Analytic continuation of the doubly-periodic Barnes zeta function

Applied Mathematics and Computation 221 (2013) 598–609

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Analytic continuation of the doubly-periodic Barnes zeta function Guglielmo Fucci a,⇑, Klaus Kirsten b a b

Department of Mathematics, East Carolina University, Greenville, NC 27858, USA Department of Mathematics, Baylor University, Waco, TX 76798, USA

a r t i c l e

i n f o

a b s t r a c t The aim of this work is to study the analytic continuation of the doubly-periodic Barnes zeta function. By using a suitable complex integral representation as a starting point we find the meromorphic extension of the doubly periodic Barnes zeta function to the entire complex plane in terms of a real integral containing the Hurwitz zeta function and the first Jacobi theta function. These allow us to explicitly give expressions for the derivative at all non-positive integer points. Ó 2013 Elsevier Inc. All rights reserved.

Keywords: Analytic continuation Barnes zeta function Special functions

1. Introduction The Barnes zeta function, introduced for the first time in [1,2], represents a higher dimensional generalization of the Hurwitz zeta function

fH ðs; aÞ ¼

1 X ðn þ aÞs

for Rs > 1:

ð1:1Þ

n¼0

Namely, let s 2 C; l 2 Rþ , and r 2 Rdþ . For Rs > d the Barnes zeta function is defined through the series

fB ðs; ljrÞ ¼

X

ðl þ n  rÞs

ð1:2Þ

n2Nd0

and it can be analytically continued in a unique way to a meromorphic function in the entire complex plane possessing only simple poles at s ¼ 1; 2; . . . ; d. In this work we shall be interested in the meromorphic extension of a function closely related to the two dimensional Barnes zeta function. Let s 2 C n R; Xm;n ¼ n þ sm with m; n 2 Z, and a 2 C n Xm;n . For Rs > 2 we consider the following zeta function

fðs; a; sÞ ¼

X

ða þ n þ smÞs ;

ð1:3Þ

ðm; nÞ2Z2

which is the kind of zeta function resulting from Dirac operators on the two-torus as considered in generalized Thirring models [8]. Note that since fðs; a; sÞ ¼ fðs; a; sÞ we can assume, without loss of generality, that Is > 0. The zeta function defined above is doubly-periodic with respect to the variable a. In fact for k 2 Z one has the relations fðs; a þ k; sÞ ¼ fðs; a; sÞ and fðs; a þ ks; sÞ ¼ fðs; a; sÞ, and due to this double-periodicity we assume, without loss of generality, that 0 6 Ra < 1 þ Rs ⇑ Corresponding author. E-mail addresses: [email protected] (G. Fucci), [email protected] (K. Kirsten). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.06.092

G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

599

and 0 < Ia < Is. Since the zeta function (1.3) is analytic in the semi-plane Rs > 2 and since Is – 0, one can conclude that (1.3) belongs to the class of elliptic functions [11]. The main idea of the present work is to represent the doubly-periodic Barnes zeta function (1.3) for Rs > 2 in terms of a contour integral in the complex plane. The desired analytic continuation of (1.3) to the region Rs 6 2 is then achieved by a suitable deformation of the integration contour. This process yields an expression for (1.3) valid in the entire complex plane in terms of an integral over the interval ½0; 1 of the Hurwitz zeta function and the logarithmic derivative of the first Jacobi theta function. The analytically continued expression for fðs; a; sÞ will allow us to very easily compute its values at all integer points, s 2 Z. In addition, we will also provide an explicit expression for the derivative of fðs; a; sÞ with respect to s at all nonpositive integer points. We would like to point out that one of the main advantages of our study of the analytic continuation of (1.3) is that its double-periodicity property remains manifest in all the formulae. This is an aesthetically pleasing feature that can also be desirable if one wishes to implement these expressions in a computer program for numerical evaluations. The study of a zeta function closely related to the doubly-periodic Barnes zeta function considered here has appeared in [4] where a method for obtaining its analytic continuation was used which differs from the one we employ in this work. The outline of the paper is as follows. In the next section we construct a contour integral representation for fðs; a; sÞ valid for Rs > 2. From this representation we perform, in Section 3, the analytic continuation in the entire complex plane. Section 4 is devoted to the explicit computation of the derivative of fðs; a; sÞ with respect to the first variable at all negative integer points and at s ¼ 0.

