A study of inverse trigonometric integrals associated with three-variable Mahler measures, and some related identities

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1 A stdy of inverse trigonometric integrals associated with three-variable Mahler measres, and some related identities Mathew D. Rogers Department of Mathematics University of British Colmbia Vancover, BC, V6T-Z, Canada Abstract We prove several identities relating three-variable Mahler measres to integrals of inverse trigonometric fnctions. After deriving closed forms for most of these integrals, we obtain ten explicit formlas for three-variable Mahler measres. Several of these reslts generalize formlas de to Condon and Lalín. As a corollary, we also obtain three q-series expansions for the dilogarithm. Key words: Mahler measre, special fnctions, polylogarithms, trilogarithm, elliptic fnctions Introdction In this paper we will ndertake a systematic stdy of each of the inverse trigonometric integrals Tv, w = Sv, w = TSv, w = tan vx tan wx dx, x sin vx sin wx dx, x tan vx sin wx dx. x This class of integrals arises when trying to find closed form expressions for the Mahler measres of certain three-variable polynomials. Recall that the Mahler measre of an n-dimensional polynomial, P x,..., x n, Preprint sbmitted to Elsevier Science 3 November 5

2 can be defined by m P x,..., x n =... log P e πiθ,..., e πiθn dθ... dθ n. In the last few years, nmeros papers have established explicit formlas relating mlti-variable Mahler measres to special constants. Smyth proved the first reslt [3] with m + x + y + z = 7 π ζ3, where the Riemann zeta fnction is defined by ζs = n= n s. In this paper, we will prove a nmber of new formlas relating threevariable Mahler measres to the aforementioned trigonometric integrals. Many of or identities generalize previosly known reslts. We will list a few of or main reslts in this introdctory section. For or first example, we can se varios properties of Tv, w to show that x x m v 4 + y + v z + x + x = 4 tan d 8 π π T v, + x. v m v 4. + x This redces to one of Lalin s formlas [8] when v = : m + y + z + zx y = 7 log ζ3 +.. π We can se the doble arcsine integral, Sv, w, to prove that if v [, ]: m v + x + y + z = sin d 4 Sv, π π = 4 Li3 v Li 3 v. π.3 The second eqality has been proved by Vandervelde [6]. Slightly more complicated argments lead to expressions that inclde m x /6 + y + z = π This fractional Mahler measre is defined by sin d π S,.4 m x /6 + y + z = m e πi/6 + y + z d,

3 notice that m x /6 + y + z m x+y +z. We can simplify the righthand side of Eq..4 by either expressing S, as a linear combination of L-fnctions, or in terms of a famos binomial sm: S, = 4 n= n 3 n n. Condon [6] proved an identity that Boyd and Rodrigez Villegas conjectred: m + x + xy + z = 8 ζ3..5 5π His proof also showed in a slightly disgised form that TS, = π We have generalized Condon s identity to show that m + x + v xy + z = π tan d 7 ζ tan d 4 TSv,,.7 π where Eq. 4.8 expresses TSv, in terms of polylogarithms. We can se this reslt to prove a nmber of new formlas, inclding: m x + v 4 + x + y + v + x z = π tan d 4 π TSv, + m x + v + x 4..8 When v = this redces to an interesting identity for ζ3 and the golden ratio: m x + + x + + x + y z = 8 ζ3 + log 5π We will show that all of the integrals TSv, w, Tv, w, and Sv, w have closed form expressions in terms of polylogarithms. The special case of TSv, will warrant extra attention, as it is related to an interesting family of binomial sms. Or closed forms are all derived throgh elementary methods. 3

4 Preliminaries: A description of the method, and some two dimensional Mahler measres Althogh there are many conjectred formlas for mlti-variable Mahler measres, most are extremely difficlt, if not impossible, to prove. Rather than attempting to prove any of these conjectres, we will take an easier approach. By investigating promising fnctions, and rewriting them as Mahler measres, we can recover a nmber of sefl formlas. Or first step was to determine a class of fnctions that we cold relate to Mahler s measre. We chose the three integrals TSv, w, Sv, w, and Tv, w, based on Condon s evalation of TS,, Eq..6. Condon s formla natrally sggested the existence of a generalized Mahler measre formla involving TSv,. From there, it was a small step to consider the similar fnctions TSv, w, Tv, w, and Sv, w. We will se the following method to express TSv,, Sv,, and Tv, /v as three-variable Mahler measres. First, a simple integration by parts changes each fnction into a two-dimensional integral, containing either a nested arcsine or arctangent integral. Recall that the following integrals define the arctangent and arcsine integrals respectively: w tan d, sin d. A typical formla for TSv,, Eq. 3.8, can be proved with little troble: TSv, = π tan d π/ sinθ tan z dzdθ. z Next, sbstitting a two-dimensional Mahler measre for the nested arctangent or arcsine integral will allow s to obtain a three-dimensional Mahler measre evalation. Theorem 3., Proposition 5., and Theorem 7.3 contain or main reslts from sing this method. Expressing the arcsine and arctangent integrals in terms of Mahler s measre represents the main difficlty in this approach. In the remainder of this section we will establish for two-variable Mahler measres for the arctangent integral, and one two-variable Mahler measre for the arcsine integral. Since many of or reslts involve polylogarithms, this will be a good place to define the polylogarithm. Definition. If z <, then the polylogarithm of order k is defined by Li k z = n= z n n k. 4

5 We call Li z the dilogarithm, and we call Li 3 z the trilogarithm. Theorem. reqires a formla of Cassaigne and Maillot []. In particlar, Cassaigne and Mallot showed that D a b πma + bx + cy = eiγ + α log a + β log b + γ log c, if π log max { a, b, c }, otherwise. The condition states that a, b, and c form the sides of a triangle. If is tre, then α, β, and γ denote the radian measres of the angles opposite to the sides of length a, b, and c respectively. In this formla, Dz denotes the Bloch-Wigner dilogarithm. As sal, Dz = Im Li z + log z arg z. Now that we have stated Cassaigne and Maillot s formla, we will prove Theorem.. Theorem. If v and w, then w sin tan d = π m + w + y + w z π 4 log + w. d = π mv + y + z. Proof. To prove Eq.. first recall the sal formla for this arcsine integral, sin d = Im Li e i sin v + sin v logv,.3 which is valid whenever v. Now apply Cassaigne and Maillot s formla to mv + y + z; we are in the case since v. It follows from a little trigonometry that Since πmv + y + z = D e i sin v + sin v logv, e i sin v =, D e i sin v = Im Li e i sin v, hence we obtain πmv + y + z = Im Li e i sin v + sin v logv. Comparing this last formla to Eq..3, we have π v mv + y + z = sin d. 5

