Some trigonometric integrals involving log Γ ( x) and the digamma function. Donal F. Connon. 19 May 2010

Transcription

1 Some trigoometric itegrals ivolvig log Γ ( ) ad the digamma fuctio Doal F. Coo dcoo@btopeworld.com 9 May Abstract This paper cosiders various itegrals where the itegrad icludes the log gamma fuctio (or its derivative, the digamma fuctio ψ ( )) multiplied by a trigoometric or hyperbolic fuctio. Some apparetly ew itegrals ad series are evaluated; these iclude si pγ [ + log( )] si p p p log Γ ( )cos pd= + ψ ψ p 4p + ( cos p)[ γ + log( )] p ( cos p) log + cot p 4p + 4 p ( cos p)[ γ + log( )] ( cos p) p p log Γ ( )si pd= + ψ ψ p 4p + si pγ [ + log( )] p si p log cot p 4p 4 p si p Si( p) ( cos p) Ci( ) log Γ ( )cos pd= log( ) + + p p 4 p + psi p si( ) 4 p Ci( ) log Γ ( + ) cot d= where si( ) ad Ci( ) are the sie ad cosie itegrals.

2 CONTENTS Page. A coectio with the sie ad cosie itegrals Represetatios of log Γ( ) i terms of the sie ad cosie itegrals 7 Applicatios of Nielse s represetatio of log Γ ( ) 43. Aother approach to the log Γ( ) itegrals 66 A observatio by Glasser 9 3. Some applicatios of the Fourier series for the Hurwitz zeta fuctio 95 Aother proof of Hurwitz s formula for the Fourier series epasio of the Hurwitz zeta fuctio The 5. The log Γ( ) cos p d itegral log Γ( )si p d itegral 6. Some itegrals ivolvig logsi 8 Appedices A. Some series for log Γ ( ) 36 B. Fourier coefficiets for log 39 C. Some aspects of the multiple gamma fuctios 43 D. Table of relevat itegrals ad series 45. A coectio with the sie ad cosie itegrals The frequetly thumbed table of itegrals compiled by Gradshtey ad Ryzhik [33] cotais oly a hadful of defiite itegrals ivolvig log Γ ( ). Of these, oe of the apparetly more comple eamples is the followig formula recorded i [33, p.65, ] which is stated to be valid for a > log Γ ( + a) si kd= log a+ cos( ka) Ci( ka) si( ka) si( k a) k [ ] but, as show below, the corrected itegral which is i fact valid for Propositio. a is (.) log Γ ( + a)si kd= log a cos( ka) Ci( ka) si( ka) si( k a) k [ ]

3 where si( ) ad Ci( ) are the sie ad cosie itegrals defied [33, p.878] by (.) si t si( ) = dt t ad for > (.3) cost cost Ci( ) = dt = γ + log + dt t t where γ is Euler s costat. Note that Si( ) is a slightly differet sie itegral which is defied i [33, p.878] ad also i [, p.3] by (.4) si t Si( ) = dt t We have si t si t si t si( ) = dt = dt dt t t t ad usig the well-kow itegral from Fourier series aalysis (.5) si t = dt t we therefore see that the two sie itegrals are itimately related by (.6) si( ) = Si( ) Proof: We start with equatio (A.8) which is derived i Appedi A to this paper + a Γ + = (.7) log ( a) log( a) γ ( a) log log ( a ) ad multiply this by si p ad itegrate to obtai 3

4 log Γ ( + a)si p d= log( + a)si p d γ ( + a)si p d + a + log log ( + a+ ) + si p d where we have assumed that the iterchage of the order of itegratio ad summatio is valid. The three compoet itegrals are dealt with i tur below. We have usig itegratio by parts log( a+ )cos p cos p log( a + )si p d = + d p p a+ ad with the substitutio t = a + we get cos p cos p acos p t si p asi p d + t dt p = a+ pt cos pt = cos pa dt+ si pa pt si pt pt dt cos pa cosu si pa si u = du du p + u p u ad referece to the defiitios of the sie ad cosie itegrals shows that this is equivalet to cos p aci( u) + si p asi( u) = p Hece we obtai cos p aci[ p( a+ )] + si pasi[ p( a+ )] log( a+ )cos p log( a+ )si p d= p which may be easily verified by differetiatio sice d si Si( ) = ad d d cos Ci( ) = d 4

5 The defiite itegral becomes (.8) log( a+ )si p d { + } + { + } cos pa Ci[ p( a )] Ci[ pa] si pa Si[ p( a )] Si[ pa] = p log( a+ ) cos p log a p We also have the defiite itegral (where k is a iteger) (.9) log( a+ )si k d { + } + { + } cos ka Ci[ k( a )] Ci[ ka] si ka si[ k( a )] si[ ka] = k log( a+ ) log a k where we ote from (.6) that Si( u) Si( v) = si( u) si( v). We also have the elemetary itegral si p ( + a)cos p ( + a)si p d= p p ad therefore si p ( + a)cos p a ( + a)si p d= p p With p = k we get ( + a)sik d= k We see by lettig a + a i (.8) that 5

6 { } cos k( a+ ) Ci[ k( a+ + )] Ci[ k( a+ )] log( + a+ )si k d= k { } si k( a+ ) si[ k( a+ + )] si[ k( a+ )] + k log( a+ + ) log( a+ ) k { } cos ka Ci[ k( a )] Ci[ k( a )] = k { } si ka si[ k( a )] si[ k( a )] k log( a+ + ) log( a+ ) k Due to telescopig we see that We also see that cos ka cos kaci[ k( a+ )] { Ci[ k( a + + )] Ci[ k( a + )] } = k k si ka si kasi[ k( a+ )] { si[ k( a + + )] si[ k( a + )] } = k k N log( a ) log( a ) log( N a) HN log( a) k = k [ ] ad as N we obtai γ + log( + a) log( a ) log( a ) k = k Fially, we ote that the itegral ivolvig thereby obtaied log si k d vaishes ad we have 6

7 log Γ ( + a)si k d { + } + { + } cos ka Ci[ k( a )] Ci[ ka] si ka si[ k( a )] si[ ka] = k log( a+ ) log a cos k aci[ k( a+ )] si kasi[ k( a+ )] + + k k k γ γ + log( a + ) + k k which simplifies to log Γ ( + a)si kd= log a cos( ka) Ci( ka) si( ka) si( k a) k [ ] This therefore shows that oe of the sigs i (.) is recorded icorrectly i Gradshtey ad Ryzhik [33, p.65, ] (ad also i Itegrals ad Series, Volume, p.6 by Prudikov et al. [48]) ad, i this regard, I ote that both Havil [37, p.6] ad Elizalde [9] defie Ci( ) as the egative of (.3); it seems that this lack of cosistecy i the defiitio of Ci( ) is likely to be the source of the error. The defiitios used i this paper correspod with those employed by Nielse [44] ad Nörlud [46] (ecept that those authors use the otatio ci( ) for Ci( )). The similarity betwee the itegral defiitios i (.) ad (.3) does ideed support the otatio ci( ) istead of Ci( ) which, ufortuately, has bee used i this paper ad elsewhere. Equatio (.) also applies i the limit as a because so that (.9.) y cost lim[cos y Ci( y) log y] = lim γ cos y + log y[cos y ] + cos y dt y y t lim[cos yci( y) log y] = γ y sice, applyig L Hôpital s rule, we see that cos y lim[log y(cos y )] = lim ylog y y y y 7

8 cos y = lim[ ylog y] lim y y y [ y y] [ y] = lim log lim si = y y ad we therefore obtai the well kow Fourier coefficiets [33, p.65, ] (.9.) γ + log k log Γ ( )si k d= k With a = / we have (.) k log Γ + si k d= log + ( ) Ci( k ) k Usig the defiitio of Ci( ) i (.3) we see that cost Ci( a) cos( a)log = γ + log a cos( a)log + dt t a We cosider the limit cost = γ + log a log [ cos( a) ] + dt t a cos( a) lim[cos( a) ]log = lim log cos( a) = lim log lim ad usig L Hôpital s rule we obtai cos( a) a si( a) lim = lim which shows that (..) lim[cos( a) ]log = 8