2. Contour integral representation of fðs; a; sÞ As we have already mentioned in the Introduction, the main idea of our approach is to represent the doubly-periodic Barnes zeta function (1.3) in terms of a contour integral. We introduce, for convenience, the function

fm ðn; sÞ ¼

1 ; ðam þ nÞs

ð2:1Þ

with n 2 Z and am ¼ a þ sm 2 C n f0g, where m 2 Z. Obviously, in terms of the newly introduced function fm ðn; sÞ, the zeta function (1.3) reads

fðs; a; sÞ ¼

XX fm ðn; sÞ:

ð2:2Þ

m2Z n2Z

By utilizing Cauchy’s residue theorem we rewrite the sum over the index n in (2.2) as a contour integral. More precisely one has

X

X 1 1 fm ðn; sÞ ¼ s ¼ 2 pi a þ nÞ ð m n2Z n2Z

Z

dz f m ðz; sÞ

p cotðpzÞ;

ð2:3Þ

C

where C is a contour that encloses counterclockwise all the (simple) poles of the function p cotðpzÞ. Let us point out that the representation (2.3) is well defined in the region Rs > 2. Before specifying the integration contour C in detail, we would like to observe that the function fm ðz; sÞ, obtained from (2.1) by replacing n with z, possesses branch cuts extending from the points z ¼ am . The exact position of the cut will depend explicitly on the summation index m. First, note that from the assumptions stated below (1.3) one has the inequalities Is > 0; Ra P 0, and Ia > 0. This allows us to conclude that Iam > 0 for m P 0 and Iam < 0 for m < 0. The last remark shows that the cut lies in the lower half complex plane when m P 0 and in the upper half complex plane when m 6 1. The contour C has to be chosen in such a way as to enclose only the poles of cotðpzÞ but not the branch points of the function fm ðz; sÞ. More precisely, the contour is the union C ¼ Cþ [ C where Cþ satisfies the property 0 < ICþ < Iam for m 6 1 while C satisfies Iam < IC < 0 for m P 0 (which simply means that the contour is closer to the real axis than the cut). The contour C is depicted in Fig. 1 with aP0 and a61 denoting representatives of the set of branch points with index m P 0 and m 6 1, respectively. With the contour of integration completely determined we can express the left hand side of (2.3) as a sum of two contributions

X

1 1 s ¼ 2 pi a þ nÞ ð m n2Z

Z

dz f m ðz; sÞ p cotðpzÞ þ

Cþ

1 2pi

Z

dz f m ðz; sÞ

p cotðpzÞ:

ð2:4Þ

C

The next step of our approach is to rewrite the integrand in (2.4) in a way that will allow a separate treatment of the integral over Cþ and over C . By utilizing the complex exponential representation of the function cotðpzÞ and after some straightforward algebraic manipulations we obtain

 cotðpzÞ ¼ i 1 þ

2 e2piz  1

 ;

for Iz > 0

ð2:5Þ

600

G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

ℑz

α≥0

−α≤−1

Γ

ℜz Γ− α≤−1

−α≥0

Fig. 1. Integration contour used for the representation (2.3).

and

 cotðpzÞ ¼ i 1 þ

 2 ; e2piz  1

for Iz < 0:

ð2:6Þ

Since ICþ > 0 and IC < 0, we use the representation (2.5) for the integral over Cþ and the representation (2.6) for the integral over C . Proceeding in this fashion leads to the results

1 2pi

Z

dz f m ðz; sÞ p cotðpzÞ ¼  Cþ

1 2

Z

dz f m ðz; sÞ 

Z

Cþ

dz

Cþ

fm ðz; sÞ e2piz  1

ð2:7Þ

and

1 2pi

Z

dz f m ðz; sÞ p cotðpzÞ ¼ C

1 2

Z

dz f m ðz; sÞ þ

C

Z

dz C

fm ðz; sÞ : e2piz  1

ð2:8Þ

The next step is the deformation of the integration contours which is the subject of the following lemma. Lemma 1. Let C ¼ Cþ [ C be the integration contour described above. Then, in the semi-plane Rs > 2, one obtains

1 2pi

Z

dz f m ðz; sÞ p cotðpzÞ ¼  C

Z

c

du

us e2piðua61 Þ

1

þ

Z

c

du

us e2piðuaP0 Þ

1

;

ð2:9Þ

where c is a contour enclosing in the clockwise direction the negative real axis including the point u ¼ 0. Proof. The proof of this result is based on a suitable deformation of the integration contour C. Before deforming the contour, we focus on the first term on the left hand side of (2.7) and (2.8). From the definition of the function fm ðz; sÞ in (2.1) we can write