6 To prove Eq.. first recall that if w, then w tan d = Im Li iw. Next observe that by Cassaigne and Maillot s formla πm + w + wy + z = D e πi/ w + tan w logw + π log + w = Im Li iw + π 4 log + w. Making a change of variables in the Mahler measre, it is clear that m + w + wy + z = It follows that for w we have w { m + w + + wyiz + m + w + wyiz } = m + w + + wy z = m + w + y + w z. tan d = π m + w + y + w z π 4 log + w. We can extend this formla to the entire positive real line. Sppose that w = /w where w, then /w tan d = π m + w + y + z π4 w log + w = π m + w + y + w z π 4 log + w π logw. Since the arctangent integral obeys the fnctional eqation [3] w tan d = π /w logw + tan d,.4 it follows that w tan d = π m + w + y + w z π 4 log + w. Therefore Eq.. holds for all w. The next theorem proves that Eq.. is not niqe. Using reslts from Theorem 6.5, we can derive three more Mahler measres for the arctangent integral. 6

7 Theorem.3 Sppose that w, then w w tan d = π 4 m + w + y + w yz + z,.5 tan d = π m y y + wz + z,.6 w tan d = π 4 m 4 + y z + z + w + z z + y w π 4 log π log + w..7 Proof. Since all three of these formlas have similar proofs, we will only prove Eq..5 and Eq..7. It is necessary to remark, that while Eq..5 follows from Eq. 6.5, and Eq..7 follows from Eq. 6.9, we mst start from Eq. 6.6 to prove Eq..6. Now we will proceed with the proof of Eq..5. From Eq. 6.5 we have where k = π 4k log π 4k log + k k k Im Li sin ir = k d, r, and < k <. After an integration by parts this becomes +r + k k k Im Li ir It follows immediately that = π + k 4k log log k k Im Li ir = 4 = 8 π/ π + k k + k sint log dt k sint log + + k sint k sint dt. d. Changing the log + term into a Mahler measre, which we can do by Jensen s formla, yields Im Li ir = π 4 m y + + k z+z. k z+z 7

8 Since k = r+r, we have Im Li ir = π 4 m y + r + r + z + z r + r z + z = π 4 m + yr + r + yz + z π 4 m r + r z + z = π 4 m + yr + r + yz + z π log + r + log + 4 r In order to sbstitte the arctangent integral for Im Li ir, we will assme that < r <. With this restriction, the formla becomes r tan d = π 4 m + yr + r + yz + z π 4 log r = π 4 m + y + r + r yz + z.8 We can manally verify that Eq..8 holds when r = and r =, and sing Eq..4 we can extend Eq..8 to all r >. Therefore, Eq..5 follows immediately. Next we will prove Eq..7. Using Eq. 6.6, we can show that Im Li ip = π logp + sin k d, where k = p +p, and < k <. To satisfy this restriction on k, we will assme that < p <. After several elementary simplifications, the right-hand side becomes = π logp + π log d + + log + k k = π logp + π + p π/ log cosθ + log + dθ p k sin θ = π + p log + π log + + cosθ k sin θ dθ 8

9 Since cosπ θ = cosθ, we have Im Li ip = π + p log + π log cosθ k sin θ dθ + π log + 4 cosθ k sin θ dθ. Applying Jensen s formla yields Im Li ip = π + p log + π m + y cos θ 4 k sin dθ θ = π + p log + π m + y z + z 4 + k z z = π + p log π 4 m + k z z + π m 4 + y z + z + k + y z z We can simplify the one-dimensional Mahler measre as follows: m 4 + k z z = m + ik z z = log + k + p = log. + p Eliminating k yields Im Li ip = π + p + p log π log + p + π m 4 + y z + z p + + y z z + p = π log π log + p + π m 4 + y z + z + p + + y z z. p 9

10 Since < p <, it follows that p tan d = π m 4 + y z + z + p + + y z z p π log π log + p..9 It is relatively easy to verify that Eq..9 holds when p = and p =. Using Eq..4, we can also extend Eq..9 to p >, which completes the proof of Eq Relations between TSv, and Mahler s measre, and a redction of TSv, w to mltiple polylogarithms The first goal of this section is to establish five identities relating TSv, to three-variable Mahler measres. We will prove these formlas in Theorem 3., sing the methods otlined in Section. Corollary 3.3 examines a few special cases of these reslts. Theorem 3.5 accomplishes the second goal of this section, which is to express TSv, w in terms of mltiple polylogarithms. This reslt, which appears to be new, is stated in Eq The importance of Eq. 3.4 lies in its easy proof, and more importantly in the fact that it immediately redces TSv, to mltiple polylogarithms. Finally, Proposition 3.6 will demonstrate that the mltiple polylogarithms in Eq. 3.4 always redce to standard polylogarithms. We will need the following simple lemma to prove Theorem 3.. Lemma 3. Assme that v and w are real nmbers with v > and w, ], then w TSv, w = tan v TSv, w = sin w sin z dz z tan d tan v sin w w v tanθ w sinθ sin z dzdθ, 3. z tan z dzdθ. 3. z Proof. To prove Eq. 3. first integrate TSv, w by parts to obtain: w TSv, w = tan v d d sin z dz z tan v w sin z dzd. z

11 Making the -sbstittion θ = tan v we have: w TSv, w = tan v sin z dz z which completes the proof of the identity. tan v w v tanθ The proof of Eq. 3. follows in a similar manner. sin z dzdθ, z The fact that Lemma 3. expresses TSv, w as a doble integral in two different ways, makes TSv, w more versatile than either Sv, w or Tv, w. These two different expansions will allow s to combine TSv, w with Mahler measres for both arctangent and arcsine integrals. Theorem 3. The following Mahler measres hold whenever v : m + x + v xy + z = π m x + v 4 + x + y + v + x z tan d 4 TSv, 3.3 π = tan d 4 π π TSv, m x + v + x 4 m + y + v 4 x + x + v y x + x z + z = 4 tan d π π TSv, m z z + v x + x y + y = π tan d π TSv, 4 + y z + z + v x + x 4 m + z z + y v x + x 4 = 4 tan d 8 π π TSv, + 4 π/ log + v sinθ dθ π + log 3.7 Proof. We will prove Eq. 3.3 first, since it has the most difficlt proof. Letting w = in Eq. 3. yields TSv, = π log tan v tan v tanθ/v sin z dzdθ. z

12 Since tanθ, we may sbstitte Eq.. for the nested arcsine integral v to obtain TSv, = π log tan v π = π log tan v π + π π/ tan v m tan v π/ m v tanθ + y + z dθ. m v tanθ + y + z dθ v tanθ + y + z dθ In the right-hand integral tanθ v, hence by Cassaigne and Maillot s formla m v tanθ + y + z = log v tanθ. Sbstitting this reslt yields: π/ TSv, = π log tan v + π tan v π π/ m v tanθ + y + z dθ = π tan d π π/ m = π log v tanθ dθ tanθ + v y + z dθ tan d π 4 m + x + v xy + z. Eq. 3.3 follows immediately from rearranging this final identity. The proofs of eqations 3.4 throgh 3.7 are virtally identical, hence we will only prove Eq Letting w = in Eq. 3., we have TSv, = π tan d π/ sinθ Sbstitting Eq..5 for the nested arctangent integral yields TSv, = π tan d π/ tan z dzdθ. 3.8 z π m + y + v sin θ + v sinθ yz + z dθ 4 = π tan d π 8 m + y v 4 x x + v i yx x z + z.

