Solving Some Definite Integrals Using Parseval s Theorem
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1 Ameican Jounal of Numeical Analysis 4 Vol. No Available online at Science and Education Publishing DOI:.69/ajna---5 Solving Some Definite Integals Using Paseval s Theoem Chii-Huei Yu * Depatment of Management and Infomation Nan Jeon Univesity of Science and Technology Tainan City Taiwan *Coesponding autho: chiihuei@nju.edu.tw Received Decembe 9 4; Revised Mach 4; Accepted Mach 3 4 Abstact This aticle takes advantage of the mathematical softwae Maple fo the auiliay tool to study si types of definite integals. The infinite seies foms of these definite integals can be obtained by using Paseval s theoem. In addition we popose some eamples 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 Solving Some Definite Integals Using Paseval s Theoem. Ameican Jounal of Numeical Analysis vol. no. (4): doi:.69/ajna Intoduction 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 si types of definite integals which ae not easy to obtain thei answes using the methods mentioned above. π sinh( cos )cosh( cos ) π sin( sin )cos( sin ) π π sinh( cos )cosh( cos ) sinh ( cos ) π sin( sin )cos( sin ) π () () (3) (4) (5) (6) whee is a 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 Theoems ). The study of elated integal poblems can efe to [-6]. 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 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 comple numbe whee i = a b 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 Suppose f () is a continuous function defined on [ π ] the Fouie seies epansion of f () is a + ( ak cosk + bk sin k) k = whee π π a = f ( ) and a = π k f ( )cosk π π b k = π f ( )sin k fo all positive integes k.
2 Ameican Jounal of Numeical Analysis 6.3. Fomulas Poof.3.. Eule s Fomula e i = cos + i sin whee is any eal numbe..3.. DeMoive s Fomula n (cos + i sin ) = cosn + i sin n whee n is any intege and is any eal numbe ([7]) sinh( p + iq) = sinh pcosq + i cosh psin q p q whee ae eal numbes. (y Fomulas.3.3 and.3.4).3.4. ([7]) cosh( p + iq) = cosh pcosq + isinh psin q p q whee ae eal numbes. And.3.5. Taylo Seies Epansion of Hypebolic Tangent Function ([8]) tanh( z ) = n n ( ) (n)! n z n whee z π is a comple numbe z < and n ae enoulli numbes fo all positive integes n Taylo Seies Epansion of Hypebolic Cotangent Function ([8]) n n coth( z ) = + n z whee z is a z (n)! comple numbe < z < π. Net we intoduce an impotant theoem used in this study..4. Paseval s Theoem ([9]) If f () is a continuous function defined on [ π ] and f ( ) = f (π ). Suppose the Fouie seies epansion of f () is a + ( an cos n + bn sin n) then π a f ( ) = + ( ) π a n + b n. efoe deiving the fist majo esult of this pape we need a lemma. In the following we find the infinite seies foms of the definite integals () () and (3)..6. Theoem Suppose is a eal numbe with < π /. Then the definite integals (9) ().5. Lemma Suppose sinh p q ae eal numbes with p + cos q. Then Poof ecause () (7) (8)
3 6 Ameican Jounal of Numeical Analysis (y Fomula.3.5) = Re n n ( ) (n)! (y DeMoive s fomula) = ( ) (n)! n n n n n (y Eule s fomula) () y Paseval s theoem we obtain n e i(n ) cos(n ). Poof (5) And Similaly because In the following we detemine the infinite seies foms of the definite integals (4) (5) and (6). (y Fomula.3.5) = n n n n ( ) (n)! Also using Paseval s theoem we have sin(n ) (3).8. Theoem Suppose is a eal numbe with definite integals < < π. Then the (6) On the othe hand fom the summation of Eq. (9) and () and using Eq. (8) we obtain (7) efoe deiving the second majo esult of this study we also need a lemma..7. Lemma Suppose sinh p q ae eal numbes with p + sin q. Then Poof ecause (8) (4)
4 Ameican Jounal of Numeical Analysis 63 (y Fomula.3.6) Using Paseval s theoem we have Similaly because (9) () Net we use Maple to veify the coectness of Eq. (). >evalf(int((sinh(/3*cos())*cosh(/3*cos()))^/((sinh (/3*cos()))^+(cos(/3*sin()))^)^=..*Pi)8); >evalf(pi*sum(^(4*n)*(^(*n)- )^*(benoulli(*n))^/((*n)!)^*(/3)^(4*n- )..infinity)8); Similaly if = / in Eq. () we have (y Fomula.3.6) Also by Paseval s theoem we obtain. () () >evalf(int((sin(/sqt()*sin())*cos(/sqt()*sin()))^ /((sinh(/sqt()*cos()))^+(cos(/sqt()*sin()))^)^ =..*Pi)8); >evalf(pi*sum(^(4*n)*(^(*n)- )^*(benoulli(*n))^/((*n)!)^*(/sqt())^(4*n- )..infinity)8); Finally let = 3/ 4 in Eq. () then In addition fom the summation of Eq. (6) and (7) and using (5) we have 3. Eamples In the following fo the definite integals in this study we povide some eamples and use Theoems and to detemine thei infinite seies foms. On the othe hand we employ Maple to calculate the appoimations of these definite integals and thei solutions fo veifying ou answes. 3.. Eample Taking = / 3 into Eq. (9) we obtain the definite integal (3) >evalf(int(((sinh(3/4*cos()))^+(sin(3/4*sin()))^)/(( sinh(3/4*cos()))^+(cos(3/4*sin()))^)=..*pi)8); >evalf(*pi*sum(^(4*n)*(^(*n)- )^*(benoulli(*n))^/((*n)!)^*(3/4)^(4*n- )..infinity)8); Eample Let = 3 in Eq. (6) we obtain the definite integal (4) >evalf(int((sinh(3*cos())*cosh(3*cos()))^/((sinh(3* cos()))^+(sin(3*sin()))^)^=..*pi)8);