Z

dz f m ðz; sÞ ¼

Z

Cþ

1

s

dx ðx þ am þ iICþ Þ :

ð2:10Þ

1

Since, by assumption, the analysis is restricted to the region Rs > 2 one can conclude that for all m

Z

dz f m ðz; sÞ ¼ 0:

ð2:11Þ

Cþ

By using a similar argument one can prove that for all m the following relation holds

Z C

dz f m ðz; sÞ ¼

Z

1

1

dx ðx þ am  iIC Þ

s

¼ 0:

ð2:12Þ

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G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

Let us consider next the second integral on the right hand side of (2.7), which can be rewritten as

Z

dz

Cþ

fm ðz; sÞ ¼ e2piz  1

Z

1

dx ðx þ am þ iICþ Þ 1

s

1 e2piðxþam Þ e2pICþ  1

ð2:13Þ

and is convergent for Rs > 2. If the branch cut extends from the points aP0 , which simply means that it lies in the lower half plane, the contour Cþ can be shifted away to infinity in the upper half plane and does not contribute. In fact

  ðx þ aP0 þ iICþ Þs 

 h i2s  1 1 6 ðx þ aP0 Þ2 þ ðICþ Þ2 !0;  2 p iðxþ a Þ 2 p I C 2 p I C þ þ P0 e 1 e 1 e

ð2:14Þ

as ICþ ! 1. If, on the other hand, the branch cut extends from the points a61 in the upper half plane, we shift the contour Cþ around the branch cut as shown in Fig. 2. The second integral over C in (2.8) is convergent for Rs > 2 and can be expressed as Z Z 1 fm ðz; sÞ 1 s dz 2piz ¼ dx ðx þ am  iIC Þ 2piðxþam Þ 2pIC : ð2:15Þ e e 1 e 1 1 C By using arguments similar to the ones outlined above one can prove that if the cut extends from the branch points a61 in the upper half plane, then we can shift the contour C away to infinity, namely IC ! 1, and the integral (2.15) vanishes. If, instead, the branch cut extends from the points aP0 , in the lower half plane, we shift the contour C as shown in Fig. 2. We can therefore conclude that for m 6 1 only the integral over the deformed Cþ gives a non-vanishing contribution and by making the substitution u ¼ z þ a61 we obtain

1 2pi

Z

dz f m ðz; sÞ p cotðpzÞ ¼ 

Z

du

c

Cþ

us e2piðua61 Þ

1

ð2:16Þ

;

where c is a contour enclosing in the clockwise direction the negative real axis including the point u ¼ 0 as shown in Fig. 3. By using a similar argument, when m P 0, we obtain

1 2pi

Z

dz f m ðz; sÞ p cotðpzÞ ¼

C

Z

du c

us : e2piðuaP0 Þ  1

ð2:17Þ

By substituting the results (2.16) and (2.17) in the relation (2.4) we find the claim (2.9). h Lemma 1 immediately allows to rewrite the doubly-periodic Barnes zeta function. First note, that from the expressions (2.1) and (2.2) we can write

fðs; a; sÞ ¼

Z 1 X dz f m ðs; zÞ cotðpzÞ; 2pi m2Z C

ð2:18Þ

ℑz

Γ −α≤−1

α≥0

ℜz Γ−

−α≥0

α≤−1

Fig. 2. Integration contour used for the representations (2.13) and (2.15).

602

G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

ℑu

γ

Cε ℜu

Fig. 3. Integration contour used for the representations (2.16) and (2.17).

which by using Lemma 1 gives 1 Z X

us

1 Z X

us du 2piðuasmÞ 2piðuasmÞ  1 e e 1 m¼1 c m¼0 c   Z Z 1 s X u 1 1 ; du 2piðuaÞ þ du us 2piðuaÞ2pism  2piðuaÞ2pism ¼ e e  1 m¼1 c 1 e 1 c

fðs; a; sÞ ¼ 

du

þ

ð2:19Þ

where the integral in the second line represents the contribution due to m ¼ 0. By introducing a new variable q 2 C defined as q ¼ eips and by combining the two exponentials in the last line of (2.19) we find the expression

fðs; a; sÞ ¼

Z

du c

us e2piðuaÞ

1

þ

1 Z X m¼1

 du us 2isinð2p½u  aÞ

c

 q2m ; 2m 4m 1  2 cos ð2p½u  aÞq þ q

ð2:20Þ

which, once again, is valid for Rs > 2. The second integral appearing in (2.20), although rather cumbersome, can be expressed in terms of a simple special function. In fact, let q 2 C with jqj < 1 and z 2 C; the first Jacobi theta function #1 ðz; qÞ has the following representation as an infinite product [6] 1 Y