13 Letting x ix, we obtain TSv, = π tan d π 8 m + y + v 4 x + x + v yx + x z + z. Eq. 3.5 follows immediately from rearranging this final eqality. Finally, we will remark that the while Eq. 3.5 follows from sbstitting Eq..5 into Eq. 3.8, we mst sbstitte Eq.. to prove Eq. 3.4, Eq. 3.6 follows from sbstitting Eq..6, and Eq. 3.7 follows from sbstitting Eq..7. Corollary 3.3 The formlas in Theorem 3. redce, in order, to the following identities when v = : m + x + xy + z = 8 ζ3, 3.9 5π m x + + x + + x + y z = ζ3 + log, 3. 5π m + x + z + x + z + y + x z + x z = 56 5π ζ3, 3. m z z + x + x y + y = 8 ζ3, 3. 5π 4z + y + z + 3x + x m + z + y + x + x = 56 6 ζ3 + 5π 3π G + log. 3.3 In Eq. 3.3, and throghot the rest of the paper, G denotes Catalan s constant. In particlar, G = Proof. As we have already stated, Condon proved Eq. 3.9 in [6]. His proof also showed that TS, = π tan d 7 5 ζ3. Using this formla, eqations 3. throgh 3.3 follow immediately from Theorem 3.. Theorem 3. shows that we can obtain closed forms for several threevariable Mahler measres by redcing TSv, to polylogarithms. We have proved a convenient closed form for TSv, in Eq Corollary 4.6 3

14 also shows that this closed form immediately implies Condon s evalation of TS,. We will postpone frther discssion of Eq. 4.8 ntil Section 4. We will devote the remainder of this section to deriving a closed form for TSv, w in terms of mltiple polylogarithms. For convenience, we will se a slightly non-standard notation for or mltiple polylogarithms. Definition 3.4 Define F j x by F j x = and define F j,k x, y by F j,k x, y = x n+ n= n + = Li jx Li j x j n= x n+ n + j n m= y m+ m + k., We will employ this notation throghot the rest of the paper. Theorem 3.5 If v i, i] [i, i and w [, ], then we can express w v w TSv, w in terms of mltiple polylogarithms. Let R =, and let vw S = iw + w, then + + TSv, w =F 3 R F 3 RS F 3 R/S 4F, R, + F, R, S + F, R, /S + i sin w {F RS F R/S F, R, S + F, R, /S}. 3.4 Proof. First note that by -sbstittion TSv, w = sin w tan v w sinθ cotθθdθ. 3.5 Since w [, ], it follows that or path of integration is along the real axis. Next sbstitte the Forier series tan v w sinθ = n= R n+ sin n + θ, 3.6 n + into Eq Swapping the order of smmation and integration, we have TSv, w = n= R n+ sin w sin n + θ cotθθdθ. n + Uniform convergence jstifies this interchange of smmation and integration. In particlar, Eq. 3.6 converges niformly whenever R < and θ R. It 4

15 is easy to show that R < except when v i, i] [i, i, in which w case R =. If R =, then Eq. 3.6 no longer converges niformly, and hence the following argments do not apply. Evalating the nested integral yields TSv, w = 4 n= R n+ { sin w n + n sin k + sin w k= k + n cos k + sin w }, k + k= 3.7 where n ak = a + +a n + an. Simplifying Eq. 3.7 completes or proof. k= Eq. 3.4 deserves a few remarks, since it is a fairly general reslt. Firstly, observe that a closer analysis of Eq. 3.6 wold probably allow s to relax the restriction that w [, ]. Secondly, Eq. 3.4 most likely has applications beyond the scope of this paper. For example, we can se Eq. 3.4 to redce the right-hand side of the following eqation n w n n+ n k+ n + n k n= = TS, w π 4 w k= sin t dt + log t w sinh t dt, t 3.8 to mltiple polylogarithms. We can se the final reslt of this section, Proposition 3.6, to redce TSv, w to reglar polylogarithms. This proposition allows s to eqate TSv, w with a formla involving arond twenty trilogarithms. While a clever sage of trilogarithmic fnctional eqations might simplify this reslt, it seems more convenient to simply leave Eq. 3.4 in its crrent form. Proposition 3.6 The fnctions F, x, y and F, x, y can be expressed in terms of polylogarithms, we have: x + y 4F, x, y =Li + x Li x + y x x y Li + x x y + Li x. 3.9 To redce F, x, y to polylogarithms, apply Lewin s formla, Eq. 7.5, for 5

16 times to the following identity: F, x, y =F 3 xy log x F xy + x log log +y y d Proof. To prove Eq. 3., first swap the order of smmation to obtain F, x, y = F 3 xy + F xf y n= Sbstitting an integral for the nested sm yields y n+ n + n k= x k+ k +. y n+ x n+ F, x, y = F 3 xy + F xf y d n= n + x = F 3 xy + F yd. Integrating by parts, the identity becomes F, x, y =F 3 xy log x F xy + x log log +y y d, 4 which completes the proof of Eq. 3.. We can verify Eq. 3.9 by differentiating each side of the eqation with respect to y. Finally, observe that we can obtain simple closed forms for F, x, and F,, x from Eq An evalation of TSv, sing infinite series This evalation of TSv, generalizes a theorem de to Condon. Condon proved a formla that Boyd and Rodrigez Villegas conjectred: m + x + xy + z = 8 5π ζ3. Condon s reslt is eqivalent to evalating TS, in closed form. As Theorem 3. has shown, generalizing this Mahler measre depends on finding a closed 6