9 ad the takig the limit as we obtai lim Ci( a) cos( a)log = γ + loga (..) [ ] Hece by lettig a i (.8) we obtai (..3) (..4) Ci( p ) γ log p log si p d= p Ci( k ) γ log k log si k d= k With a = i (.) we have (.) Ci( k ) log Γ ( + )si k d= k which is (idirectly) reported with the correct sig i [33, p.65]. A alterative proof of (.) is show below. We have log Γ ( + ) si kd= log si kd+ log Γ( ) si k d ad equatio (.) results by combiig (.) ad (..4). We ow cosider the itegral log Γ ( + a)si(k+ ) d. Propositio. k log Γ ( ) si(k+ ) d= log + + (k+ ) k+ j= j+ Proof We multiply (.7) by si(k+ ) ad itegrate to obtai 9

10 log Γ ( + a)si(k+ ) d= log( + a)si(k+ ) d γ ( + a)si(k+ ) d + a + log log ( + a+ ) + si(k+ ) d These three itegrals are dealt with i tur below. Usig (.8) we get (.) log( a+ )si(k+ ) d { } cos(k + ) a Ci[(k + ) ( a + )] Ci[(k + ) a] = (k + ) { } si(k + ) a si[(k + ) ( a + )] si[(k + ) a] + (k + ) log( a+ ) + log a + (k + ) ad log( + a+ )si(k+ ) d { } cos[(k+ )( + a) ] Ci[(k+ ) ( + a+ )] Ci[(k+ ) ( + a)] = (k + ) { } si[(k+ )( + a) ] si[(k+ ) ( + a+ )] si[(k+ )( + a) ] + (k + ) log( + a+ ) + log( + a) + (k + ) Sice cos[(k+ )( + a) ] = ( ) cos(k+ ) a si[(k+ )( + a) ] = ( ) si(k+ ) a

11 we obtai log( + a+ )si(k+ ) d = { } cos[(k+ ) a]( ) Ci[(k+ ) ( + a+ )] Ci[(k+ ) ( + a)] (k + ) { } si[(k+ ) a ]( ) si[(k+ ) ( + a+ )] si[(k+ )( + a) ] + (k + ) log( + a+ ) + log( + a) + (k + ) We saw above that si p ( + a)cos p a ( + a)si p d= p p ad hece we have a + ( + a)si(k+ ) d= (k + ) + we have Sice ( ) [ ] = ( ) (.3.) ( ) { Ci[(k ) ( a )] Ci[(k ) ( a )] } ad = Ci[(k + ) ( a + )] ( ) Ci[(k + ) ( a + )] (.3.) ( ) { si[(k ) ( a )] si[(k ) ( a )] } = si[(k + ) ( a + )] ( ) si[(k + ) ( a + )]

12 We will see later i (.7) that (.3.3) ψ ( a) = log a + [cos( a) Ci( a) + si( a) si( a)] a ad we have [5, p.4] (by differetiatig the correspodig epressio for the log gamma fuctio) ψ ( a+ k) = ψ ( a) + k j= a+ j = ψ ( a) + k j= a+ j = ψ + () ( a) Hk ( a) where ( m H ) ( a) is the geeralised harmoic umber fuctio defied by H ( m) k ( a) = k j= ( a+ j) m With a = / we obtai [5, p.] (.4) k k + ψ = γ log + j= j + ad we the see from (.3.3) that k+ k+ ψ = log + cos[(k+ ) ] Ci[(k+ ) ] k + or equivaletly k ( ) [( + ) ] = log log k+ j+ (.5) Ci k γ ( k ) j= We et eed to cosider the series a + log log( a ) log( a) k = Usig (.7)

13 + a log Γ ( + a) + log( + a) + γ ( + a) = log log ( + a+ ) + which, with +, becomes + + a log Γ ( + + a) + log( + + a) + γ ( + + a) = log log ( + a+ + ) + Addig these two equatios together ad otig the fuctioal equatio results i log Γ ( + + a) = log( + a) + log Γ ( +a ) log Γ ( + a) + log( + a) + log( + + a) + γ ( + a) + γ = a+ + = log log ( + a+ + ) log ( + a+ ) + ad with = we see that (.6) log Γ ( + a) + log( + a) + γ ( + a) a + = log log( + a+ ) log( + a) + With a = we obtai the familiar formula log log( + ) + = γ We therefore obtai by combiig the various itegrals log Γ ( + a)si(k+ ) d { } cos(k + ) a Ci[(k + ) ( a + )] Ci[(k + ) a] = (k + ) { } si(k + ) a si[(k + ) ( a + )] si[(k + ) a] (k + ) 3

14 log( a+ ) + log a (a+ ) γ (k+ ) (k+ ) cos(k+ ) aci[(k+ ) ( a+ )] cos(k+ ) a + + ( ) Ci[(k + ) ( a + )] (k+ ) (k+ ) si(k+ ) asi[(k+ ) ( a+ )] si(k+ ) a + + ( ) si[(k + ) ( a + )] (k+ ) (k+ ) log Γ ( + a) + log( + a) + γ ( + a) + (k + ) which simplifies to (.7) log Γ ( + a)si(k+ ) d cos(k+ ) aci[(k+ ) a] + si(k+ ) asi[(k+ ) a] = (k + ) log Γ ( + a) γ a log a + (k + ) cos(k+ ) a + ( ) Ci[(k + ) ( a + )] (k + ) si(k+ ) a + ( ) si[(k + ) ( a + )] (k + ) With a = we obtai log Γ ( )si(k+ ) d Ci[(k + ) a] log a = lim + ( ) Ci[(k + ) ] a (k+ ) (k+ ) ad referrig to (.5) ad (.9) this becomes 4

15 k γ + log[(k + ) ] = + γ log log ( k + ) + + (k+ ) (k+ ) k+ j= j+ Hece we obtai the kow result (see also Sectio 4) (.8) k log Γ ( ) si(k+ ) d= log + + (k+ ) k+ j= j+ Referrig to (.3) we see that cost Ci( u) = γ + logu + log + dt t u ad we have (.9) [ ] lim Ci( u) log = γ + log u We also see that u cost cos( u) Ci( u) = [ γ + log u] + cos( u)log + cos( u) dt t ad thus cost = [ γ + log u] + log + [cos( u) ]log + cos( u) dt t u [ ] lim cos( u) Ci( u) log = [ γ + log u]cos( u) + lim[cos( u) ]log The usig (..) we obtai (.) [ ] lim cos( u) Ci( u) log = γ + logu ad i particular we have (as used above) (.) lim a Ci[(k + ) a] log a γ + log[(k + ) ] = (k+ ) (k+ ) With a = i (.7) we obtai 5

16 (.) log Γ ( + ) si(k+ ) d = (k ) Ci[(k ) ] γ ( ) Ci[(k ) ( )] + = (k ) m Ci[(k ) ] γ ( ) Ci[(k ) m)] + m= = (k ) Ci[(k ) ] γ ( ) Ci[(k ) )] + Lettig p = k + i (..3) gives us Ci[(k + ) ] γ log[(k + ) ] log si(k+ ) d= (k + ) ad hece we have log Γ ( + )si(k+ ) d= log si(k+ ) d+ log Γ ( )si(k+ ) d We the have = (k ) + Ci[(k ) ] γ ( ) Ci[(k ) )] Ci[(k + ) ] γ log[(k + ) ] + log Γ ( )si(k+ ) (k + ) d or (.3) log[(k + ) ] log Γ ( )si(k+ ) d= ( ) Ci[(k+ ) )] + (k+ ) (k+ ) which correspods with (.8). Therefore we have (.3.) k ( ) Ci[(k + ) )] + log[(k + ) ] = log + + k+ j= j+ ad with k = this becomes 6

17 (.3.) ( ) Ci( ) = log There is however a ueplaied differece betwee this result ad (.7) ( ) Ci( ) = γ log We ow cosider the itegral log Γ ( + a)cos k d. Propositio.3 Proof log Γ ( + a)coskd= si( ka) Ci( ka) + cos( ka) si( k a) k We multiply (.7) by cos p ad itegrate to obtai [ ] log Γ ( + a)cos p d= log( + a)cos p d γ ( + a)cos p d + a + log log ( + a+ ) + cos p d These three itegrals are dealt with i tur below. As before, itegratio by parts gives us log( a+ )si p si p log( a + )cos p d = d p p a+ ad with the substitutio t = a + we get si p cos p asi p t si p acos p d t dt p = a+ pt 7

