetical infeences of this study. In fact, the applications of this theoem ae extensive, and can be used to easily solve many difficult poblems; we endeavo to conduct futhe studies on elated applications. In addition, Maple also plays a vital assistive ole in poblem-solving. In the futue, we will extend the eseach topic to othe calculus and engineeing mathematics poblems and solve these poblems by using Maple. These esults will be used as teaching mateials fo Maple on education and eseach to enhance the connotations of calculus and engineeing mathematics. Refeences [] C. Oste, Limit of a definite integal, SIAM Review, vol. 33, no., pp. 5-6, 99. [] A. A. Adams, H. Gottliebsen, S. A. Linton, and U. Matin, Automated theoem poving in suppot of compute algeba: symbolic definite integation as a case study, Poceedings of the 999 Intenational Symposium on Symbolic and Algebaic Computation, pp. 53-6, Canada, 999. [3] M. A. Nyblom, On the evaluation of a definite integal involving nested suae oot functions, Rocky Mountain Jounal of Mathematics, vol. 37, no. 4, pp. 3-34, 7. [4] C. -H. Yu, Using Maple to study two types of integals, Intenational Jounal of Reseach in Compute Applications and Robotics, vol., issue. 4, pp. 4-, 3. [5] C. -H. Yu, Solving some integals with Maple, Intenational Jounal of Reseach in Aeonautical and Mechanical Engineeing, vol., issue. 3, pp. 9-35, 3.

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