17 form for TSv,. Eq. 4.8 accomplishes this goal by expressing TSv, in terms of polylogarithms. This calclation of TSv, is based on several series transformations. The first step is to expand TSv, in a Taylor series; observe that the following formla holds whenever v < : TSv, = π k= k k + vk+ k= k k + 3 v k+ k k. 4. We can easily prove Eq. 4. by starting from Eq Formla 4. shows that TSv, is analytic in the open nit disk. Unfortnately Eq. 4. does not converge when v =, and hence it can not be sed to calclate TS,. It will be necessary to find an analytic contination of TSv, in order to carry ot any sefl comptations. The following family of fnctions will play a crcial role in or calclations. Definition 4. Define h n v by the infinite series, h n v = k= k k + n v k+ k k. 4. Using the definition of h 3 v, combined with the identity k= k k + vk+ = it follows that Eq. 4. can be rewritten as TSv, = π tan d, tan d h 3v. 4.3 Finding a closed form for TSv, we will entail finding a closed form for h 3 v. Theorem 4.5 accomplishes this goal, however first we need to prove several axiliary lemmas. The idea behind or proof is very simple: first we will find a closed form for h v and then integrate it to find a closed form for h 3 v. Batir recently sed this method in an interesting paper [] to obtain a formla that is eqivalent to Eq Unfortnately Batir seems to have missed Eq. 4., so we will provide a fll derivation of this important reslt. Lemma 4. The fnction h v is analytic if v i, i] [i, i. Fr- 7

18 thermore, we can express h v in terms of the dilogarithm, k+ v h v = 4 k= k v v = Li + v Li + v +. + v 4.4 Proof. We se the following elementary identity to prove Eq. 4.4, 4k j k! k + k = k j= j + k + j +!k j!. 4.5 Sbstitting Eq. 4.5 into the definition of h v, we have h v = = 4 k= k= k v k+ k + k k k v k+ j k! j + k + j +!k j!. j= If we assme that v <, then the series converges niformly, hence we may swap the order of smmation to obtain h v = 4 = 4 j= j= j + j + where x n = Γx+n. Bt then we have Γx h v = 4 j= k+j k + j! v k+j+ k= k + j +!k! v j+ j + kj + k v k, k= j + k k! v j+ j + F [ j+,j+ j+ v ], where F [ a,b c x ] is the sal hypergeometric fnction. A standard hypergeometric identity [7] shows that F [ j+,j+ j+ v ] = j+ + + v j+, from which we obtain j+ v h v = 4 j= j + +, + v conclding the proof of the identity. 8

19 We can se Eq. 4.4 to analytically contine h v to a larger domain. Recall that Li r Li r is analytic whenever r, ] [,, and v + is analytic whenever v i, i] [i, i. Since we have already +v assmed that v i, i] [i, i, we simply have to show that the range v of r = + does not intersect the set {, ] [, }. +v v Some elementary calcls shows that r = + +v for all v C, with eqality occrring only when v i, i] [i, i. It follows that h v is analytic on C { i, i] [i, i }. Since we have now expressed h v in terms of dilogarithms, we can find a closed form for h v by differentiating Eq. 4.4: h v = + v log v + + v. 4.6 In Theorem 4.5, we will integrate Eq. 4.4 to find a closed form for h 3 v involving trilogarithms. To prove this theorem, we first need to establish two lemmas. Lemma 4.3 evalates a necessary integral, while Lemma 4.4 expresses F,, x in terms of polylogarithms. v +, then we have the fol- +v Lemma 4.3 If j is an integer, and r = lowing identity: + + j+ d = log + r + rj+ j r j + r k+ k= k Proof. To evalate the integral w j v = j+ + d, + first make the sbstittion z = +. In particlar we can show that = + z and d = +z. Therefore we have z dz z w j v = = r r + z z j dz z r z dz z j r z dz z j+ dz. z Next sbstitte the geometric series zj z = j k= zk into each of the right- 9

20 hand integrals, and swap the order of smmation and integration to obtain r w j v = = log j z dz k= + r r r k+ k + j + rj+ j + j Lemma 4.4 The following doble polylogarithm k= k= r k+ k + r k+ k +. F,, x = n= n + can be evalated in closed form. If x <, n k= x k+ k + x 8F,, x =4Li 3 x Li 3 x 4Li 3 x 4Li 3 + 4ζ3 + x + x + log Li x + π π log + x + log x x log3 + x logx log x Proof. We will verify Eq. 4.9 by differentiating each side of the identity. First observe that the infinite series in Eq. 4.8 converges niformly whenever x, hence term by term differentiation is jstified at all points in the open nit disk. It follows that d dx F x n+,, x = n + x whenever x <. n= = π 8 x x x Li x 4 Li x, 4. Let ϕx denote the right-hand side of Eq Taking the derivative of ϕx we obtain: dϕ dx = 4 x Li x x Li x + 4 x Li x 4 x x Li + + x + x x Li x log + x log x + π π x + x 6 x + + x log + x x log x + 4 logx log x x 4.

21 We can simplify Eq. 4. by eliminating Li x and Li x +x with the fnctional eqations: Li x = π 6 logx log x Li x, x Li = + x log + x + Li x Li x. Sbstitting these identities into Eq. 4. and simplifying, we are left with dϕ dx = π x = d dx {8F,, x}. 8x Li x x 4 Li x Eq. 4. jstifies this final step. Since the derivatives of 8F,, x and ϕx are eqal on the open nit disk, and since both fnctions vanish at zero, we may conclde that 8F,, x = ϕx. The proof of Eq. 4.9 reqires a remark. Despite the fact that the righthand side of Eq. 4.9 is single valed and analytic whenever x <, the individal terms involving Li 3 x and logx are mltivaled for x,. To avoid all ambigity, we can simply se F,, x = F,, x to calclate the fnction at negative real argments. Theorem 4.5 The fnction h 3 v is analytic on C { i, i] [i, i }. If v i, i] [i, i, then h 3 v can be expressed in terms of polylogarithms. v Let r = +, then +v h 3 v = r Li 3r + 4Li 3 r + 4Li 3 4ζ3 + r + r log Li r π r 3 log r 3 log3 + r + logr log r. We can recover an eqivalent form of Condon s identity by letting v = : 4. h 3 = 4 ζ Proof. This proof is very simple since we have already completed all of the hard comptations. Observe from Eq. 4. that if v <, h 3 v = h d. 4.4 Lemma 4. shows that h v is analytic provided that v i, i] [i, i. If we assme that the path of integration does not pass throgh either of