18 si pt = cos pa dt si pa pt cos pt pt dt cos pa si u si upa cos = du du p u p u ad referece to the defiitios of the sie ad cosie itegrals shows that this is equivalet to cos p asi( u) si p aci( u) = p Hece we obtai si p aci[ p( a+ )] cos pasi[ p( a+ )] + log( a+ )si p log( a+ )cos p d= p The defiite itegral becomes (.4) log( a+ )cos p d { + } { + } si pa Ci[ p( a )] Ci[ pa] cos pa si[ p( a )] si[ pa] = p log( a+ ) si p + p We also have the defiite itegral (where k is a iteger) (.5) log( a+ ) cos k d { + } { + } si ka Ci[ k( a )] Ci[ ka] cos ka si[ k( a )] si[ ka] = k With a = we obtai (.6) log cos k d= Si[ k ] k We also have 8

19 cos p ( + a)si p ( + a)cos p d= + p p ad therefore cos p ( + a)si p ( + a)cos p d= + p p With p = k we get ( + a)cosk d= We see by lettig a + a i (.5) that { + } si k( a+ ) Ci[ k( a+ + )] Ci[ k( a )] log( + a+ )cos k d= k As before, due to telescopig we see that { + } cos k( a+ ) si[ k( a+ + )] si[ k( a )] k { } si ka Ci[ k( a )] Ci[ k( a )] = k { } cos ka si[ k( a )] si[ k( a )] k si ka si kaci[ k( a+ )] { Ci[ k( a + + )] Ci[ k( a + )] } = k k cos ka cos kasi[ k( a+ )] { si[ k( a + + )] si[ k( a + )] } = k k Fially, we ote that the itegral ivolvig thereby obtaied log cos k d vaishes ad we have 9

20 log Γ ( + a)cos k d { + } { + } si ka Ci[ k( a )] Ci[ ka] cos ka si[ k( a )] si[ ka] = k si k aci[ k( a+ )] cos kasi[ k( a+ )] + k k which simplifies to k (.7) log Γ ( + a)coskd= [ si( ka) Ci( ka) + cos( ka) si( k a) ] This is the compaio itegral to (.) ad this corrects the etry reported i [33, p.65, ] for a > ; the sig error referred to i (.) is also replicated here. This sig error also features i (.76) ad (.78). We may also ote that (.7) is also valid for a = because si u lim[si u.log u] = lim ulog u u u = u ad therefore usig (.3) we see that lim si( k a) Ci( k a) =. a Sice from (.6) si() =, we therefore obtai the well kow Fourier coefficiets [33, p.65, ] (.8) log Γ ( ) cos k d= 4k Further derivatios of this itegral are cotaied i (.8) ad (3.9) below. Lettig a = / results i (.8.) k + ( ) log Γ + cos k d= si( k ) k With a = i (.7) we get for k

21 (.9) si( k ) log Γ ( + )cos k d= k This may be easily cofirmed as follows. We have log Γ ( + ) cos k d = log cos k d + log Γ( ) cos k d ad (.9) results by usig (.6) ad (.8). Usig the duplicatio formula for the gamma fuctio log Γ ( ) = log Γ ( ) + log Γ + + ( ) log log we have log Γ ( ) cos kd= log Γ ( ) cos kd+ log Γ + cos kd We ote that ( ) cosk d= ( ) + log cos k d k + ( ) log Γ ( )cos k d= + si( k ) 4k k log Γ ( )cos kd= log Γ( u)cos kudu log Γ ( u)cos kudu = log Γ ( u)cos kudu+ log Γ( u)cos kudu log Γ ( u)cos kudu = log Γ ( + )cos k( + ) d

22 k = ( ) log Γ ( + )cos k d k log Γ ( )cos kd= log Γ ( )cos kd+ ( ) log Γ ( + )cos k d k k = [ + ( ) ] log ( )cos kd ( ) logcoskd Γ + We ote from Appedi B that Si( k ) log cos k d= k ad ufortuately we simply ed up with k ( ) [ + ( ) ] log ( ) cos k d Γ = + 4k 4k k We recall (.) ad (.7) log Γ ( + a)si d= log a cos( a) Ci( a) + si( a) si( a) [ ] log Γ ( + a)cosd= si( a) Ci( a) + cos( a) si( a) [ ] ad multiplyig each by si( a) ad cos( a) respectively we get si( a) log Γ ( + a)si d= ad log asi( a) cos( a)si( a) Ci( a) si ( a) si( a) cos( a) log Γ ( + a)cos d=

23 + cos( a)si( a) Ci( a) cos ( a) si( a) Addig these two equatios results i log Γ ( + a)[si( a)si + cos( a)cos ] d or equivaletly = + [ log asi( a) si(4 a) Ci( a) cos(4 a) si( a) ] log Γ ( + a) cos[ ( a)] d = + With a = we immediately obtai (.8) [ log asi( a) si(4 a) Ci( a) cos(4 a) si( a) ] log Γ ( ) cos d= 4 ad a = / gives us (.9) log Γ ( + / ) cos d= ( ) + si( ) Similarly subtractio gives us si( a) log Γ ( + a)si d cos( a) log Γ ( + a)cos d = log asi( a) cos( a)si( a) Ci( a) si ( a) si( a) + + cos( a) si( a) Ci( a) cos ( a) si( a) 3

24 = + [ log asi( a) si( a) ] ad we have (.9.) log Γ ( + a)cos[ ( + a)] d= [ log asi( a) si( a) ] The substitutio u = + a gives us a+ (.9.) log Γ ( u) cos udu = [ log asi( a) si( a) ] a ad differetiatio with respect to a results i the very obvious idetity (.9.3) log Γ ( a+ ) cos ( a+ ) log Γ ( a) cos a = log acos( a) si( a) si( a) + a a = log acos( a) Fially, we cosider the itegral Propositio.4 log Γ ( + a) cos(k+ ) d. log( ) + γ log log Γ ( )cos(k+ ) d= + (k+ ) 4 (k+ ) A attempted proof As before we have log Γ ( + a)cos(k+ ) d= log( + a)cos(k+ ) d γ ( + a)cos(k+ ) d + a + log log ( + a+ ) + cos(k+ ) d Usig the defiite itegral (.4) we see that 4

25 (.3) log( a+ ) cos(k+ ) d { } si(k + ) a Ci[(k + ) ( a + )] Ci[(k + ) a] = (k + ) { } cos(k + ) a si[(k + ) ( a + )] si[(k + ) a] (k + ) ( + a)cos(k+ ) d= (k + ) log( + a+ )cos(k+ ) d { } si[(k+ )( + a) ] Ci[(k+ ) ( a+ + )] Ci[(k+ ) ( a+ )] = (k + ) { } cos[(k+ )( + a) ] si[(k+ ) ( a+ + )] si[(k+ ) ( a+ )] (k + ) Sice si[(k+ )( + a) ] = ( ) si(k+ ) a cos[(k+ )( + a) ] = ( ) cos(k+ ) a this becomes We have as i (.3.) { } ( ) si(k+ ) a Ci[(k+ ) ( a+ + )] Ci[(k+ ) ( a+ )] = (k + ) { } ( ) cos(k+ ) a si[(k+ ) ( a+ + )] si[(k+ ) ( a+ )] (k + ) ( ) [( + ) ( + + )] [( + ) ( + )] { Ci k a Ci k a } 5

26 ad = Ci[(k + ) ( a + )] ( ) Ci[(k + ) ( a + )] ( ) [( + ) ( + + )] [( + ) ( + )] { si k a si k a } = si[(k + ) ( a + )] ( ) si[(k + ) ( a + )] Fially, we ote that the itegral ivolvig thereby obtaied log cos(k+ ) d vaishes ad we have log Γ ( + a) cos(k+ ) d { } si(k + ) a Ci[(k + ) ( a + )] Ci[(k + ) a] = (k + ) { } cos(k+ ) a si[(k+ ) ( a+ )] si[(k+ ) a] + (k + ) γ + (k + ) si(k+ ) aci[(k+ ) ( a+ )] si(k+ ) a + + ( ) Ci[(k + ) ( a + )] (k+ ) (k+ ) cos(k+ ) a si[(k+ ) ( a+ )] cos(k+ ) a ( ) si[(k + ) ( a + )] (k+ ) (k+ ) (k+ ) which simplifies to si(k+ ) aci[(k+ ) a] cos(k+ ) asi[(k+ ) a] = (k + ) γ + (k + ) 6

