Evaluation of Some Non-trivial Integrals from Finite Products and Sums

Transcription

1 Turkish Joural of Aalysis umber Theory 6 Vol. o Available olie at Sciece Educatio Publishig DOI:.69/tjat--6-5 Evaluatio of Some o-trivial Itegrals from Fiite Products Sums F. M. S. Lima * Istitute of Physics Uiversity of Brasilia P.O. Box Brasilia DF Brazil *Correspodig author: fabio@fis.ub.br Received September 3 6; Revised ovember 9 6; Accepted ovember 9 6 Abstract I this ote by maipulatig the sums obtaied from certai fiite products of trigoometric fuctios at ratioal multiples of I put them i the form of Riema sums. By takig the limit as the umber of (equallyspaced subitervals teds to ifiity I have foud exact closed-form results for some o-trivial itegrals e.g. l ( si θ d / / / θ l ( si θ l ( cos θ l ( ta θ dθ. I also show how the method applies l cos l x dx for the prompt evaluatio of more complex itegrals such as Γ ( x dx ( x Γ ( cos ( x l Γ ( x dx si ( x l Γ ( x dx ( x Ψ ( x dx ( k ( si θ Cl θ d θ. si Sice this approach does ot ivolve ay search for primitives it ca be a good alterative to more complex itegratio techiques. Keywords: Products of sies Ratioal multiples of Itegratio techiques Cite This Article: F. M. S. Lima Evaluatio of Some o-trivial Itegrals from Fiite Products Sums. Turkish Joural of Aalysis umber Theory vol. o. 6 (6: doi:.69/tjat Itroductio The cotiuity (eve differetiability of the fuctios l taθ except at some l ( siθ l ( cosθ ( isolated poits suggests that the evaluatio of the itegrals l ( si d / / θ θ l si θ l cos θ ( ( / l ( taθ dθ via the Fudametal Theorem of Calculus should be a straightforward task. However all these itegrals are improper as their itegrs have at least oe ifiite discotiuity i the respective itegratio iterval the correspodig idefiite itegrals caot be solved i a (fiite closed-form i terms of elemetary fuctios oly. Their exact evaluatio usually dems advaced itegratio techiques such as the series expasio of the itegr followed by a term-by-term itegratio (see Sec..9 of Ref. [8] or the evaluatio of a suitable cotour itegral o the complex plae (associated to the Cauchy's residue theorem as see e.g. i Secs... of Ref. []. However these methods preset some disadvatages whe applied to itegrals of `log-trig' fuctios because the expasio of the itegr l siθ i a Taylor series is ot possible for ( l ( taθ. For ( l cosθ though a series expasio exists it is ot easy to fid a closed-form expressio for the geeral term which makes it difficult to recogize the umber it represets. Ideed the evaluatio of a cotour itegral o the complex plae has the icoveieces of requirig the choice of a suitable path of itegratio usually a difficult task results i logarithms of complex umbers o-elemetary fuctios (e.g. dilogarithm elliptic hypergeometric fuctios which ofte makes the fial result obscure. This is just what oe gets with the use of mathematical softwares e.g. Maple (release 5 Mathematica (release which l si x dx : retur the followig stodgy result for ( ( ix x ( ( ix e + i + x x x e i Li l si l where Li ( : z z / z is the dilogarithm fuctio. I this work the itegrals l ( si θ d θ / / / l ( si θ l ( cos θ l ( ta d θ θ will be promptly evaluated from the logarithm of certai products of trigoometric fuctios at ratioal multiples of which yields fiite sums that ca be writte i the form of Riema sums whose limit as the umber of terms teds to ifiity will result i closed-form expressios without ay search for primitives. Iterestigly it will be show that this techique also works for more complex itegrals such as l Γ ( x dx cos ( x l Γ ( x dx ( ( cos x l Γ x dx si ( l ( x Γ x dx si ( x Ψ ( x dx