22 these branch cts, then it is easy to see that Eq. 4.4 provides an analytic contination of h 3 v to C { i, i] [i, i }. Next we will prove Eq. 4.. Sbstitting Eq. 4.4 into Eq. 4.4 yields an infinite series for h 3 v that is valid whenever v i, i] [i, i. We have v n+ h 3 v = 4 n + + d. + n= The nested integrals can be evalated by Lemma 4.3. Letting r = is clear that + r h 3 v =4 log + rn+ n n= n + r n + r j+ j= j + = π log + r r + 4Li 3 r Li 3r 8F,, r, v + it +v 4.5 where F,, r has a closed form provided by Eq Since r < whenever v i, i] [i, i, we may sbstitte Eq. 4.9 to finish the calclation. Observe that when v =, we have r = 5. It is easy to verify that 3 5 = r = r = r. Using Eq. 4., it follows that +r h 3 = Li 3 4ζ3 3 log Li + 4π log log3. Eq. 4.3 follows immediately from sbstitting the classical formlas for Li and 3 5 Li into Eq Notice that Eq. 4.3 is eqivalent to a new evalation of the 4 F 3 hypergeometric fnction, [ ],, 4F, 3 3, 3, 3 4 = 7 ζ Corollary 4.6 Let r = then TSv, = π TS, = π tan + ζ3 + log + r r v + and sppose that v i, i] [i, i, +v d r 4 Li 3r Li 3 r Li 3 + r Li r + π log r log3 + r logr log r, 4.8 tan d 7 ζ

23 Proof. Eq. 4.8 follows immediately from sbstitting Eq. 4. into Eq. 4.3, while Eq. 4.9 follows from combining Eq. 4.3 with Eq The fact that we can redce h v, h v and h 3 v to standard polylogarithms is somewhat miraclos. Integrating Eq. 4.5 again, we can show that h 4 v = π r r + r log r log + Li Li 4 + r + π 4. F r + 4F 3 r 8F 3,, r 8F,, r + 6F,,,, r. Considering the complexity of these mltiple polylogarithms, it seems nlikely that h n v will redce to standard polylogarithms for n 4. 5 Relations between Sv, and Mahler s measre, and a closed form for Sv, w. In this section we will stdy the doble arcsine integral, Sv, w. Recall that we defined Sv, w with an integral: Sv, w = sin vx sin wx dx. x First, we will show that both Sv, and Sv, v redce to standard polylogarithms. Next, we will discss several interesting reslts relating Sv, and Sv, v to Mahler s measre and binomial sms. Finally, Theorem 5.4 concldes this section by expressing Sv, w in terms of polylogarithms. Theorem 5. Assme that v, then Sv, v and Sv, both have simple closed forms: Sv, = π sin x Li3 v Li 3 v dx, 5. x Sv, v = Li 3 e i sin v + Li 3 e i sin v ζ3 4 + sin v Li e i sin v Li e i sin v 5. i + sin v logv. Proof. To prove Eq. 5., we will sbstitte the Taylor series for sin vx into the integral Sv, = dx. After swapping the order of sin vx sin x x 3

24 smmation and integration, we have Sv, = = π = π n= n= n + n + v n n+ sin xx n dx n v n n+ v n+ n n= n + 3 Li3 v Li 3 v. sin x dx x To prove 5. make the -sbstittion x = sint, and then integrate by v parts as follows: Sv, v = sin vx x dx = sin v t cottdt = sin v sin v logv t logsintdt. Next sbstitte the Forier series for logsint into the previos eqation, recall that cosnt logsint = log n is valid for < t < π. Integrating by parts a second time completes the proof. The fnction Sv, v provides a connection to a second family of interesting binomial sms. If we recall the formla sin x = n= x n, n= n n n then it is immediately obvios that if v we mst have Sv, v = 4 v n. 5.3 n= n 3 n n Comparing Eq. 5.3 with Eq. 5. yields a classical formla: S, = 4 n= n 3 n n = n= cos πn 3 ζ3 n 3 + π 6 n= sin πn n 4

25 Proposition 5. If v [, ] and w, ], we have Sv, w = sin w π sin w sin d m v w sinθ + y + z dθ 5.5 Proof. This proof is similar to the proof of Proposition 3.. After an integration by parts, and the -sbstittion = sinθ/w, we obtain Sv, w = sin w sin d π sin w w sinθ sin z dzdθ. z Since v and < w, it follows that v sinθ. Therefore w we may complete the proof by sbstitting Eq.. for the nested arcsine integral. Corollary 5.3 We can recover Vandervelde s formla by letting w = in Eq. 5.5: m v + x + y + z = sin d 4 Sv, π π = Li3 v Li 3 v π Notice that if v = w = in Eq. 5.5, we have S, = π / sin d π π/6 m sinθ + y + z dθ 6 = π / sin d π 6 m x /6 + y + z 5.7 Comparing Eq. 5.7 to Eq. 5.4 allows s to express a famos binomial sm as the Mahler measre of a three-variable algebraic fnction. The final reslt of this section allows s to express Sv, w in terms of standard polylogarithms. Theorem 5.4 Sppose that v < w, and let θ = sin w sin v. 5

26 Then we have Sv, w =Sv, v + Sw, w S sinθ, sinθ v v + Li 3 w eiθ + Li 3 w e iθ v Li 3 w v v iθli w eiθ + iθli w e iθ + θ log + v w v w cosθ. 5.8 Notice that Eq. 5. redces Sv, v, Sw, w, and S sinθ, sinθ to standard polylogarithms. Proof. The details of this proof are not particlarly difficlt. First observe the following trivial formla: Sv, v Sv, w + Sw, w = sin w sin v d. Rearranging, and then applying the arcsine addition formla yields Sv, w =Sv, v + Sw, w sin w v v w d. 5.9 This sbstittion is jstified by the monotonicity of the arcsine fnction. In particlar, v < w implies that sin w sin v π for all [, ]. Next we will make the -sbstittion z = w v v w. In particlar, we can show that = and we can easily verify that z w + v vw z, d dz = z vwz v + w vw z z. 6

27 Observe that the new path of integration will rn from z = to z = sinθ = w v v w. Therefore, Eq. 5.9 becomes Sv, w =Sv, v + Sw, w sinθ sin z z vwz v + w vw z dz z =Sv, v + Sw, w S sinθ, sinθ sinθ + sin z vwz v + w vw z z dz. If we let t = sin z, then this last integral becomes Sv, w =Sv, v + Sw, w S sinθ, sinθ θ + t vw sint v + w vw cost dt. 5. Since v < w, a formla from [7] shows that vw sint v + w vw cost = n= v w n sinnt. 5. The Forier series in Eq. 5. converges niformly since v < w. It follows that we may sbstitte Eq. 5. into Eq. 5., and then swap the order of smmation and integration to obtain: Sv, w =Sv, v + Sw, w S sinθ, sinθ v n θ + t sinntdt. n= w 5. Simplifying Eq. 5. completes the proof of Eq q-series for the dilogarithm, and some associated trigonometric integrals In this section we will prove several doble q-series expansions for the dilogarithm. While these formlas are relatively simple, it appears that they are new. The first of these formlas, Eq. 6.8, follows from a few simple maniplations of Eq. 5.. The remaining formlas follow from integrals that we have evalated in Theorem 6.5. Recall that Theorem 6.5 figred prominently in the proof of Theorem.3. In this section, the twelve Jacobian elliptic fnctions will play an important role or calclations. Recall that the Jacobian elliptic fnctions are dobly 7