27 si(k+ ) a + ( ) Ci[(k + ) ( a + )] (k + ) cos(k+ ) a ( ) si[(k + ) ( a + )] (k + ) (k+ ) With a = we have log Γ ( ) cos(k+ ) d γ = k+ (k+ ) (k+ ) (k+ ) ( ) si[(k ) ] This is obviously icorrect because of the appearace of the diverget series; the author would appreciate receivig a corrected versio of the proof. Usig a differet approach we ote from (4.9) that log( ) + γ log log Γ ( )cos(k+ ) d= + (k+ ) 4 (k+ ) Represetatios of log Γ( ) i terms of the sie ad cosie itegrals The followig aalysis is etracted from a earlier paper []; it idicates how the log Γ( ) fuctio is itself itimately coected with the sie ad cosie itegrals. Whittaker & Watso [56, p.6] posed the followig questio: Prove that for all values of a ecept egative real values we have (.4) si( ) log Γ ( a) = log( ) + a log a a+ ( + a) d ad this result was attributed by Stieltjes to Bourguet. Equatio (.4) may also be derived usig the Euler-Maclauri summatio formula (see for eample Kopp s book [39, p.53]). By differetiatio we ca easily see that 7

28 d si( ) ( cos( a) Si[ ( + a)] si( a) Ci[ ( + a)] ) = d + a ad we therefore have M si( ) d a Si a a Ci a + a = + + ( cos( ) [ ( )] si( ) [ ( )]) { } { } = cos( a) Si[ ( M + a)] Si[ a] si( a) Ci[ ( M + a)] Ci[ a] From (.4) ad (.5) we see that lim Si[ ( M + a)] = M ad from (.3) we see that lim Ci[ ( M + a)] = M Hece we obtai as M M si( ) d = cos( a ) Si ( a ) + si( a ) Ci ( a ) + a ad referece to (.6) shows that this is equal to = cos( a) si( a) + si( a) Ci( a) Therefore usig Bourguet s formula we have (.4) lo g Γ ( a) = log( ) + a log a a+ [si( a) Ci( a) cos( a) si( a)] Nielse [44, p.79] also reports a similar formula which is valid for < a < (.4.) log Γ ( a) = log( ) log a+ [si aci( ) cos asi( )] ad a derivatio of this is cotaied i Appedi B. Note that i this case, the sie ad cosie itegral fuctios do ot cotai the variable a i their argumets. 8

29 Equatio (.4) may be epressed as log Γ ( a) = log( ) + a log a a+ ad we have with a a log Γ ( a) = log( ) + a log( a) a+ C C Sice ) C C = C + ( we have log Γ ( a) = a log( a) a+ log Γ( a) a log a+ ( ) C ad usig Legedre s duplicatio formula for the gamma fuctio [5, p.7] Γ ( a) = Γ( a) Γ a+ this results i (.4) log Γ ( a + / ) = a ( ) log( ) + alog a a+ [si( a) Ci( a) cos( a) si( a)] which is also reported by Nörlud [46, p.4]. Sice si() = / this idetity may be easily verified for a =. With a = / we obtai (.43) si ( ) = log which is cotaied i [44, p,8]. Lettig a = we have (.44) si( ) 3 ( ) = log Elizalde [9] reported i 985 that for a > 9

30 (.45) ς (, a) = ς + + (, a)log a a [cos( ) ( ) si( ) ( )] a Ci a a si a 4 where ς ( sa, ) is the Hurwitz zeta fuctio. Sice si( ) = Si( ) this may be writte as (.46) si( a 4 4 = ς (, a) = ς(, a)loga a + + ) [cos( a ) Ci ( a ) + si( a ) Si ( a )] Elizalde [9] gave o idicatio of the source of the above idetity but differetiatio of (.45) sheds more light o the subject: we have (, a ) a log a a a a 6 a a ς () ς = + + a [ si( a ) Ci ( a ) + cos( a ) si ( a ) ] sice d cos Ci( ) = ad d d si si( ) =. d This simplifies to ς (, a ) a = log [si( ) ( ) cos( ) ( )] a + a a Ci a a si a We have ( sa, ) ( s, a) s ( s, a) a s ς = ς + ς + ad hece (, a) (, a) (, ) a ς = ς + ς a The usig Lerch s idetity 3

31 (.46.) ς (, a) = log Γ( a) log( ) this becomes We the obtai (, a) (, a) log ( a) log( ) a ς = ς + Γ ς (, a) + log Γ( a) log( ) = a log a + [si( a) Ci( a) cos( a) si( a)] which, for < a <, simplifies to (.47) log Γ ( a) = log( ) + a log a a+ [si( a) Ci( a) cos( a) si( a)] cos( a) = log( ) + a log a a+ = + [si( a ) Ci ( a ) cos( a ) Si ( a ) ] ad usig the Fourier series [55, p.48] (.47.) cos( a) = log[si( a)] this becomes (.48) log Γ ( a) = log( ) + a log a a log[si( a)] + [si( a ) Ci ( a ) cos( a ) Si ( a ) ] 3

32 which correspods with (.4) above. The formula (.47) was give by Nörlud i [46, p.4]. Whe a = / we obtai (.49) si( ) ( ) = log This is a particular case of Nielse s formula [44, p.83] (.49.) si( ) ( ) = log It may be oted that Nielse idicated that (.49.) was oly valid for (, ). With a = /4 i (.48) we get (.5) log Γ = log( ) + log log [si( /) Ci ( /) cos( /) Si ( /) ] We also have usig (.47) (.5) log Γ ( a ) = a log log ( ) log a a a + Γ + + a ( ) + [si( a ) Ci ( a+ ) cos( a ) si ( a+ )] ad with a = / we obtai (.5) si( ) = log( ) ad this cocurs with Nielse s result [44, p.79]. A completely differet derivatio of (.54) is give i (.5) below. Usig (.47.), which is valid for < a <, equatio (.48) may be writte as (.53) log Γ ( a) = log( ) + a log a a log[si( a)] 3

33 + [si( a ) Ci ( a ) cos( a ) Si ( a ) ] This may be writte more compactly as Γ ( a)si( a) log [si( a) Ci( a) cos( a) Si( a)] a = a Itegratio of (.47) results i log Γ ( a) = ε ( )log( ) log log ε + ε ε ε si( ) si(ε ) ( ) ( ) ε ε cos( ) ( ) + si( ) ( ) log( ) [ Ci Si ] + cos( ) ( ) + si( ) ( ) log( ) [ ε Ci ε ε Si ε ε ] Therefore as ε, ad usig (.9.) (.6) log Γ ( ada ) = lim[cos yci( y) log y] = γ, we have y si( log( ) + ( ) log γς ( ) [ cos( ) Ci( ) + si( ) Si( ) log( ) ] + si( ) = log( ) + + ( ) log + + log 4 4 = ) 33

34 ς() log( ) ς () γς() [ cos( ) Ci( ) + si( ) Si( ) ] + + ad with a little algebra ad usig the well kow formula (.6) ς ( ) = ( γ log ) + ς () (which is obtaied by differetiatig the fuctioal equatio for the Riema zeta fuctio) we obtai (.6) log Γ ( ada ) = si( ) = log( ) + + ( ) log + + log 4 4 [ cos( ) Ci( ) + si( ) Si( ) ] + ς ( ) Lettig = i (.6) gives us (.63) Ci( ) loga ς ( ) = = ad with =/ we get ( ) log Γ ( ada ) = log( ) + + log Ci ( ) + ς ( ) ad usig equatio (6.7b) i [] we see that Ci( ) ( ) = log+ + ς 48 ( ) (.64) 5 3 log Γ ( ada ) = log + log + log A 4 4 which is reported i [5, p.35]. With = /4 we obtai 34

35 4 log Γ ( ada ) = 5 G = log( ) log [ cos( / ) Ci( / ) + si( / ) Si( / ) ] + ς ( ) From [5, p.35] we have (.54.) 4 9 G log Γ ( ada ) = log + log + log A ad we therefore obtai 5 cos( /) Ci ( /) + si( /) Si ( /) = + log loga (.65) [ ] Usig (.46) we may write (.6) as Gosper s itegral (.66) log Γ ( ada ) = ( ) + log( ) + ς (, ) ς ( ) I fact, we ote that equatig (.6) with (.66) results i Elizalde s formula (.45). Itegratig Elizalde s idetity (.45) gives us ς (, ada ) = 3 cos( ) [ ( )( )log ] = + (3) [si( ) Ci( ) cos( ) Si( )] 8 ς = where we have used the itegrals (easily derived usig itegratio by parts) 35

