Some integrals related to the Basel problem

Transcription

1 November, 6 Some itegrals related to the Basel problem Khristo N Boyadzhiev Departmet of Mathematics ad Statistics, Ohio Norther Uiversity, Ada, OH 458, USA k-boyadzhiev@ouedu Abstract We evaluate several arctaget ad logarithmic itegrals depedig o a parameter This provides a closed form summatio of certai series ad also itegral ad series represetatio of some classical costats Key words ad phrases, Basel problem, arcta itegrals, dilogarithm, trilogarithm, Catala costat Mathematics Subject Classificatio Primary M6; 3B3; 4A5 Itroductio The famous Basel problem posed by Pietro Megoli i 644 ad solved by Euler i 735 asked for a closed form evaluatio of the series 3 (see [7], [9], []) Euler proved that () 3 6 I the meatime, tryig to evaluate this series, Leibiz discovered the represetatio

2 log ( t) dt, t 3 but was uable to fid the umerical value of the itegral (see commets i [7]) How to relate this itegral to /6 is discussed i [7] ad []; it is show that by usig the comple logarithm oe ca solve the Basel problem There eist, however, other itegrals which ca be used to quickly prove () without comple umbers Possibly, the best eample is the itegral () for Here arcsi d Li ( ) Li ( ) Li ( ) is the dilogarithm [] Settig i (), the left had side becomes while the RHS is arcsi 8 Li () Li ( ) 3 Li 3 () ( ) ad () follows immediately This proof was recetly published by Habib Bi Muzaffar i [3] It is possibly oe of the best solutios to the Basel problem Euler also used the arcsi fuctio i oe of his proofs Euler s approach is eplaied o pp i [5] I this short paper we follow the idea from [3] ad cosider some other itegrals that ca be associated to the Basel problem, either solvig it, or leadig to similar results I the process we evaluate a umber of itegrals from the tables [8] ad [4] We start with three arctaget itegrals i Sectio, ad the we also discuss two logarithmic itegrals Amog other thigs, i sectio we fid the curious represetatio (equatio (8) below) 3 ( )

3 I Sectio 3 we focus o several itegral ad series represetatio of some classical costats (see equatios (5) ad (7) below) I particular, we list two itegral represetatios of (3) ad evaluate oe special arctaget itegral (see (6) ad (9) below) Special itegrals with arctagets ad logarithms At the ed of [3] it was metioed that istead of arcsi ( ) i () oe could use arcta( ) However, as we shall see here, itegrals with arctagets are ot so simple Two atural cadidates for the described method are clear (3) arcta arcta d, ad d Whe, these itegrals evaluate to a multiple of ad i order to prove () we eed to evaluate them also as a multiple of the series Li () Here is how the fist oe works Propositio For ay, arcta (4) log log Li d ( ) Li ( ) It easy to see (by usig limits) that the RHS eteds to ad The fuctio log log becomes zero for ad With we compute immediately 3 4 (arcta ) Li () ad hece Li () 6 Proof Let J( ) be the LHS i (4) By differetiatio (for ) J( ) d d ( )( ) log log 3

4 Thus, sice J( ) is defied for J ( ) Itegratig by parts we fid log t dt t log( t) log( t) J( ) loglog dt t t log log Li ( ) Li ( ) This itegral was recetly evaluated i [] It is missig from the popular table [8], but appears i [4] as etry 74 () However, it appears there i a differet form arcta d 3 log ( ) Li Li ( ) which is less helpful for provig () Now we look at the secod itegral i (3) Although it does ot lead directly to the proof of (), it provides the closed form evaluatio of oe iterestig series Propositio For every we have arcta d, (5) log H where ( ) H,,,; H, 3 are the skew-harmoic umbers (see [4]) I particular, for we fid from (5) 6 (6) log H Note that H are the partial sums i the epasio of log ad the series i (6) is alteratig Proof Usig the Taylor series for arcta ( ) we write 4

