International Journal of Pure and Applied Mathematics Volume 17 No , EVALUATIONS OF THE IMPROPER INTEGRALS. sin 2m+1 (αx) x 2n+1

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1 Internatonal Journal of Pure and Appled Mathematcs Volume 17 No. 3 4, EVALUATIONS OF THE IMPROPER INTEGRALS sn m (α) AND n Qu-Mng Luo sn m+1 (α) Department of Mathematcs Jaozuo Unversty Jaozuo Cty, Henan, 4543, P.R. CHINA e-mal: luoqm@jzu.edu.cn Abstract: In ths artcle, usng the l Hosptal rule, mathematcal nducton, the trgonometrc power formulae and ntegraton by parts, some ntegral formulae for the mproper ntegrals sn m (α) and sn m+1 (α) n are establshed, where m n are all postve ntegers and α. AMS Subject Classfcaton: 6A4, 6A6, 3E Key Words: evaluaton, mproper ntegral, ntegral formula, nequalty, ntegraton by parts, l Hosptal rule, Drchlet ntegral, mathematcal nducton 1. Introducton The followng mproper ntegral s well-known and s synonymous wth names of Laplace and Drchlet sn = π. (1) In fact, n 1781, t was frst obtaned usng the resdue method by Euler. It can be found n standard tetbooks for undergraduate students, for eamples [9, pp. 6 7] and [13, pp ]. Receved: June 3, 4 c 4, Academc Publcatons Ltd.

2 9 Q.-M. Luo Dependng on the partal fracton decomposton 1 sn t = 1 t + ( 1 ( 1) t nπ + 1 ), () t + nπ =1 an elegant calculaton of formula (1) s provded n [5, pp ] and [6, pp ], due to the noted geometrcan N.I. Lobatscheusk. Another polshed proof of dentty (1) s gven n [6, pp ]. As eercses n [1, p. 53, p. 147 and p. 335] and [1, p. 495], usng the Laplace transform, the Parseval denttes of sne and cosne Fourer transforms and the resdue theorem, the followng formulae are requested to compute: sn t t dt = π, sn 4 t t dt = π, sn 3 t t 3 dt = 3π 8. (3) In [14, pp and p. 84], usng the Melln transform and by approaches n theory of Fourer analyss or theory of resdues, the followng formulae are obtaned: cos(t) z = Γ(z)t z cos πz, Re(z) >, t > ; (4) sn(t) z = Γ(z)t z sn πz, Re(z) > 1, t > ; (5) sn z = Γ(1 z)cos πz = π Γ(z)sn πz. (6) Especally, takng t = 1 and z n (5) or takng z = 1 n (6) produces (1). The followng generalsaton of formula (1) can be found n [] and [3, p. 458, No ]: π ( ) sn n cos(b) [r] ( ) n = n( n 1 n!) 1 ( 1) k (n b k) n 1, (7) k where b < n, n 1, r = n b, and [r] s the largest nteger contaned n r. In [11], some general results related to formulae (1) and (7) were obtaned. In [1] and [7, p. 66], the followng nequalty s gven: snt p t dt π, p ; (8) p k=

3 EVALUATIONS OF THE IMPROPER INTEGRALS Equalty s vald only f p =. The ntegral (1) and other ntegral formulae stated above are useful and arsng n research of dampng vbraton and other scence or engneerng. Ths was mentoned n [13, p. 17]. Recently, Q.-M. Luo and B.-N. Guo n [8] obtaned the followng theorem. Theorem A. (see [8]) For k N and α, we have ( ) sn(α) k+1 = sgn α k = ( 1) (k + 1) k Ck+1 4 k α k π (k)!, (9) ( ) sn(α) k = sgn α k 1 = ( 1) (k ) k 1 Ck α k 1 π (k 1)!. (1) If takng k = 1 n (9), the formula (1) follows. In ths artcle, usng the l Hosptal rule, mathematcal nducton, trgonometrc power formulae and ntegraton by parts, we wll establsh ntegral formulae of the mproper ntegrals m (α) sn and n sn m+1 (α), where m n are all postve ntegers and α. The followng theorem holds. Theorem 1. Let m,n N, m n, and α. Then sn m+1 (α) = ( 1)m+n sgn α m = ( 1) (m + 1) n Cm+1 4 m α n π (n)!, (11) sn m (α) n = ( 1)m+n sgn α m 1 = ( 1) (m ) n 1 Cm 4 m n α n 1 π (n 1)!. (1) As a drect consequence of Theorem 1, the followng ntegral formula holds.