36 Ci ( ) d = Ci ( ) si ad Si( ) d = Si( ) + cos I evaluatig the itegral at a =, we have used the fact that Si () = ad from (.3) we have cost si Ci( ) = γ si + si log + si dt t si cost = γ si + log + si d t t We therefore see that limsi Ci( ) = Sice the Hurwitz zeta fuctio is aalytic i the whole comple plae ecept for s, its partial derivatives commute i the regio where the fuctio is aalytic: we therefore have ς(,) st = ς(,) st = [ sς( s+,)] t t s s t s = ς( s+, t) s ς ( s+, t) s ad upo itegratig with respect to t we see that v v v s ς ( s+, t) dt = ς( s, t) dt+ ς( s+, t) d t s t We therefore get v s ς ( s+, t) dt = ς ( s, v) ς ( s,) + ς( s+, t) dt v ad with s = we have v ς (, t) du = ς (, v) ς (, ) + ς(, t) dt v The, usig the well-kow result [6, p.64] 36

37 B (, ) () v ς v = for we obtai v B+ B+ () v (.66) ς (, t) dt = + ς (, v) ς (, ) ( + ) We have lim[ ς ( sa, ) a s ] = ς ( s) a lim[ ς( sa, ) a s ] = ς( s) a lim[ ς( sa, ) a s ] = ς ( s) s a lim[ (, ) s s ς sa a] = lim[ ς ( sa, ) a log a] s a a = ς ( s,) Therefore we have ς ( s,) = ς ( s) ad thus we obtai v B+ B+ () v (.67) ς (, t) dt = + ς (, v) ς ( ) ( + ) This itegral was origially derived by Adamchik [] i a differet maer i 998. We have for = ς ada B3 ς ς (, ) = ( ) + (, ) ( ) Therefore we obtai 3 cos( ) [ ( )( )log ] = + (3) [si( ) Ci( ) cos( ) Si( )] 8 ς = 37

38 = B 3( ) + ς (, ) ς ( ) This is easily simplified to (.68) 5 cos( ) ς ( )( ) log + (3) + 3 [si( ) Ci ( ) cos( ) Si ( )] = B 3( ) + ς (, ) ς ( ) Equatio (.68) could the be itegrated to produce a idetity ivolvig ς ( 3, ) ad so o. Differetiatig (.4) term by term (ad boldly assumig that the procedure is valid) easily results i (.7) ψ ( a) = log a + [cos( a) Ci( a) + si( a) si( a)] a which appears i Nörlud s book [46, p.8]. Lettig a = results i (.7) γ = Ci( ) ad this corrects the correspodig formula give by Nielse [44, p.8]. It appears that Nielse s aalysis is icorrect because he effectively used the Fourier series (.4.) which does ot hold at the poit a =. This formula was also used i []. With a = / we get ψ = log + ( ) Ci( ) ad hece we have 38

39 (.7) ( ) Ci( ) = γ log With a a/ we see that (.73) ψ ( a/ ) = log( a/ ) + [cos( a) Ci( a) + si( a) si( a)] a ad we also have ψ ( a+ ) = log( a+ ) ( a + ) + cos[ ( a+ )] Ci[ ( a+ )] + si[ ( a+ )] si[ ( a+ )] Equatio (.7) may be epressed as ψ ( a) = loga + a a ad we have with a a ψ ( a) = log( a) + a 4a Sice ) a a = a + ( we have ψ ( a) = log( a) + ψ ( a) log a+ + ( ) a 4a 4a ad usig the duplicatio formula for the gamma fuctio [5, p.5] (which may be obtaied by differetiatig (.6.)) this results i ψ ( a) = log + ψ( a) + ψ a+ (.74) ψ a+ = loga+ ( ) [cos( a) Ci( a) + si( a) si( a )] 39

40 Differetiatig (.4.) results i (which is valid for < a < ) (.75) ψ ( a) = + [cos aci( ) + si asi( )] a or equivaletly ψ ( a+ ) = [cos aci( ) + si asi( )] which may be compared with (.7) ψ ( a) = log a + [cos( a) Ci( a) + si( a) si( a)] a Differetiatig (.) with respect to a log Γ ( + a)si d= log a cos( a) Ci( a) si( a) si( a) we obtai [ ] cos( a) cos( a) + si( a) Ci( a) a a ψ( + a)sid= si( a) si( a) cos( a) si( a) a ad therefore we have (.76) ψ ( + a)si d= si( a) Ci( a) + cos( a) si( a) which corrects the etry i [33, p.65, ]. With a = we get (.77) ψ ( + )si d= si() ad with a = we get 4

41 (.77.) ψ( )si d= si() = which is correctly reported i [33, p.65, ]. We see that si ψ( + )si d= ψ( )sid+ d = ψ( )si d + Si( ) Si() ad this cocurs with (.77) ad (.77.). Similarly, differetiatig (.7) we obtai (.78) ψ ( + a)cosd= si( a) si( a) + cos( a) Ci( a) which corrects the etry i [33, p.65, ]. With a = we have (.78.) ψ ( + )cos d= Ci( ) Usig itegratio by parts we see that for a ψ ( + a)si d= si log Γ ( + a) Γ ( + a)cos d We have for a si lim si log Γ ( + a) = lim log Γ ( + a) = ad hece we have for a (.79) ψ ( + a)si d= Γ ( + a)cos d 4

42 Usig (.7) we agai obtai log Γ ( + a)cosd= si( a) Ci( a) + cos( a) si( a) [ ] ψ ( + a)si d= si( a) Ci( a) + cos( a) si( a) Similarly we have for a > (.8) ψ ( + a)cos d= log Γ ( + a) log Γ ( a) + Γ ( + a)si d ad usig (.) log Γ ( + a)si d= log a cos( a) Ci( a) si( a) si( a) we agai obtai [ ] ψ ( + a)cos d= cos( a) Ci( a) + si( a) si( a) Nielse [44, p.8] reports these itegrals i the case a = as (.8) ψ( + )sid= log Γ ( + )cos d (.8) ψ( + )cos d = log Γ ( + )si d Applyig Parseval s idetity with (.78) ad (.78.) we obtai ψ ( + ) d = [ Ci ( ) + si ( ) ] With regard to this, see also Lewi s moograph [43, p.4] where he showed that 4

43 u e Li ( ) d= Ci ( u) + si ( u) u ad with u = we have e Li ( ) d= Ci ( ) + si ( ) We make the summatio = + e Li ( ) d [ Ci ( ) si ( ) ] Usig the derivative of the geometric series we have so that z z = ( z) e = ( e ) e Li ( ) d Li ( ) d ad hece we have the curious equality e ψ ( e ) Li ( ) d = ( + ) d A differet represetatio of Ci u + si u ( ) ( ) as a itegral is give i [44, p.3]. Applicatios of Nielse s represetatio of log Γ ( ) We recall Nielse s represetatio of log Γ( ) i equatio (.4.) log Γ ( ) = log( ) log + [si Ci( ) cos si( )] ad multiplyig this by cos p ad itegratig gives us 43

44 (.9) si p Si( p ) log Γ ( )cos pd= log( ) + p p ( cos p ) Ci( ) psi p si( ) p 4 p where we have employed the itegral log cos p d= [log si p Si( p)] p (.9) Si( p ) log cos p d= p Lettig p = i (.9) results i Si( p ) log Γ ( d ) = log( ) + lim p p ad usig L Hôpital s rule we see that log Γ ( d ) = log( ) Si( p ) lim = givig us the familiar result p p With p = we have (.9) Si( ) 4 Ci( ) log Γ ( ) cos d= + 4 ad p = k results i Si( k) si( k) log Γ ( ) cos k d= = k k 4k i agreemet with (3.). The followig limits have bee used i the above calculatio ( cos p) Ci( ) si pci( k) lim lim 4 p = ( p) = p k p k 44

45 psi p si( ) ( p cos p si p ) si( k ) si( k ) lim + = lim = p k p k p k 4 p k ( ) With p = k + we have (.93) Si[(k + ) ] 4 Ci( ) log Γ ( )cos(k+ ) d= + (k+ ) 4 (k+ ) We see from the defiitio (.3) that cost Ci( ) = γ + log( ) + dt t ad we have the summatio Ci( ) log cost = [ γ + log( )] + + dt t log cost = [ γ + log( )] + + dt 4 4 t Usig cost cos( y) dt = dy t y the last series becomes cost cos( y) dt = dy 4 t 4 y cos( y) = lim dy 4 a = y = lim a a cos( y) dy (4 ) y cos( y) = lim dy a + y (4 ) y a = It is a eercise i Apostol s book [7, p.337] to show that for < < 45