5 arcta ( ) d d Etry 34() i [8] says that ( ) d d ( ), where ( ) is the icomplete beta fuctio (see 837, pp i [8] ad also [5]) Accordig to equatio 8375 () i [8] we have ( ) ( ) log H Puttig all these pieces together we arrive at (5) The proof is completed It would be iterestig to see what happes whe i Propositio we replace arcta by (arcta ), which leads to With the otatio we have 3 o the LHS h (,), h 3 5 Propositio 3 For every, (7) (arcta ) h ( ) d 3 4 I particular, whe, we have the curious represetatio (8) 3 h ( ) Proof Usig the Taylor series for, 3 5 ( ) arcta, 3 5 5

6 it easy to compute the epasio ( ) (arcta ) h Therefore, accordig to etry 34() i [8] d h d (arcta ) ( ) ( ) h, where, as above, ( ) is the icomplete beta fuctio Accordig to the represetatio () z k k ( ) z k (see 837 () i [8]) we compute ( ) ( ) , ad this fiishes the proof Net we preset two logarithmic itegrals which ca aturally be associated to the arctaget itegrals i (3) Usig the same techique, differetiatio o a parameter, they ca be evaluated to somewhat similar outcomes Propositio 4 For every, log( ) (9) d log log( ) Li ( ) ( ) Whe this is the itegral log( ) d, ( ) 6 which is equivalet to etry 4958 i [8] ad is also a particular case of etry 65 i [4] Proof Settig h( ) to be the LHS we have 6

7 h ( ) d log ( )( ) log log From here, itegratig by parts, Doe! log t log( t) h( ) dt log log( ) dt t t log log( ) Li ( ) Propositio 5 For every, () log( ) d Li Li ( ) Whe, this is etry 49 i [8] ad etry 68 i [4] log( ) d Li log ( ) Proof Settig g( ) to be the LHS we have g ( ) d log log ( )( ) At the same time we otice that so the coclusio is d Li log, d g d d ( ) Li Therefore, for some costat C we have 7

8 g( ) CLi With we compute C Li ad the proof is fiished We ca evaluate the itegral i () also i terms of a power series i Usig the Taylor series for log( ) we compute d d ( ) log( ) ( ) ( ) ( ) d ( ) (log H ) Comparig this to () we come to the followig result Corollary 6 For every, I Particular, with, (See also [4]) (log H ) Li Li (log H ) Li log 3 Two itegral represetatios for (3) ad a special arctaget itegral For, let Li p( ) p be the polylogarithm [] Lema 7 For every ad p, 8

9 p p () (log ) log d ( ) ( p ) Li p( ) Li p( ) () (log ) p log ( ) ( ) p d ( p )Li p ( ) Equatio () is etry 696 i [4] Proof With e t the first itegral becomes ( ) p p log( t ) log( t ) t e e dt p p t p ( ) p t ( ) t e dt ( ) t e dt p ( ) ( ) ( p ) p p p ( ) ( p ) Li p( ) Li p( ) The same substitutio i the secod itegral provides p p t p p t ( ) t log( e ) dt ( ) t e dt p p p p ( ) ( ) ( ) ( p)li p( ) ad the lemma is proved Corollary 8 We have the represetatios (3) (4) 8 dt (3) arcta t arcta 7, t t dt (3) log( t) log t t, where the first itegral is equivalet to 9

10 (5) (arcta t) 7 dt G (3) t 8 The most remarkable itegral i (5) brigs together three importat costats,, the Catala costat G, ad (3) This result is kow; see etry 8 i the list [] ad also p 8 i [6] Proof The startig poit is equatio (4), where i the itegral we make the substitutio t to brig it to the form arcta t dt t log log Li ( ) Li ( ) Here we divide both sides by ad itegrate for from to, d t dt d d t arcta log log Li ( ) Li ( ) Evaluatig these itegrals (usig the above lemma for the secod oe) we come to the equatio dt 7 arcta t arcta Li 3() Li 3( ) (3) t t 4 We shall trasform ow this itegral First we split it this way: oe we make the substitutio to get t dt dt arcta t arcta arcta t arcta t t t t, ad the i the last This proves the first represetatio above, equatio (3), dt 7 arcta t arcta (3) t t 8 Net we use the idetity ( t ) arcta arcta t, t ad the well-kow fact that