28 periodic and meromorphic on C. The Jacobian sine fnction, sn, inverts the incomplete elliptical integral of the first kind. If C is an arbitrary nmber, then nder a sitable path of integration: = sn dz z k z. The Jacobian amplitde can be defined by the eqation sn = sinam, and the Jacobian cosine fnction is defined by cn = cosam. As sal the complementary sine fnction is given by dn = k sn. Notice that every Jacobian elliptic fnction implicitly depends on k; this parameter k is called the elliptic modls. Following standard notation, we will denote the real one-qarter period of sn by K. Since snk =, we may compte K from the sal formla K := Kk = = π F [, dz z k z k ]. Let K = K k K π, and finally define the elliptic nome by q = e K. Proposition 6. If k,, then we have the following integral: K amcnd = π sin k k Li k Li k. 6. k Proof. Taking the derivative of each side of Eq. 5., we obtain: d dk Sk, = sin x k x dx = π sin k Li k Li k. 6. k k Making the -sbstittion x = sn completes the proof. We will need the following two inversion formlas for the elliptic nome. Lemma 6. Let q be the sal elliptic nome. Sppose that q,, then q is invertible sing either of the formlas: n q n+/ k = sin 4, 6.3 n= n + + q n+ q n+/ k = tanh n + q n+ n= 8

29 Proof. To prove Eq. 6.3 observe that sin dx k = k k x = k K Recall the Forier series expansion [7] for cn: cn = π kk n= cnd 6.5 q n+/ πn + cos + qn+ K. 6.6 Since < q <, this Forier series converges niformly. It follows that we may sbstitte Eq. 6.6 into Eq. 6.5, and then swap the order of smmation and integration to obtain: sin k = π q n+/ K πn + cos K n= + q n+ K d n q n+/ = 4 n + + q n n= Eq. 6.3 follows immediately from taking the sine of both sides of the eqation. Eq. 6.4 can be proved in a similar manner when starting from the integral tanh dx k = k k x. Next we will tilize the Forier-series expansions for the Jacobian elliptic fnctions to prove the following theorem: Theorem 6.3 If q is the sal elliptic nome, then the following formla holds for the dilogarithm: Li k Li k 8 q n+/ = n= n + + q n+ + 4 n= m= n + m q n+m+/ + q m + q n+ 6.8 Proof. We have already stated the Forier series expansion for cn in Eq We will also reqire the Forier series [7] for am: am = π K + n= q n πn n + q sin n K

30 Sbstitting Eq. 6.6 and Eq. 6.9 into the integral in Eq. 6., and then simplifying yields: Li k Li k 8 = π π 8 sin k n q n+/ n + + q n+ + n= + 4 n= m= n= n + q n+/ + q n+ n + m q n+m+/ + q m + q n+. 6. This proof is nearly complete, the final step is to sbstitte the identity sin k = 4 n q n+/ n + + q n+ n= into Eq. 6.. This formla for sin k follows immediately from Lemma 6.. The fact that Eq. 6.8 follow easily from an integral of the form K amϕd, sggests that we shold try to generalize Eq. 6.8 by allowing ϕ to eqal one of the other eleven Jacobian elliptic fnctions. Theorem 6.5 proves that ten of these eleven integrals redce to dilogarithms and elementary fnctions. First, Theorem 6.4 will prove that the one exceptional integral can be expressed as the Mahler measre of an elliptic crve. Theorem 6.4 The following formlas hold whenever k, ]: 4 m k + x + x +y + y k = log + + k π k = log + + k π K sin x x dx k x 6. am cn d. sn 6. Proof. First observe that if k R and < k, then m 4 k + x + x + y + = log y k + m + k x x + y +. y 3

31 For brevity let ϕk = m + k 4 x + + y + x y. Making the change of variables x, y x/y, yx, we have: ϕk =m + k x + x y + y 4 k =m y + y + m x + 4 x + 4 k y + y k = log + m x + 4 x +. 4 k y + y Applying Jensen s formla with respect to x redces ϕk to a pair of onedimensional integrals: k ϕk = log + π log + + k cos θ 4 π k cosθ dθ π log + k cos θ π k cosθ dθ. The right-hand integral vanishes nder the assmption that < k. Therefore, it follows that Eq. 6.3 redces to ϕk = log k + π/ 4 π log + k cos θ dθ. k cosθ With the observation that π/ log cosθ dθ = π log, this formla becomes: ϕk = π/ log + k cos θ dθ. 6.4 π Making the -sbstittion of x = cosθ, we obtain ϕk = π log + k x dx. x Integrating by parts to eliminate the logarithmic term yields: + k ϕk = log + sin x k x dx π x k x + k = log + sin x π x k x dx sin x dx. π x Since sin x dx = π log, it follows that x + k ϕk = log + 4 π sin x x k x dx, 3

32 from which we obtain 4 m k + x + x + y + k = log y + + sin x k π x k x dx. To prove Eq. 6. simply make the -sbstittion x = sn. The elliptic crve defined by the eqation 4/k + x + /x + y + /y = was one of the simplest crves that Boyd stdied in [4]. Rodrigez Villegas derived q-series expansions for a wide class of fnctions defined by the Mahler measres of elliptic crves in [4]. We can recover one of his reslts by sbstitting the Forier series expansions for am and cn/sn into Eq. 6.. If we let k = sinθ, and then integrate Eq. 6. from θ = to θ = π, we can prove that m 8 + z + x + z x + y + = 4 y π G + 4 π sin x Kxdx. 6.5 x Using Mathematica, we can redce the right-hand integral to a rather complicated expression involving balanced hypergeometric fnctions evalated at one. Theorem 6.5 We will assme that < k < and that each Jacobian elliptic fnction has modls k. Let p = k, r = k +k +, and s = k k k, then K K K K K amsnd = sin k d = Li is Li is ki sin amcnd = k d = π amdnd = π 8 am sn d = sin k k Li k Li k k sin k d 6.9 = π log p + Li ip Li ip i am d = 6. cn 3