46 4 cos( ) (.94) si = 4 ad i fact, sice =, this holds true for <. It seems likely that (.94) 4 may also be determied usig the differetiatio theorems for Fourier series appearig i Tolstov s book [55, p.4]. We have so that cos( y) ( si y) = 4 4 cos( y) siy lim dy lim dy Si( ) a + y (4 ) a y = = 4 y 4 a = a Hece we see that Ci( ) log (.95) = [ γ + log( )] + Si( ) ad referrig to (.93) Si( ) 4 Ci( ) log Γ ( ) cos d= + 4 we obtai log log Γ ( )cosd= log( ) γ which we shall derive i a differet maer i (4.5.). We may write (.95) as Ci( ) log( ) log( ) log + log( ) + Si( ) = γ ad take the limit as right-had side.. Havig regard to (.3) we see that the limit equates to the Comparig (.93) with (4.6) results i 46

47 4 γ + log( ) Si[(k+ ) ] 4 Ci( ) (.95.) = + 4 (k+ ) (k+ ) 4 (k+ ) Itegratig (.95) results i Ci( ) si( ) log = [ γ + log( )] ad hece we obtai si( ) (.96) = [+ cos( ) ] 4 Si( ) [cos( ) ] 4 4 which cocurs with the result directly obtaied by itegratig (.94). I passig, we ote that it is a eercise i Bromwich s book [5, p.37] to show that for < < cos si a( ) (.96.) = ( ) a 4a cos a + Multiplyig (.4.) by si p ad itegratig gives us (.97) cos p Ci( p) γ + log( p) log Γ ( )si pd= log( ) + p p p si p Ci( ) p( cos p) si( ) + 4 p 4 p where we have employed the itegral (..3) Ci( p ) γ log( p) log si p d= p With p = we have 47

48 (.98) Ci( ) γ + log si( ) log Γ ( )sid= log( ) ad comparig this with (5.7) we deduce that si( ) 4 (.99) = [ 3 + Ci( ) γ log(4 )] Lettig p = k results i log Γ( )si k d Ci( k ) γ + log( k) cos pci( k) ( + psi p cos p) si( k) = + lim + lim k k p k p p k ( ) p k ( ) ad hece we obtai aother derivatio of γ + log( k) log Γ ( )si k d= k With p = k + we have (.) Ci[(k + ) ] γ + log[(k + ) ] log Γ ( ) si(k+ ) d= log( ) + (k+ ) (k+ ) (k+ ) (k+ ) si( ) + 4 (k+ ) which may be compared with (5.6) k log Γ ( ) si(k+ ) d= log + + (k+ ) k+ j= j+ Further idetities may be obtaied by multiplyig (.4.) by ad itegratig as before. si p or cos Equatio (.99) may also be derived as follows. Referrig to (.9) we have p 48

49 Si( ) log cos d= ad we make the summatio cos Si( ) log d= 4 4 si( ) = ad usig (.5) this becomes si( ) = log Assumig that the iterchage of summatio ad itegratio is valid cos cos log d = log d 4 4 Employig (.94) this becomes ad usig (..3) we have = log ( si d ) 4 = [ + Ci ( ) γ log ] 4 Therefore we obtai si( ) (.) = [3 + Ci( ) γ log(4 )] 4 Alteratively we cosider si( ) Si( ) =

50 Si( ) = (log ) 4 where we have used (.5). Si( ) sit dt = 4 4 t = si( t) dt 4 t ad usig (.96.) this becomes cos( t) + t = dt t We ote from (.3) that cost Ci( ) = γ + log + log + dt t cos( t) = γ + log + log + dt t Therefore we obtai aother derivatio of Si( ) = [3 + Ci( ) γ log(4 )] 4 Equatig (.9) ad (4.4.) gives us si p Si( p) ( cos p) Ci( ) psi p si( ) log( ) p p 4 p 4 p si pγ [ + log( )] si p p p = + ψ ψ p 4p + 5

51 ( cos p)[ γ + log( )] p ( cos p) log + cot p 4p + 4 p Equatig (.97) ad (5.5.) results i cos p Ci( p) γ + log( p) si p Ci( ) log( ) + p p p 4 p + p( cos p ) si( ) 4 p ( cos p )[ γ + log( )] ( cos p ) p p = + ψ ψ p 4p + + si pγ [ + log( )] p si p log cot p 4p 4 p Ci( ) ad hece we have two simultaeous equatios ivolvig ad 4 p = si( ). 4 p We showed i [] that (.) p( )cotd b a b = p( )sid a which is valid for a wide class of suitably behaved fuctios. Specifically we require that p( ) is a twice cotiuously differetiable fuctio ad that either (i) both si( / ) ad cos( / ) have o zero i [ ab, ] or (ii) if either si( a / ) or cos(a / ) is equal to zero the p(a) must also be zero. Coditio (i) is equivalet to the requiremet that si has o zero i [ ab, ]. Sice log Γ () = log Γ () =, it is clear that p ( ) = log Γ ( + ) satisfies the ecessary coditios o the iterval [,]. Therefore we have log Γ ( + ) cotd= log Γ ( + ) si d 5

52 ad usig (.) this becomes (.3) Ci( ) log Γ ( + ) cot d= This is a iterestig result if oly for the fact that Mathematica is ot able to evaluate eve the seemigly simpler itegral We ote that itegratio by parts gives us log cot d. log Γ ( + )cot d = log ( )logsi ψ( )logsi d Γ + + ad usig L Hôpital s rule we obtai log si Γ + = lim[log ( )log si ] lim /log Γ ( + ) = lim Γ + ψ ( + ) cotlog ( ) We also have Γ + = lim ψ()si log ( ) log Γ ( + ) ψ ( + ) lim log ( ) lim lim log ( ) Γ + si = Γ + = cos Similarly we have log Γ ( + ) log Γ ( + ) lim log ( ) lim log ( ) Γ + = Γ + si si ad sice log Γ ( + ) ψ ( + ) lim lim = we obtai lim[log Γ ( + ) log si ] = Hece we have 5

53 log Γ ( + ) cot d= ψ( + ) log si d It was also show i [] that uder the same coditios we have (.4) b a b p( d ) = p ( )cosαd a ad we employ this with p ( ) = log Γ ( + ) ad (.7) to give us log ( ) d log ( ) cos d Γ + = Γ + = si( ) We therefore get (.5) si( ) = log( ) which we have previously see i (.54). A umber of other closed form Dirichlet series ivolvig the sie ad cosie itegrals are give i my earlier paper []. Motivated by a problem posed by Furdui [3] i 9, we fid a more direct derivatio of (.63) which is set out below. We see from (.3) that cost Ci( ) = γ + log + dt t ad we make the summatio Ci( ) log cost = ( γ + log ) + + dt t cost = ς()( γ + log ) ς () + dt t 53

54 We see that cost cos y dt = dy t y ad thus cost cosy dt = dy t y cosy = lim dy a = y a cos y = lim dy a a y We have the well kow Fourier series [55, p.48] so that ad cos y y y = cos y = + y 6y 4 cos y y = + y 4 y We the easily see that (.6) Ci( ) = ς()( γ + log ) ς () + 8 ad with = we obtai (.63) agai (.7) Ci( ) = ς ()[ γ + log( )] ς () More geerally we have 54

55 (.8) Ci( ) cosy = ς()( s γ + log ) ς () s + lim s dy y s a a Similarly we may cosider the sie itegral fuctio si t Si( ) = dt t si t Si( ) = dt t Si( ) sit = dt t s s Si( ) sit = s dt t s ad we ca certaily evaluate this for s = 3 usig the Fourier series [55, p.48] si t 3 = 3 ( t 3 t + t) I this regard, see also the problems proposed by Choulakia [8] i 998. Usig (..3) we make the summatio log si(k ) + Ci[(k+ ) ] log[(k ) ] d γ + = k+ (k+ ) k= k= ad substitutig the Fourier series [5, p.49] k = si(k+ ) = k + 4 we obtai Ci[(k + ) ] log(k + ) = ( γ + log ) 4 (k ) (k ) ( ) k= + k= + k= k+ 55

56 or equivaletly [( ) ] log( ) = + Ci k + k + ( log ) γ 4 k= (k+ ) 8 k= (k+ ) which may be compared with (B.) [ Ci[( + ) ] log( + ) ] ( γ + log ) log= 4 = ( + ) As a matter of iterest, Abramowitz ad Stegu [, p.3] defie auiliary fuctios si y f ( ) = cos si( ) + si Ci( ) = dy y + cos y g ( ) = cos Ci ( ) si si ( ) = dy y + ad report that for Re ( ) > f ( ) = u e + u du g ( ) = ue + u u du The above results may be derived by cosiderig the double itegral ( a+ y) I e si y ddy = where itegratig with respect to gives us e ( a+ y) d= a + y ad thus we have 56