5 64 Ameican Jounal of Numeical Analysis >evalf(pi*(6/9+sum(^(4*n)*(benoulli(*n))^/((*n )!)^*3^(4*n-)..infinity))8); In addition if taking = 5 into Eq. (7) then (5) >evalf(int((sin(sqt(5)*sin())*cos(sqt(5)*sin()))^/(( sinh(sqt(5)*cos()))^+(sin(sqt(5)*sin()))^)^=..* Pi)8); >evalf(pi*(4/45+sum(^(4*n)*(benoulli(*n))^/((*n )!)^*(sqt(5))^(4*n-)..infinity))8); On the othe hand let = 3/ 6 in Eq. (8) then (6) >evalf(int(((sinh(3/6*cos()))^+(cos(3/6*sin()))^ )/((sinh(3/6*cos()))^+(sin(3/6*sin()))^)=..*pi) 8); >evalf(*pi*(36/69+69/34)+*pi*sum(^(4*n)*(be noulli(*n))^/((*n)!)^*(3/6)^(4*n- )..infinity)8); Conclusion In this pape we use Paseval s theoem to detemine some types of definite integals. In fact the applications of this theoem ae etensive 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 etend 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 [] M. A. Nyblom On the evaluation of a definite integal involving nested squae oot functions Rocky Mountain Jounal of Mathematics vol. 37 no. 4 pp [] 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 Vancouve Canada 999. [3] C. Oste Limit of a definite integal SIAM Review vol. 33 no. pp [4] C. -H. Yu A study of two types of definite integals with Maple Jökull Jounal vol. 64 no. pp [5] C. -H. Yu Evaluating two types of definite integals using Paseval s theoem Wulfenia Jounal vol. no. pp [6] C. -H. Yu Some types of integal poblems Ameican Jounal of Systems and Softwae vol. no. pp [7] C. -H. Yu and. -H. Chen Solving some types of integals using Maple Univesal Jounal of Computational Mathematics vol. no. 3 pp [8] C. -H. Yu Solving some definite integals by using Maple Wold Jounal of Compute Application and Technology vol. no. 3 pp [9] C. -H. Yu Application of Paseval s theoem on evaluating some definite integals Tukish Jounal of Analysis and Numbe Theoy vol. no. pp [] C. -H. Yu Evaluation of two types of integals using Maple Univesal Jounal of Applied Science vol. no. pp [] C. -H. Yu Studying thee types of integals with Maple Ameican Jounal of Computing Reseach Repositoy vol. no. pp [] C. -H. Yu The application of Paseval s theoem to integal poblems Applied Mathematics and Physics vol. no. pp [3] 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] C. -H. Yu Solving some integals with Maple Intenational Jounal of Reseach in Aeonautical and Mechanical Engineeing vol. issue. 3 pp [5] C. -H. Yu A study on integal poblems by using Maple Intenational Jounal of Advanced Reseach in Compute Science and Softwae Engineeing vol. 3 issue. 7 pp [6] C. -H. Yu Evaluating some integals with Maple Intenational Jounal of Compute Science and Mobile Computing vol. issue. 7 pp [7] C. -H. Yu Application of Maple on evaluation of definite integals Applied Mechanics and Mateials vols pp [8] C. -H. Yu A study of some integal poblems using Maple Mathematics and Statistics vol. no. pp [9] C. -H. Yu Application of Maple on the integal poblems Applied Mechanics and Mateials vols pp [] C.-H. Yu Application of Maple on the integal poblem of some type of ational functions Poceedings of the Annual Meeting and Academic Confeence fo Association of IE D357-D36. [] C. -H. Yu Using Maple to study the integals of tigonometic functions Poceedings of the 6th IEEE/Intenational Confeence on Advanced Infocomm Technology no [] C.-H. Yu Application of Maple on some type of integal poblem Poceedings of the Ubiquitous-Home Confeence pp. 6-. [3] C. -H. Yu A study of the integals of tigonometic functions with Maple Poceedings of the Institute of Industial Enginees Asian Confeence 3 Spinge vol. pp [4] C.-H. Yu Application of Maple on some integal poblems Poceedings of the Intenational Confeence on Safety & Secuity Management and Engineeing Technology pp [5] C.-H. Yu Application of Maple on evaluating the closed foms of two types of integals Poceedings of the 7th Mobile Computing Wokshop ID 6. [6] C.-H. Yu Application of Maple: taking two special integal poblems as eamples Poceedings of the 8th Intenational Confeence on Knowledge Community pp [7] R. V. Chuchill and J. W. own Comple vaiables and applications 4th ed. McGaw-Hill New Yok p [8] Hypebolic functions online available fom [9] D. V. Widde Advanced calculus nd ed. Pentice-Hall New Jesey p
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