33 K K K K K K K am dn d = sin k d π = k 8 + Li r Li r 6. am sn d = cn 6. am sn dn d = sin k d = Li r Li r k k am cn sn d = sin k d = π log r + π m 4 k + x + x + y + y am cn dn d = sin k d = π k logp Li ir Li ir ki am dn d = G 6.6 sn am dn d = 6.7 cn Proof. First observe that Eq. 6., Eq. 6., and Eq. 6.7 all follow from the fact that cnk =. Similarly, Eq. 6.8 and Eq. 6.6 both follow from the formla d am = dn. d We already proved Eq. 6.7 in Proposition 6., and Eq. 6.4 was proved in Theorem 6.4. This leaves a total of five formlas to prove. To prove Eq. 6.6, observe that after letting = z, we have sin is sin z d = k ki is z dz. If < k /, then s. With this restriction on k, we may expand the 33

34 sqare root in a Taylor series to obtain: = / m is m+ sin z ki m= m z m dz = is m+ ki m + m= = Li is Li is. 6.8 ki Notice that Eq. 6.8 extends to < k <, since both sides of the eqation are analytic in this interval. Therefore, Eq. 6.6 follows immediately. To prove Eq. 6.9 make the -sbstittion = sin z k +z = tan z k, we obtain sin k d = tan z k. Recalling that +z z k z + z Using Mathematica to evalate this last integral yields: = π logp + [ ] k 3F,, 3, 3 k n+ = π logp + k n= n + n n = π logp + Li ip Li ip, i where Eq. 4.4 jstifies the final step. To prove Eq. 6. observe that after the -sbstittion = sinθ we have sin k d = π/ θ k sin θ dθ. Now sbstitte the Forier series k k sin θ = + m m k + cosmθ 6.9 k m= into the integral, and simplify to complete the proof. The proof of Eq. 6.3 follows the same lines as the derivation of Eq. 6.. Observe that dz. sin k d = π/ θ sinθ k sin θ dθ. 34

35 Now sbstitte the Forier series k k sinθ m+ k sin = m k θ m= + sin m + θ 6.3 k into the integral, and simplify to complete the proof. Finally, we are left with Eq Expanding / k in a geometric series yields: sin k d = = k n n= k n n= sin n d π/ n + = π 4k log k + k n n + n n h ik ik. Sbstitting the closed form for h ik provided by Eq. 4.4 completes the proof. We can obtain each of the following q-series by applying the method from Theorem 6.3 to the formlas in Theorem 6.5. Corollary 6.6 Let p = k, and let r = k +k +. The following formlas k hold for the dilogarithm: Li k Li k 8 Li r Li r 4 Li ip Li ip 8i q n+/ = n= n + + q n+ + 4 n= m= n + m q m+n+/ + q m + q n+, 6.3 q n+/ = n= n + + q n+ + 4 n= m= m n + m q m+n+/ + q m + q n+, 6.3 = G 4 + π 6 logp n= m= n= n n + q n+ q 4n+ n+m n + m q m+n+ + q m q 4n Proof. As we have already stated, each of these formlas can be proved by sbstitting Forier series expansions for the Jacobian elliptic fnctions into Theorem

36 Using the method described, we have already proved Eq. 6.3 in Theorem 6.3. Eq. 6.3 follows in a similar manner from Eq Eq is a little trickier to prove. Expanding Eq. 6.6 in a q-series yields the identity q n n + q n n= n j= j j + n q n+ = n= n + + q n+ + 4 n= m= n+m n + m q m+n+ + q m + q n Next expand Eq. 6.9 in the q-series Li ip Li ip 4i = G + π 8 logp q n n + q n n= n= m= n j= j j + + n= n n + q n+ q n+ n+m n + m q m+n+ + q m q n+, and then combine it with Eq to complete the proof of Eq It is important to notice that the nine convergent integrals in Theorem 6.5 only prodce three interesting q-series for the dilogarithm. The other q-series we may obtain from Theorem 6.5 really jst restate known facts abot the elliptic nome. For example, if we expand Eq. 6. in a q-series, we will obtain Eq. 6.3 with q replaced by q and k replaced by r. This is eqivalent to k the fact that q + = q k. If we let l = k, k + k then clearly k and l satisfy a second degree modlar eqation []. 7 A closed form for Tv, w, and Mahler measres for T v, v Recall that we defined Tv, w sing the following integral: Tv, w = tan vx tan wx dx. 7. x Since this integral involves two arctangents, rather than one or two arcsines, Tv, w possesses a nmber of sefl properties that Sv, w and TSv, w appear to lack. 36

37 First observe that Tv, w obeys an eight term fnctional eqation. If we let Tv = tan x dx, then we can se properties of the arctangent fnction x to prove the following formla: Tv, w + T v, w T w v, = π Tv + T v T w, w v T T. w 7. If v < and w <, we can sbstitte arctangent Taylor series expansions into Eq. 7. to obtain: n w n+ n Tv, w = n= n + m= n v n+ n + n= n + m= v/w m+ m + w/v m+ m Eq. 7.3 immediately redces Tv, w to mltiple polylogarithms. Theorem 7. improves pon this reslt by expressing Tv, w in terms of standard polylogarithms. Theorem 7. If v and w are real nmbers sch that w/v, then w 4Tv, w =Li 3 Li 3 w vi + vi + Li 3 + Li 3 v v wi + wi + vi vi Li 3 Li 3 wi + wi w vi w + vi Li 3 Li 3 v wi v + wi w + vi w vi + Li 3 + Li 3 v wi v + wi + v w + log Li + w Li w v v 4 tan Li wi Li wi v i 4 tan Li vi Li vi w i + v π log tan w + 4 logv tan v tan w. + w

38 Proof. Sbstitting logarithms for the inverse tangents, we obtain 4Tv, w = = iw + iv + d log log iv iw + v w log + d v log. w The identity then follows more or less immediately from for applications of Lewin s formla x log z log cz dz z cx =Li 3 + Li 3 + Li 3 x c cx Li 3 cx Li 3 x Li 3 c x [ ] + log cx Li Li x [ c ] + log x Li cx Li + π c 6 + logc log x, 7.5 which was proved in []. Condon has discssed the intricacies of applying this eqation in [6]. This closed form for Tv, w is qite complicated. Notice that a slight change in the integrand in Eq. 7. prodces a remarkably simplified formla: tan vx tan wx x n+ v + w +v dx = π + +w. 7.6 n= n + To prove Eq. 7.6, make the -sbstittion x = sinθ, and then apply Eq. 3.6 twice. There are two special cases of Eq. 7.4 worth mentioning. First observe that T v, v redces to a very simple expression. If we let w /v in Eq. 7.4, and perform a few tortros maniplations, we can show that T v, = π v Im [Li iv] Li3 v Li 3 v + logv Li v Li v. 7.7 Lalín obtained an eqivalent form of Eq. 7.7 sing a different method. See Appendix in [8]. Lalín s formla for Tv, + T/v, redces to Eq