57 I = si y dy a+ y Similarly, itegratig with respect to y gives us a ( a+ y) e y iy iy a e e si ydy = e ( e e ) dy = i + Therefore we see that a e si y d = dy + a+ y ad the validity of the operatio ( a+ y) d e si y dy = ( a+ y) dy e si y d is cofirmed by [8, p.8]. The formula a e cos y d = dy + a+ y may be derived i a similar fashio. Lettig t = y we see that si( y) si t d = dt + a t+ ay I accordace with the above, we have from [33, p.338] v ve (.9) dv = [cos( a) Ci( a) + si( a) si( a)] a + v ad we make the summatio v ve a + v dv = [cos( a) Ci( a) + si( a) si( a)] The geometric series gives us 57

58 v ve dv = ve v ( )( v ) a + v a + v e dv ad hece we have v [ Ci( a)cos( a) + si( a)si( a)] = dv v ( a + v )( e ) We ow let a a ad v v to obtai t (.) [cos( a) Ci( a) + si( a) si( a)] = dt t ( a + t )( e ) Usig (.7) ψ ( a) = log a + [cos( a) Ci( a) + si( a) si( a)] a we the see that t (.) ψ ( a) = loga t a dt ( a + t )( e ) Equatio (.) is well kow ad, iter alia, is reported i [56, p.5]. Itegratig (.) gives us av da dv = a[cos( a) Ci( a) + si( a) si( a)] da ( a + v )( e ) v We have i a scitilla temporis usig the Wolfram Olie Itegrator a[cos( a) Ci( a) + si( a) Si( a)] da [ si( a) + cos( a)] Ci( a) [acos( a) si( a)] Si( a)] log a = ( ) where we ote that Mathematica defies SiItegral[ ] as Si( ). This itegral may of course be easily obtaied usig itegratio by parts. 58

59 We have a[cos( a) Ci( a) + si( a) Si( a)] da cos( ) Ci( ) + si( ) Si( ) log γ + log() = ( ) si( ) Ci( ) cos( ) Si( ) + where we have used (.9.). We have a[cos( a) Ci( a) + si( a) Si( a)] da ad = a[cos( a) Ci( a) + si( a) si( a)] da+ asi( a) da si( ) cos( ) asi( a) da= ( ) a[cos( a) Ci( a) + si( a) si( a)] da cos( ) Ci( ) + si( ) Si( ) log γ + log() = ( ) si( ) Ci( ) cos( ) Si( ) + si( ) cos( ) + ( ) cos( ) Ci( ) + si( ) si( ) log γ + log( ) = ( ) si( ) Ci( ) cos( ) si( ) + 59

60 Therefore we have a[cos( a) Ci( a) + si( a) si( a)] da cos( ) Ci( ) + si( ) si( ) () (log log( ) ) ς = + γ ( ) 4 4 si( ) Ci( ) cos( ) si( ) + Usig (.4) this becomes cos( ) Ci( ) + si( ) si( ) () (log ) ς = + γ ( ) 4 4 log ( ) log( ) + Γ log + 4 Itegratig (.) gives us av v dv a I = da dv = da ( a + v )( e ) e a + v v v log( + ) log v v v v = dv dv e e v v We have [4] (.) ( ) log 3 + log ( + ) = log log( ) ς ( ) v + + e v v G dv where G ( ) is the Bares double gamma fuctio. This result was origially obtaied by Adamchik [3] i 4. With = we have vlog v (.3) v dv = ς ( ) e Therefore we obtai 6

61 Hece we obtai 3 = log log( ) log ( ) I G cos( ) Ci( ) + si( ) si( ) (.4) ( ) 3 ς () = (log log( ) + γ) + log log( ) log G( ) log Γ ( ) + log Alteratively, we multiply (.) by a ad itegrate to obtai 3 aψ a da = G ( ) log log log( ) log ( ) Itegratio by parts results i aψ ( a) da = log Γ( ) log Γ( a) da ad we obtai Aleeiewsky s theorem [5, p.3] (.5) log Γ ( ada ) = log Γ( ) log G( + ) + + log( ) We also have from [33, p.338] μv e dv = [si( μ ) Ci ( μ ) cos( μ ) si ( μ ) ] + v ad we make the summatio μv e dv = [si( μ ) Ci ( μ ) cos( μ ) si ( μ ) ] + v 6

62 Assumig that iterchagig the order of summatio ad itegratio is valid μv e μv = + v + v dv e dv ad usig (obtaied by lettig α = iv i (4.7)) (.6) cothv= + v v + v we have vcothv μv (.7) e dv [si( μ ) Ci ( μ ) cos( μ ) si ( ) ] = μ v Mathematica caot evaluate this itegral. With μ = we have vcothv v ( ) e dv si( ) = v ad referrig to (.49) we obtai vcothv v (.8) e dv log = v With μ = we have vcothv si( e dv = v ) v ad referrig to (.54) we obtai (.9) vcoth v v e dv v = log( ) We ote [33, p.336, 3.34] that 6

63 ep( p ta d ) = si pci( p) cos psi( p) so that ep( a ta ) d= si aci( a) cos asi( a) Referrig to (.4) log Γ ( a) = log( ) + a log a a+ [si( a) Ci( a) cos( a) si( a)] ad usig ep( a ta ) = log[ ep( a ta )] we obtai the itegral represetatio (.9.) log Γ ( a) = log( ) + a log a a log[ ep( ata )] d ad we also ote that asec d = log[ ep( a ta )] d ep( ata ) As oted by Nielse [44, p.7] ad Bartle [8, p.345] we have the Fourier series (provided is ot a iteger) (.) si ( ) cost = + + cos t + si ( ) = cost + ad with t = we obtai 63

64 ( ) = + si + We the have + + si ( ) ( ) cost = cos t + si which we write as + si ( ) cost = (cos t ) We ow divide by t ad itegrate to obtai u + u cost si ( ) cos dt = t t dt It is easily see that u u cost cos y dt = dy t y ad hece we have u cost dt = Ci( u) γ log( u) t Therefore we obtai + si ( ) Ci( u) γ log( u) = [ Ci( u) γ log( u)] si ( ) si ( ) = [ Ci( u) log ] ( γ + log u) si ( ) si = [ Ci( u) log ] ( γ + log u) where we have used 64

65 si si ( ) = + Hece we obtai (.) + si ( ) ( + log u)si γ Ci( u) log = [ Ci( u) log ] + which corrects a misprit i Nielse s book [44, p.7]. I eamiig the limit as equal., usig (.9) we see that both sides of the equatio are Usig L Hôpital s rule we easily see that both sides are also equivalet i the case where = k where k is a iteger. I passig, we ote that we may write (.) as (.) cost si ( ) si + = cos t ad the take the limit as. Applyig L Hôpital s rule three times gives us the well kow Fourier series [55, p.48] With t + ( ) cost = ( 3 t ) = i (.) we obtai cos si = si = which is the same as (4.7). Differetiatig (.) with respect to t results i + si t ( ) = si t si ad applyig L Hôpital s rule as gives us si t t cost t lim = lim = si cos 65

66 ad we ed up with the well-kow Fourier series [55, p.48] t ( ) = + si t This could also be employed to derive (.49). Itegratig (.) with respect to t results i si t t si ( ) si + = si t We ote that [37a] gives a closed form for the followig Fourier series i respose to a problem posed by Fettis ad Glasser for <, ( pq, ) =, q + + si t = si ( + k ) log ( p/ q) pq q si( p/ q) si(/ q)( + k ) ) q ( ) p si( k p/ q) si(/ q)( k ) ) k= We have [5, p.4] ψ ( ) = log+ log + k = + k + k which could the be used to evaluate itegral ψ( )si pd usig (.8), (..3) ad the si p d = cos( kp ) Si [ p ( + k )] si( kp ) Ci [ p ( + k )] + k. Aother approach to the log Γ( ) itegrals Part of the followig aalysis is take from the 86 book Eposé de la théorie, propriétés, des formules de trasformatio, et méthodes d évaluatio des itégrales défiies by David Bieres de Haa [35, p.69], a copy of which is available o the iteret courtesy of the Uiversity of Michiga Historical Mathematics Collectio. The volumious itegrals evaluated by Bieres de Haa were a importat source for the table of itegrals subsequetly compiled by Gradshtey ad Ryzhik [33]. Bieres de Haa (8-895) was, iter alia, a mathematicia, a actuary ad a historia; a short 66