2 Turkish Joural of Aalysis umber Theory 73 si ( k Cl ( d x θ θ θ where ( : t x t e dt Γ is the d ψ Γ is dx classical Euler's gamma fuctio ( x : l ( x i the digamma fuctio Cl ( θ : I { Li( e θ } θ l si ( ϕ / d ϕ is the Clause itegral [5].. Some Products of Trigoometric Fuctios I Appedix A.3 of Ref. [7] o presetig a elemetary proof for the Euler result / / 6 those authors prove a idetity ivolvig cot θ θ beig a ratioal multiple of. Let us take their proof as our startig poit. Lemma (Product of tagets. For ay iteger ta +. + Proof. The proof i Ref. [7] follows from a compariso of the De Moivre's theorem ( cosθ + isiθ m cos( mθ + isi ( mθ m m the biomial theorem for m m cos( mθ + isi ( mθ si θ( cot θ + i. For completeess let us preset the mai steps. Let F( x be a polyomial equatio with j + j F( x ( x j + j ( ( x θ + cot where θ : /( the siθ >... roots of F( x are just the umbers cot θ. Sice < θ < / which implies that the The well-kow rule for the product of roots of a polyomial equatio the yields ( cot θ (. + + The iverse of both sides reads ta θ + from which the desired result promptly follows. There i Appedix A.3 of Ref. [7] oe also fids the followig idetity. Lemma (Product of sies. For ay iteger > si. Proof. The roots of the polyomial equatio z are the -th roots of uity ω exp( / with.... The Fudametal Theorem of Algebra yields z ( zω valid for all z. Sice ω a divisio by z gives z +... z + z + + z ( z ω. z The limit as z yields ( ω ( ω. ( From Euler's formula for complex expoetials oe has ω cos i si si si i cos. Sice si α ± i cosα α the ω si si si / > for all.... The because ( substitutio of this result i Eq. ( completes the proof. Other similar products of trigoometric fuctios may be easily derived. Lemma 3 (Product of squares of sies cosies. For ay iteger > si cos. Proof. From the symmetry relatio si ( / + α si ( / α it follows from Lemma that ( si si si si si valid for all > from which the product of squares of sies promptly follows. The product of squares of cosies follows from the substitutio α / i the trigoometric idetity siα cos ( / α α [ ] 3. Evaluatio of Defiite Itegrals /. The geeral idea uderlyig my method is to rewrite a kow fiite product or sum i the form of a Riema

3 7 Turkish Joural of Aalysis umber Theory sum with equally-spaced subitervals whose limit as the umber of terms teds to ifiity is a defiite itegral. We begi by applyig this procedure to the products of trigoometric fuctios established i the previous sectio. Theorem (Some `log-trig' itegrals. The followig exact closed-form results hold: l siθ d θ l ( / / l siθdθ l l cos θ (3 / l taθdθ. ( Proof. By takig the logarithm of each side of the product foud i Lemma oe has l si l ( l. (5 This implies that l si l l. (6 This ca be rewritte as l si ( x x l (7 l where x / x /. Clearly the sum at the left-h side has the form of a Riema sum i which the grid poits x are equally spaced by x. By takig the limit as o both sides otig that lim l / as follows from l'hopital's rule oe fids which meas that lim l si l ( x x (8 l si ( l. x dx (9 The chage of variable θ x leads to the first itegral. Whe the above procedure is applied to the product of square of sies i Lemma 3 oe fids / l l si ( x x l. The limit as leads to ( l si x dx l which is equivalet to the secod itegral. Similarly the product of squares of cosies i Lemma 3 yields This result ca also be foud by applyig some trick substitutios as doe i Sec..5 of Ref. []. / l cos x l l. x The limit as yields ( l which is equivalet to the other (cosie itegral. Fially the product of tagets i Lemma leads to l cos / x dx l ta l + + ( By substitutig + M (hece M > is a odd iteger the dividig both sides by M oe has ( M / l M l ta ( x x M where x / M x / M. The limit as M yields ( M / l M lim l ta ( x x lim M M M which implies that ( l ta x dx. Theorem (A less obvious `log-trig' itegral. The exact closed-form result l si ( x + θ dx l holds for all real values of θ. Proof. This closed-form evaluatio follows from the fiite product ( θ si si + θ ( valid for all positive iteger all θ. This trigoometric idetity is proposed as a exercise i Appedix A.3 of Ref. [7] (see its Ex. 6. For all θ such that si ( θ the logarithmic versio of this product reads l si + θ ( θ ( l si l l si θ. A divisio by yields l si ( x θ ( θ + x l si l si θ l. ( ( The limit as leads to the desired result. The above procedure also works for evaluatig more complex itegrals.