#1 ðz; qÞ ¼ 2q1=4 sinz 1  2q2n cosð2zÞ þ q4n 1  q2n :

ð2:21Þ

n¼1

By taking the derivative of the logarithm of (2.21) and by using q introduced below (2.19), we obtain the formula [5] 1 X d #0 ðz; qÞ q2n ln #1 ðz; qÞ ¼ 1 : ¼ cot z þ 4sinð2zÞ 2n dz #1 ðz; qÞ 1  2q cosð2zÞ þ q4n n¼1

ð2:22Þ

Since jqj < 1, for any finite z the sum in (2.22) is absolutely convergent [9]. This implies, in particular, that the convergence of the series in (2.22) is uniform in z in the region jIzj < pIs. The last remark justifies the interchange of the sum and the integral in (2.20) to obtain, by using the expression (2.22) with the substitution z ! pðu  aÞ, the representation

fðs; a; sÞ ¼

Z

du c

us e2piðuaÞ  1

þ

i 2

Z

du us cot ðp½u  aÞ 

c

i 2p

Z c

du us

d ln #1 ðp½u  a; qÞ; du

ð2:23Þ

where the contour c is chosen so that jIcj < pIs in order to allow the use of (2.22). By combining the first two integrals in (2.23) and by using the relation

1 i 1 þ cot px ¼  ; e2pix  1 2 2 we find

fðs; a; sÞ ¼ 

i 2p

Z

du us

c

d 1 ln #1 ðp½u  a; qÞ  du 2

ð2:24Þ Z

du us :

ð2:25Þ

c

Since Rs > 2, the second integral in (2.25) gives a vanishing contribution and we are left with the following compact representation

fðs; a; sÞ ¼ 

i 2p

Z c

du us

d ln #1 ðp½u  a; qÞ: du

ð2:26Þ

G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

603

We would like to point out that the last integral representation preserves the double-periodicity property of the original sum in (1.3). In fact, let m; n 2 Z. By using (2.26) we write

fðs; a þ n þ sm; sÞ ¼ 

i 2p

Z

du us

c

d ln #1 ðp½u  a  np  smp; qÞ: du

ð2:27Þ

Now, let zm; n ¼ ðn þ msÞp denote the vertices of the fundamental parallelogram in the z-plane. Then the first Jacobi theta function is quasi-periodic on the lattice [5] 2 #1 ðz þ ðn þ msÞp; qÞ ¼ ð1Þmþn eið2mzþpm sÞ #1 ðz; qÞ:

ð2:28Þ

By utilizing (2.28) in (2.27) we obtain

fðs; a þ n þ sm; sÞ ¼ n

Z

du us þ fðs; a; sÞ ¼ fðs; a; sÞ;

ð2:29Þ

c

where, as before, the last equality follows from the fact that since Rs > 2 the integral over c vanishes. The representation (2.26) allows to easily compute fðs; a; sÞ for s ¼ n; n 2 N; n P 2. For these values of s, the contour encloses a singularity at u ¼ 0 of order n and the residue theorem immediately shows

i 2p

fðn; a; sÞ ¼ 

Z

n

du un

Ce

d pn d ln #1 ðp½u  a; qÞ ¼  ln #1 ðu; qÞju¼pa du ðn  1Þ! dun

n

pn

d ln #1 ðu; qÞju¼pa : ðn  1Þ! dun

¼

ð2:30Þ

We next use (2.26) to construct the representation of fðs; a; sÞ valid in the whole complex plane. 3. Analytic continuation The integration contour c in (2.26) consists of a union of three distinct paths, namely c ¼ cþ [ C e [ c , where C e is the circular portion of radius e centered at the origin, cþ is the straight path positioned at a distance d above the negative real axis, and c represents the straight path positioned at a distance d below the negative real axis. Furthermore, for later use, we denote by ~e the projection on the negative real axis of the intersection of cþ (or c ) with the circular portion C e of the integration path. This remark allows us to rewrite (2.26) as

fðs; a; sÞ ¼ 

i 2p

Z

du us

Ce

d i ln #1 ðp½u  a; qÞ  du 2p

Z

du us

cþ [c

d ln #1 ðp½u  a; qÞ: du

ð3:1Þ

The last representation is a suitable starting point from which we can proceed with the analytic continuation to the region Rs 6 2. The first term in (3.1), namely the integral along C e , is left unchanged; it will be dealt with later. The second term in (3.1) can be expressed as a sum I cþ þ I c , where I cþ represents the integral over the path cþ and I c the one over the path c . We will present details for the integral I cþ ; I c follows accordingly. We parameterize the path cþ as u ¼ id þ x, with x 2 ð1; ~e, and rewrite the integral over cþ as follows