39 after applying Eq. 7. with w = /v. Observe that when w = v in Eq. 7.4, we have Tv, v = [ + vi Re Li 3 Li 3 + vi ] 7 vi vi 8 ζ3 + tan vim [Li iv] logv tan v 7.8. Finally, it appears that Tv, does not redce to any particlarly simple expression. Letting w fails to simplify Eq. 7.4 in any appreciable way. Expanding Tv, in a Taylor series reslts in an eqally complicated expression: Tv, = n= + π 4 v n+ n + n k+ k= k tan x dx log Li v Li v x Theorem 7.3 relates Tv, w to three-variable Mahler measres, and generalizes one of Lalín s formlas. Once again, we will need a simple lemma before we prove or theorem. Lemma 7. Sppose that v and w are positive real nmbers, then w Tv, w = tan v T v, = π v tan d tan d tan v w v tanθ π/ tanθ tan z dzdθ, 7. z tan z dzdθ. 7. z Proof. While we can verify Eq. 7. with a trivial integration by parts, the proof of Eq. 7. is slightly more involved. To prove Eq. 7., first let w = v in Eq. 7.. This prodces T v, /v = tan v v tan v tan d tanθ v tan z dzdθ. z 7. Letting v /v in Eq. 7. gives T v, v = tan tan tan v d v π = tan tan v d π/ tanθ tan v tanθ tan z dzdθ z tan z dzdθ z 39

40 Now apply Eq..4 twice, which transforms this last identity to π T v, v = tan v v π/ tan v tanθ v tan d + π logv tan z dz π z log v tanθ dθ 7.3 To complete the proof, simply add eqations 7. and 7.3 together, and then simplify the reslting sm. Theorem 7.3 Sppose that v >, then the following Mahler measres hold: x x m v 4 + y + v z + x + x = 4 tan d 8 π π T v, + x v m v 4, + x x x m v 4 + v y z z + x + x + y = 8 tan d 6 v, π π T, v y m y x z + v z + x = 4 tan d 8 π π T v,, v 4 + y z + z x v 4 + x m + z z + y x + v 4 + x = 8 tan d 6 v, π π T + 4 π/ log + v tanθ dθ v π + log Proof. Each of these reslts follows, in order, from sbstitting Eq.., Eq..5, Eq..6, and Eq..7, into Eq. 7.. Corollary 7.4 If we let v = in Eq. 7.4, we can recover one of Lalín s formlas [8]: m + z + y + zx y = 7 log ζ π 4

41 Letting v = in Eq. 7.5, Eq. 7.6, and Eq. 7.7, yields in order: m 4 + y + y x x z z = 4 ζ3 7.9 π m + x y y + x z z = 7 ζ3 7. π m 6 + y 4 z + z + + y z z x + x = 4 π ζ π G Proof. To prove Eq. 7.8, let v = in Eq From Eq. 7.7 we know that T, = πg 7 ζ3, hence 8 7 ζ3 + log = m π Now let x, y, z x, y z, xz to obtain x + + x y + x + x = m 4x + + xy + x z. z 7 π ζ3 + log = m 4x + xy + xz x = m 4 + xy + xz = m + + xy + xz. With the final change of variables x, y, z z,, x yz yz, we have 7 + z ζ3 + log = m + + π yz completing the proof of Eq zx yz = m + z + y + zx y, The proofs of Eq. 7.9 throgh Eq. 7. follow almost immediately from or evalation of T,. The proof Eq. 7. also reqires the fairly easy fact that π/ log + tanθ dθ = G + π log 4 8 Conclsion In principle, we shold be able to apply the techniqes in this paper to prove formlas for infinitely many three-variable Mahler measres. The main difficlty, which is significant, lies in the challenge of finding infinitely many 4

42 Mahler measres for the arctangent and arcsine integrals. In Section we proved one sch formla for the arcsine integral, and for formlas for the arctangent integral. 9 Acknowledgements I wold like to thank my advisor, David Boyd, for bringing Condon s paper to my attention. He also made several sefl sggestions on evalating the integrals in Theorem 6.5. I trly appreciate his spport. Finally, I wold like the thank the Referee for the helpfl sggestions. References [] N. Batir, Integral representations of some series involving k k k n and some related series, Applied Math. and Comp. 47 4, [] B.C. Berndt, Ramanjan s Notebooks part IV, Springer-Verlag, New York, 994. [3] D.W. Boyd, Speclations concerning the range of Mahler s measre, Canad. Math. Bll. 4 98, [4] D.W. Boyd, Mahler s measre and special vales of L-fnctions, Experiment. Math , [5] D.W. Boyd, F. Rodrigez Villegas, Mahler s measre and the dilogarithm I, Canad. J. Math. 54, [6] J. Condon, Calclation of the Mahler measre of a three variable polynomial, preprint, October 3. [7] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Prodcts, Academic Press, 994. [8] M. N. Lalín, Some examples of Mahler measres as mltiple polylogarithms, J. Nmber Theory. 3 3, [9] M. N. Lalín, Mahler measre of some n-variable polynomial families, preprint 4, to appear in J. Nmber Theory. [] M. N. Lalín, Some relations of Mahler measre with hyperbolic volmes and special vales of L-fnctions, dissertation, 5. [] L. Lewin, Polylogarithms and Associated Fnctions, Elsevier North Holland, New York, 98. 4

43 [] V. Maillot, Géométrie d Arakelov des variétés toriqes et fibrés en droites intégrables. Mém. Soc. Math. Fr. N.S 8, 9pp. [3] S. Ramanjan, On the integral x tan t t dt, J. Ind. Math. Soc. 7 95, [4] F. Rodrigez Villegas, Modlar Mahler measres I, Topics in nmber theory University Park, PA, 997, 7 48, Math. Appl., 467, Klwer Acad. Pbl., Dordrecht, 999. [5] C.J. Smyth, An explicit formla for the Mahler measre of a family of 3-variable polynomials, J. Th. Nombres Bordeax, 4, [6] S. Vandervelde, A formla for the Mahler measre of axy + bx + cy + d, J. Nmber Theory, 3, 84-. [7] G.N. Watson, A Treatise on the Theory of Bessel Fnctions, Cambridge University Press, 9. 43