11 arcta t dt G, t to prove (5) For the secod represetatio (4) we use (9) i the form (with t log( t) dt log log( ) Li ( ) t( t) Dividig by ad itegratig for from to we write that is, ) log( t) d Li ( ) dt log log( ) d d t t, log( t) log dt Li 3() Li 3() (3) t t I the same way as above we trasform this itegral to log( t) log( t) log dt log dt t t t t, which yields (4) The proof of the corollary is fiished We ed with a etesio of equatio (5) to power series Propositio 9 For ay, (6) h (arcta ) d ( ) I particular, for, (7) h 7 ( ) G (3) 4 (The umbers h were defied right before Propositio 3) The series (7) is etry (59) i [6]

12 Proof By epadig (arcta ) i power series as i the proof of Propositio 3, (arcta ) ( ) ( ) d h d h With the assertio (7) follows from equatio (5) Remark The series ad the itegral i (6) ca be evaluated eplicitly i a closed form by usig a result of Ramauja Namely, Ramauja proved that for,, (8) h log log Li Li log 7 Li3 Li 3 (3) 4 (see [3], p 55) We shall use the priciple brach of the logarithm The above equatio ca be eteded by aalytic cotiuatio i the disc where the RHS is defied I particular, we ca replace by i i order to obtai a alteratig series To simplify the RHS we use the dilogarithm idetity Li Li log log Li ( ) Li ( ) 4 ad also we use the formulas log( i) log i ad i log iarcta The result is i (9) h i i ( ) Li d 3 Li3 (arcta ) i i 7 i arcta( ) Li ( i ) Li ( i ) i log( ) arcta( ) (3) 4 4 Remark Propositios 3 ad 9 admit atural etesios whe replacig (arcta ) p, for ay iteger p I this case we use the epasio (arcta ) by () p (arcta ) A(, p), where A(, p) for p ad for p

13 ! s ( k, p) A(, p) ( ) ( ) k! 3p p p k p k p k or, 3p p p! k A(, p) ( ) ( ) L(, k) s ( k, p) p,! k p (see [], Table 3) Here s ( k, p ) are the Stirlig umbers of the first kid ad! L( k, ) k k! are he Lah umbers Thus we have () p (arcta ) d A(, p) d A(, p) ; d A p d p (arcta ) (, ), that is, () p (arcta ) d A(, p) (see [8], etry 34 () Whe we fid from here (3) ( p ) A(, p) p p Refereces [] Victor Adamchik, 33 represetatios for Catala's costat, Olie paper [] Tewodros Amdeberha, Victor H Moll, ad Armi Straub, Closed-form evaluatio of itegrals appearig i positroium decay, Joural of Math Physics, 5 (9), 358 [3] Bruce C Berdt, Ramauja s Notebooks, Part, Spriger, 985 [4] Khristo N Boyadzhiev, Power series with skew-harmoic umbers, dilogarithms, ad double itegrals, Tatra Moutai Math Publicatios, 56, (3),

14 [5] Khristo N Boyadzhiev, Luis A Media ad Victor H Moll The itegrals i Gradshtey ad Ryzhik, Part : The icomplete beta fuctio SCIENTIA, Series A: Mathematical Scieces, 8 (9), 6-75 [6] David M Bradley, Represetatios of Catala's costat, CiteSeerX: 6879 () [7] David Brik, A solutio to the Basel problem that uses Euclid s iscribed agle theorem, Math Magazie, 87, No 3 (4), -4 [8] Izrail S Gradshtey ad Iosif M Ryzhik, Table of Itegrals, Series, ad Products, Academic Press, 98 [9] Da Kalma, Si ways to sum a series, College Math J 4 (993), 4 4 [] Da Kalma ad Mark McKizie, Aother way to sum a series: geeratig fuctios, Euler, ad the dilog fuctio, Amer Math Mothly, 9 () (), 4-5 [] Vladimir V Kruchii, Dmitry V Kruchii, Composita ad its properties, J Aa Num Theor, No, (4), -8 [] Leoard Lewi, Polylogarithms ad associated fuctios, North Hollad, 98 [3] Habib Bi Muzaffar, A New Proof of a Classical Formula, Amer Math Mothly,, No 4 (3), [4] Aatolii P Prudikov, Yury A Brychkov, Oleg I Marichev, Itegrals ad Series, Part, CRC, 998 [5] Veeravalli S Varadaraja, Euler through time, AMS, 6 4