4 94 Q.-M. Luo Corollary 1. Let m be a nonnegatve nteger, then we have sn m+1 (α) = sgnα (m)! 4 m (m!) π. (13) It s obvous that Theorem 1 generalzes the formula (1), Theorem A and other results above.. Some Lemmas The followng trgonometrc power formulae are the bass and key of our proof for Theorem 1. Lemma 1. ([4, p. 41 and p. 8] and [15]) For α > and k N, we have sn k+1 = 1 sn k = 1 k 1 k = [ k 1 sn(α) = π, (14) k ( 1) k+ Ck+1 sn[(k + 1)], (15) ] ( 1) k+ Ck cos[(k )] + 1 Ck k, (16) = where Cn k = ( n) k = n! (n k)!k!. The followng three combnatoral denttes can be regarded as by-products, enablng us to employ the l Hosptal rule n the proof of Theorem 1. They can also be found n [8]. Lemma. For 1 m k and k N, we have k ( 1) (k + 1) m 1 Ck+1 =, (17) = k 1 ( 1) k+ Ck + 1 Ck k =. (18) = For 1 l k 1 and k N, we have k 1 ( 1) (k ) l Ck =. (19) =

5 EVALUATIONS OF THE IMPROPER INTEGRALS Proof. By the trgonometrc power formula (15), t s easy to see that lm k = ( 1)k+ Ck+1 sn[(k + 1)] k = k sn k+1 lm k =, () thus k ( 1) Ck+1 sn[(k + 1)] = o(k ) as, (1) = then, for j k, ( k ) (j) ( 1) Ck+1 sn[(k + 1)] = o( k j ) as, () = therefore, for any natural number 1 m k, we have ( k ) (m 1) = lm ( 1) Ck+1 sn[(k + 1)] = ( ) k = lm ( 1) m 1 ( 1) (k + 1) m 1 Ck+1 cos[(k + 1)] = k = ( 1) m 1 ( 1) (k + 1) m 1 Ck+1. (3) Identty (17) follows. By the trgonometrc power formula (16), t s not dffcult to obtan = lm k 1 = ( 1)k+ C k cos[(k )] + 1 Ck k k 1 = k 1 lm sn k =, (4) k 1 hence k 1 ( 1) k+ Ck cos[(k )] + 1 Ck k = o(k 1 ) as, (5) =

6 96 Q.-M. Luo consequently { k 1 = lm ( 1) k+ Ck cos[(k )] + 1 } Ck k = and, for 1 j k 1, k 1 = ( 1) k+ Ck + 1 Ck k, (6) = ( k 1 (j) ( 1) k+ Ck cos[(k )] + 1 k) Ck = o( k j 1 ) = as, (7) then, for any postve nteger l such that 1 l k 1, we have = lm ( k 1 ( 1) k+ Ck cos[(k )] + 1 Ck k = = lm ( ( 1) l k 1 ) (l) ) ( 1) (k ) l Ck cos[(k )] = k 1 = ( 1) l ( 1) (k ) l 1 Ck. (8) = Identtes (18) and (19) follow. The proof s complete. 3. Proof of Theorem 1 Let t = α, then, by straghtforward computaton, we have sn r (α) s = α s 1 sgn α sn r t t s dt. (9) From Lemma 1 and Lemma, usng the l Hosptal rule and ntegraton by parts yelds sn m+1 = 1 m = ( 1)m+ Cm+1 sn[(m + 1)] m

7 EVALUATIONS OF THE IMPROPER INTEGRALS = ( 1)j 1 (n j)! m (n)! { m = ( 1)m+ (m + 1) j 1 Cm+1 sn [ (m + 1) + (j 1)π ] n j+1 m = ( 1)m+ (m + 1) j Cm+1 sn[ (m + 1) + jπ ] } n j+1 = ( 1)n m (n)! = ( 1)m+n m (n)! m = ( 1)m+ (m + 1) n Cm+1 sn[(m + 1)] m ( 1) (m + 1) n Cm+1 = sn[(m + 1)] = ( 1)m+n m = ( 1) (m + 1) n Cm+1 m π (n)!, where 1 j n. Combnng wth (9), the formula (11) follows. By formula (16), usng the l Hosptal rule, from Lemma, ntegraton by parts gves us sn m n = 1 m 1 = ( 1)m+ C m cos[(m )] + 1 Cm m m 1 n = 1 [ m 1 1 m 1 ] ( 1) m+ Cm cos[(m )] + 1 ( 1 ) n 1 Cm m d n 1 = [ m 1 1 = = ( 1)m+ Cm cos[(m )] + 1 Cm m (m 1) n 1 n 1 ] m 1 = + ( 1)m+ [(m )]Cm sn[(m )] n 1 1 = (m 1) n 1 (n j )! = m 1 (n 1)! = ( 1) n m 1 (n 1)! = m 1 = ( 1)m+ [(m )]C m sn[(m )] n 1 { m 1 = ( 1)m+ [(m )] j C m sn[ (m )] + (j 1)π n j 1 m 1 = ( 1)m+ [(m )] j+1 C m sn [ (m )] + jπ ( 1) n m 1 (n 1)! m 1 n j 1 ] ] } m 1 = ( 1)m+ [(m )] n 1 Cm sn[(m )] ( 1) m+ [(m )] n 1 Cm = sn[(m )]