I cþ ¼  ¼

i 2p Z

i 2p

Z

du us

cþ

d i ln #1 ðp½u  a; qÞ ¼  du 2p

1

dx ðid  xÞ

s

~e

Z

~e

dx ðx þ idÞ

1

s

d ln #1 ðp½x þ id  a; qÞ dx

d ln #1 ðp½id  x  a; qÞ: dx

ð3:2Þ

The integral in the last line of (3.2) is well defined at the lower limit of integration. To establish the convergence at the upper limit of integration it seems we would need to analyze the asymptotic behavior of the integrand as x ! 1. However, this analysis can be avoided by using the following quasi-periodicity property

#1 ðu þ p; qÞ ¼ #1 ðu; qÞ;

ð3:3Þ

which is obtained by setting m ¼ 0 and n ¼ 1 in the more general formula (2.28). In order to exploit the above property we represent I cþ as

I cþ ¼

1 Z nþ1þ~e i X s d dx ðid  xÞ ln #1 ðp½id  x  a; qÞ; 2p n¼0 nþ~e dx

ð3:4Þ

and perform the change of variables x ¼ y þ n to obtain

I cþ ¼

1 Z 1þ~e i X s d dy ðid  y  nÞ ln #1 ðp½id  y  n  a; qÞ: 2p n¼0 ~e dy

ð3:5Þ

604

G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

Next we send d ! 0, in which case ~e ! e. By using the quasi-periodicity (3.3) in the expression (3.5), we then have 1 Z 1þe i ips X d e dy ðy þ nÞs ln #1 ðp½y  a; qÞ: 2p dy n¼0 e

I cþ ¼

We note, at this point, that the series us to rewrite (3.6) as

i ips e 2p

I cþ ¼

Z

1þe

dy fH ðs; yÞ

e

P1

n¼0 ðy

ð3:6Þ

þ nÞs converges uniformly in y in the interval ½e; 1 þ e for Rs > 1. This allows

d ln #1 ðp½y  a; qÞ; dy

ð3:7Þ

where the Hurwitz zeta function fH ðs; xÞ in Eq. (1.1) has been used. As is well known, the Hurwitz zeta function can be analytically continued in a unique way to a meromorphic function in the entire complex plane possessing only a simple pole with residue 1 at the point s ¼ 1. For the integral I c over the lower part of the contour we proceed similarly to obtain

I c ¼ 

i ips e 2p

Z

1þe

dy fH ðs; yÞ

e

d ln #1 ðp½y  a; qÞ: dy

ð3:8Þ

By adding the contribution from I cþ and I c we obtain the result

I cþ [c ¼ I cþ þ I c ¼

sinðpsÞ

p

Z

1þe

e

dy fH ðs; yÞ

d ln #1 ðp½y þ a; qÞ: dy

ð3:9Þ

Noticing that the integral along C e in (3.1) is defined for s 2 C, the above expression performs the analytic continuation of fðs; a; sÞ in terms of the Hurwitz zeta function fH ðs; yÞ to the full complex plane and it is valid for e > 0 small enough. By choosing Rs < 1, the limit e ! 0 can be performed and the integral along C e in (3.1) can be seen to vanish. We can now summarize the results obtained in this section as follows: Theorem 1. Let fðs; a; sÞ denote the doubly-periodic Barnes zeta function defined in (1.3). The integral representation

fðs; a; sÞ ¼

sinðpsÞ

p

Z

1

dy fH ðs; yÞ

0

d ln #1 ðp½y þ a; qÞ; dy

ð3:10Þ

holds for Rs < 1. Due to the prefactor sinðpsÞ=p in the integral representation (3.10), the substitution s ¼ n with n 2 N0 leads to a vanishing contribution of fðs; a; sÞ when the first argument is a negative integer including zero. This remark proves the following. Corollary 1. Let n 2 N0 , then fðn; a; sÞ ¼ 0. In order to find the representation of fðs; a; sÞ valid for Rs > 1, note that for y ! 0 we have the behavior

fH ðs; yÞ ¼

1 þ fR ðsÞ þ OðyÞ; ys

ð3:11Þ

where the first term results from the n ¼ 0 contribution in (1.1). This is the term responsible for the restriction Rs < 1 found in the previous results. The analytic continuation of (3.10) to the full plane is found by observing that

fH ðs; yÞ ¼

1 þ fH ðs; y þ 1Þ: ys

ð3:12Þ

By substituting (3.12) in the representation (3.10) we find that the second term in (3.12) gives a function holomorphic for s 2 C. The resulting contribution to (3.10) coming from the first term in (3.12) gives, instead, the integral