67 iterestig biography of him appears i Talvila s paper [53] ad a more detailed presetatio is give i [49]. Let us desigate the followig itegrals as (.) (.) I( p) = log Γ( )cos p d K( p) = log Γ( )si p d The we have by lettig (.3) I( p) = cos p log Γ( )cos pd+ si p log Γ( )si p d ad we see that (.4) I(/ ) = log Γ( ) si( / ) d We have from the defiitio I( p) = log Γ ( )[coscos p+ sisi p] d The itegral J( p) = log Γ( )cos p d becomes by lettig = log Γ ( )[cos p cos p+ si psi p] d Similarly, with L( p) defied by L( p) = log Γ( )si p d 67

68 becomes by lettig = log Γ( )[si p cos p cos psi p] d I the same way as above we fid that (.5) K( p) = si p log Γ( )cos pd cos p log Γ( )si pd ad we see that (.5.) K(/ ) = log Γ( )cos( / ) d We have Euler s reflectio formula [] for all z C ecept z =, ± where N Γ( z) Γ( z) = si z ad therefore for < < (.6) log Γ ( ) + log Γ( ) = log( ) log(si ) Multiplyig this equatio by cos p ad itegratig, we obtai log Γ ( )cospd+ log Γ( )cos pd = log( ) cos pd log(si )cos pd The, usig the Fourier series [6, p.4] (.7) log si ( t / ) cos t = < t < ad, assumig that chagig the order of itegratio ad summatio is valid, we get (.8) log Γ ( )cos pd+ log Γ( )cos pd 68

69 si p = log( ) + cos {( + p) } + cos {( p) } d p { + p } { p } si p si ( ) si ( ) = log( ) + + p ( + p) ( p) Hece, provided p, as origially determied by Bieres de Haa [35, p.7] we have (where I have corrected a typographical error) (.9) si p psi p log Γ ( )cos pd+ log Γ( )cos pd= log( ) p 4 p Rather cofusigly, Bieres de Haa [35, p.7] reports the left-had side of (.9) as beig equal to I( p) + I( p). From (.9) we see that with p = (.) log Γ ( ) d+ log Γ( ) d = log( ) ad by a simple chage of variable we have log Γ( ) d = log Γ( ) d Hece we obtai Raabe s itegral [56, p.6] (.) log Γ ( d ) = log( ) From (.9) we see that with p = (.) = log Γ ( ) cos d + log Γ( ) cos d which may of course also be obtaied by a simple chage of variable. With p = we have 69

70 log Γ ( )cosd+ log Γ( )cosd = log( ) cos d log(si )cos d = log(si )cosd ad, with the substitutio y = si, we see that the itegral o the right-had side vaishes. This is easily cofirmed because, with the substitutio u =, we see that log Γ( )cosd= log Γ( u)cosudu We are therefore uable to evaluate log Γ( )cos dusig this methodology. We ote i passig that Raabe s itegral (.) may also be easily obtaied as follows by itegratig (.6), i.e. with p = log Γ ( ) d + log Γ( ) d = log( ) log(si ) d where we also use the well-kow itegral due to Euler log si d= log to determie that log Γ ( d ) + log Γ( d ) = log( ) ad fially we see that log Γ ( ) d = log Γ( ) d 7

71 I passig, we ote that a iterestig derivatio of Euler s itegral appears i a 864 paper by Jeffery [38, p.94]. We cosider log si d = log si d ad with the substitutio t = si we have logt = 4 t ( t ) dt = Buv (, ) 4 u u= v= / = Γ Γ Γ Γ 4 () = log Lettig p = / i (.9) we see that log Γ ( )cos( / ) d+ log Γ( )cos( / ) d= log( ) 4) 4 (/ With the obvious substitutio we see that = log( ) 6 log Γ( )cos( / ) d= log Γ( )si( / ) d Hece we have (.3) log Γ ( )[cos( / ) + si( / )] d= log( ) 6 With regard to the above ifiite series, Ramauja [, p.9] showed that 7

72 (.4) = 3log 6 = I fact, as reported i [, p.9], this summatio was the first problem submitted by Ramauja i 9 to the Joural of the Idia Mathematical Society. Ramauja [, p.6] also showed that (.5) = log 4 This series is cotaied i [33, p.9,] where it is attributed to Bromwich s tet [5, p.5]. A proof is also give i Ramauja s Notebooks [, p.6]. I [5] the proof is based o the defiitio of Euler s costat γ ad it is therefore quite apt that the series should be coected with the gamma fuctio. Therefore we obtai from (.3) ad (.4) + (.6) log Γ ( )[cos( / ) + si( / )] d= [ log log ] or equivaletly 4 + (.6.) log Γ( ) cos [ ] d= [ log log ] By defiitio we have I = log Γ ( )cos( / ) d = log Γ( t)cos( t) dt We have Legedre s duplicatio formula for the gamma fuctio (see [56, p.4] ad [5] for eample) (.6.) Γ( ) Γ + = Γ() 7

73 which may be epressed as log Γ ( ) = log Γ ( ) + log Γ + + ( ) log log Hece we have I(/ ) = log ( t)cos( t) dt log t cos( t) dt Γ + Γ + + ( t ) log log cos( t) dt Usig the obvious chage of variables we have log Γ t + cos( t) dt = log Γ( u) cos u du = log Γ( u)siudu = log Γ( u)siu du log Γ( u)siu du = log log ( u )si ud + Γ u where we used (.6) below. Therefore we have (.7) log Γ( t)[cost si t] dt+ log log + log + log + We defie 73

74 φ( ak, ) = + k a a ( ) ad (.8) φ( a) = lim φ( a, k) = + k a ( a) Sitaramachadrarao [5] has show that if p, q are positive itegers ad if φ ( p, q) eists, the (.9) q/ p p q qj j φ = log p cos log si + q q p p p j= pj q where the (corrected) first summatio rus over < j < p / iteger. Hece we have from (.9), ad deotes the largest p p qsi( p / q) psi( p / q) I + J = log( ) q q p q 4 ( p/ q) si( / ) si( / ) q p q q p q = log( ) p p q/ p ( q/ p) ad thus (.) p si( / ) si( / ) p q p I J q log( ) q p q q + = φ q q p p p As metioed by Sitaramachadrarao, (.9) is clearly related to Gauss s formula for the digamma fuctio (see [5, p.5] ad [5, p.8]. q p p cot pj log q cos ψ γ log si j = + q q j= q q With p = ad q= k+ i (.9) we obtai k φ = (k + )log + (k + ) k+ j= j (k+ ) or equivaletly 74

75 k φ = (k + )log (k + ) k+ j= j By defiitio we have 3 φ = + (k + ) k+ 4 (k+ ) ad hece we have (.) = log 4 (k+ ) (k+ ) k+ j+ k j= This idetity is employed i (5.6). For eample, with k = i (.) we recover Ramauja s formula (.5). We immediately otice the strog similarity with the followig itegral which is evaluated i Sectio 6 k logsisi(k+ ) d= log (k+ ) k+ j= j+ ad, with a mior adjustmet, we see that k log(si )si(k+ ) d= log (k+ ) k+ j= j+ The very fact that Ramauja deals with φ ( a) i Chapter 8 of [], which also ivolves coectios with Kummer s formula (3.) ad related matters, strogly suggests that he was well aware of equatio (.9). We ote from equatio (A.3) i the Appedi that (.) ψ ( + ) + ψ( ) + γ = which cocurs with Prudikov et al. [48, ]. I aother paper [7] we have derived rapidly covergig series for Catala s costat G = β () ad for Apéry s costat ς (3) by employig (.). These are set out below: 75

76 G = 3( log ) (6 ) 7 ς (3) = (5 4 log ) G + 6 (6 ) 3 It may be oted that Mathematica cofirms these idetities both umerically ad aalytically. The method may be easily geeralised to produce ew series represetatios for values of the Riema zeta fuctio ς ( + ) ad the Dirichlet beta fuctio β ( ). With / i (.) we get ψ + ψ = γ + ( ) as reported by Ramauja [, p.86]. With = p / i (.) we have (.3) p p ψ + + ψ + γ = p 4 p with the result that we may write (.9) as (.4) log Γ ( )cos pd+ log Γ( )cos pd For eample, with p =/ si p log( ) p p = γ ψ ψ p i (.3) we have 5 3 = γ log( ) ψ ψ ad usig [5, p.] ψ 3 = 3log 4 γ + ψ 5 = 3log 4 4 γ + 76