8 98 Q.-M. Luo = ( 1)m+n m 1 = ( 1) (m ) n 1 Cm π (m n) (n 1)!, where 1 j (n 1). On combnng wth (9), the second formula (1) follows. The proof of Theorem 1 s thus complete. Acknowledgements The author s supported n part by NNSF (#1116) of P.R. Chna, SF for the Promnent Youth of Henan Provnce, SF of Henan Innovaton Talents at Unverstes, NSF of Henan Provnce (#4518), SF for Pure Research of Natural Scence of the Educaton Department of Henan Provnce (# ), Doctor Fund of Jaozuo Insttute of Technology, P.R. Chna. References [1] Amer. Math. Monthly, 97, No. 8 (199), 663. [] E.T. Erdély, et. al., Tables of Integral Transforms, Vol. I and II, McGraw Hll, New York (1954). [3] I.S. Gradshteyn, I.M. Ryzhk, Table of Integrals, Sres, and Products, Academc Press (198). [4] Group of Complaton, Handbook of Mathematcs, The People s Educaton Press, Bejng, Chna (1979), In Chnese. [5] G. Klambauer, Mathematcal Analyss, Chnese Edton, Translated by Ben-Wang Sun, The People s Press of Hunan, Changsha Cty, Hunan, Chna (1981); Englsh Edton, Marcel Dekker, Inc., New York (1975). [6] G. Klambauer, Problems and Propostons n Analyss, Marcel Dekker, Inc., New York and Basel (1979). [7] J.-Ch. Kuang, Appled Inequaltes, Second Edton, Hunan Educaton Press, Changsha, Chna (1993). In Chnese. [8] Q.-M. Luo, B.-N. Guo, Evaluatons of a class of the frst knd of the mproper ntegrals, Math. Gaz., 87 (3), In Press.

9 EVALUATIONS OF THE IMPROPER INTEGRALS [9] W. Rudn, Real and Comple Analyss, Thrd Edton, McGraw-Hll Book Company (1987); Reprnted n The People s Republc of Chna by World Publshng Corporaton (199). [1] J.-R. She, Z.-Sh. Lu, Applcatons of Integral Transform to Engneerng, Anhu Educaton Press, Hefe Cty, Chna, In Chnese. [11] A. Sofo, On the ntegral No. 6 (1998), (sn )p+q q, Internat. J. Math. Ed. Sc. Tech., 9, [1] M. R. Spegel, Prncples of Advanced Calculus and Solutons to Problems, Schaum Publshng Corporaton (1963). Translated by H.-Zh. L, Xaoyuan Press and World Publshng Corporaton, Tawan and Bejng, Chna (1993), In Chnese. [13] Staff Room of Hgher Mathematcs at X an Jaotong Unversty, Comple Functons, Fourth edton, Hgher Educaton Press, Bejng, Chna (1), In Chnese. [14] L. Tan, Readng Notes on the Gamma Functon, Zhejang Unversty Press, Hangzhou Cty, Zhejang, Chna (1997), In Chnese. [15] E.W. Wessten, CRC Concse Encyclopeda of Mathematcs on CD-ROM (1999); Avalable onlne at [16] Qu-Mng Luo, B.-N. Guo, Feng Q, Evaluaton of a class of mproper ntegrals of the frst knd, Math. Gaz., 87 (3), [17] Qu-Mng Luo, ) Feng Q, Evaluaton of a class of mproper ntegrals n cos(b), Octogon Math. Mag., 11, No. 1 (3), ( sn (β) [18] Qu-Mng Luo, Feng Q, Evaluatons of the mproper ntegrals sn m (α) cos(b) and sn m+1 (α) n cos(b), Austral. Math. Soc. Gaz., 3 (3),

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