4 Turkish Joural of Aalysis umber Theory 75 Theorem 3 (A `log-gamma' itegral. The exact closedform result holds. Proof. By addig ( fids l ( l Γ x dx ( l to both sides of Eq. (5 oe l l si l l + l ( ( which promptly simplifies to l ( l ( l (3 si ( x where x /. O takig ito accout the reflectio property of the gamma fuctio i.e. Γ x. Γ x / si x valid for all x ( ( ( dividig both sides by oe fids ( ( l Γ x + l Γ x l l (. The limit as the yields l ( l ( l (. Γ x + Γ x dx ( l x dx Sice this itegral exps to Γ ( + ( l Γ x dx the substitutio y x i the latter itegral reduces Eq. ( to l Γ ( x dx l (. Theorem (Some `impossible' itegrals. The exact closed-form results si ( x Ψ ( x dx (5 cos( x l Γ ( x dx (6 hold. Proof. From Eq. (6. of Ref. [] a recet paper by Coo amely q j j p si p Ψ q (7 j q q q oe easily shows the first itegral result by reducig it to a Riema sum the proceedig as i the above proofs. Similarly from Eq. (6. of Ref. [] amely q j j cos p l Γ j q q p p Ψ +Ψ γ + l ( q q q (8 This ice result is stamped o the cover of the book Irresistible itegrals []. the secod itegral readily follows. The itegral i Eq. (6 above ca be further maipulated i order to geerate two other iterestig itegrals. Theorem 5 (Further log-gamma itegrals. The exact closed-form results si ( x l Γ ( x dx l ( 8 cos ( x l Γ ( x dx l ( + 8 hold. Proof. O applyig the double-agle formula ( cos θ cos θ si θ to Eq. (6 it follows that cos ( x l Γ( x dx si ( x l Γ ( x dx. (9 This implies that l Γ( x dx si ( x l Γ ( x dx. ( From Theorem 3 the first itegral follows. The secod itegral follows from the idetity ( x ( x si cos. Theorem 6 (A itegral ivolvig the Clause fuctio. The exact closed-form result si ( kθ Cl ( θ dθ k holds for all itegers k >. Proof. I Ref. [6] akamura gives a simple proof of 3 k ζ m si ( k / m ( m / + m si k Cl m m ( s where ζ ( sx : / ( + x is the Hurwitz-zeta fuctio k > is a iteger. O dividig both sides by m takig the limit as m oe fids ζ ( k / m si ( k / m lim lim m m / m / m / + si ( kx Cl ( x dx. ( Simple applicatios of the l'hopital's rule together with the substitutio θ x reduce Eq. ( to k k + si ( kθ Cl ( θ (3 from which the desired result promptly follows. Iterestigly the defiite itegral i Theorem 6 has the typical form of Fourier coefficiets which has led me to ivestigate the correspodig cosie itegrals i.e. 3 Typo corrected i the deomiator of the first term of his Eq. (. The Clause fuctio has a well-kow trigoometric expasio i.e. ( θ ( θ Cl si / foud by Clause himself [3].

5 76 Turkish Joural of Aalysis umber Theory cos( Cl ( k θ θ d θ ( which were ot explored i literature. After some umerical ivestigatios for small values of k I have arrived at some cojectures amely cosθcl ( θ d θ l (5 cos( θ Cl ( θ dθ (6 l 7 cos( 3θ Cl ( θ dθ (7 9 7 cos( θ Cl ( θ dθ (8 l 3 cos( 5θ Cl ( θ dθ ( cos( 6θ Cl ( θ dθ (3 7 all accurate to a thous of decimal places. Apparetly for odd values of k oe has l / k p/ q p q beig positive itegers whereas for eve values of k oe has egative ratioal values. The determiatio of a geeral patter seems to deped o the closed-form evaluatio of Eq. ( which seems to be a ope problem. Of course a proof for ay of the above cojectures would be valuable for this lie of research. Refereces [] M. J. Ablowitz A. S. Fokas Complex Variables (d ed. Cambridge Uiversity Press ewyork 3. [] G. Boros V. H. Moll Irresistible Itegrals Cambridge Uiversity Press ew York. [3] T. Clause Über die Fuctio siφ+(/ siφ+(/3 si3φ+etc. J. Reie Agewte Mathematik (83. [] D. F. Coo Determiatio of the Stieltjes costats at ratioal argumets. Available at arxiv: v (5. [5] L. Lewi Structural properties of polylogarithms America Mathematical Society Providece 99. [6] T. akamura Some formulas related to Hurwitz-Lerch zeta fuctios Ramauja J (. [7] I. ive H. S. Zuckerma H. L. Motgomery A Itroductio to the Theory of umbers (5 th ed. Wiley ew York 99. [8] J. Stewart Calculus (7th ed. Brooks/Cole Belmot (USA.