Io ¼

Z

1

dy ys

0

d ln #1 ðp½y þ a; qÞ; dy

ð3:13Þ

which is valid for Rs < 1. At this point, partial integration can be applied repeatedly to obtain a representation that extends expression (3.13) further to the right of Rs ¼ 1. For example, after one partial integration (3.13) reads

( )  Z 1 2  1 d sþ1 d  I0 ¼ ln #1 ðp½y þ a; qÞ  dy y ln #1 ðp½y þ a; qÞ ; 2 1  s dy 0 dy y¼1

ð3:14Þ

which is now valid for Rs < 2. After n partial integrations a representation valid for Rs < n þ 1 is found and (2.30) can be verified from there. The above representation also shows

fð1; a; sÞ ¼ p

d #1 ðpð1 þ aÞ; qÞ ln #1 ðu; qÞju¼pa  ln ; du #1 ðpa; qÞ

ð3:15Þ

G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

605

where the first term comes from I0 , and the second term from the pole of fðs; y þ 1Þ at s ¼ 1. The second term can be simplified noting

ln

#1 ðp½1 þ a; qÞ ¼ ip; #1 ðpa; qÞ

ð3:16Þ

which can be proved starting from the property (3.3). 4. The derivative of the doubly-periodic Barnes zeta function at negative integers In addition to the results already presented, from the integral representation (3.10) one can compute the derivative of the doubly-periodic Barnes zeta function with respect to the first variable. Differentiating (3.10) leads to the result

f0 ðs; a; sÞ ¼ cosðpsÞ

Z

1

dy fH ðs; yÞ

0

d sinðpsÞ ln #1 ðp½y þ a; qÞ þ dy p

Z 0

1

dy f0H ðs; yÞ

d ln #1 ðp½y þ a; qÞ; dy

ð4:1Þ

where, here and in the rest of this work, the prime indicates differentiation with respect to the variable s. By setting s ¼ 0 in (4.1) and by noting that for n 2 N0 the following relation holds [6]

fH ðn; xÞ ¼ 

Bnþ1 ðxÞ ; nþ1

ð4:2Þ

with Bn ðxÞ denoting the Bernoulli polynomials defined in terms of the Bernoulli numbers Bk as

Bn ðxÞ ¼

n   X n Bk xnk ; k k¼0

ð4:3Þ

we find

f0 ðn; a; sÞ ¼ ð1Þnþ1

 Z 1 nþ1  X nþ1 Bk d dy ynkþ1 ln #1 ðp½y þ a; qÞ: dy n þ 1 k 0 k¼0

ð4:4Þ

Integrating by parts (4.4) yields the result

 n  X nþ1 ð1Þnþ1 ð1Þnþ1 #1 ðp½1 þ a; qÞ Bk þ Bnþ1 ln ln #1 ðp½a þ 1; qÞ #1 ðpa; qÞ nþ1 n þ 1 k k¼0  Z 1 nþ1 X n  nþ1 ð1Þ  ðn  k þ 1ÞBk dy ynk ln #1 ðp½y þ a; qÞ: n þ 1 k¼0 k 0

f0 ðn; a; sÞ ¼

ð4:5Þ

At this point it is convenient to distinguish between two cases: n ¼ 0 and n P 1. The reason for this distinction lies in the relation [6]

  n  X nþ1 0 Bk ¼ k B 0 k¼0

for n P 1

ð4:6Þ

for n ¼ 0:

By exploiting the above relation and (3.16), one obtains

f0 ð0; a; sÞ ¼  ln #1 ðpa; qÞ þ

ip þ 2

Z

1

dy ln #1 ðp½y þ a; qÞ;

ð4:7Þ

0

and, when n P 1,

f0 ðn; a; sÞ ¼ ip

 Z 1 n  nþ1 ð1Þn ð1Þn X ðn  k þ 1ÞBk Bnþ1 þ dy ynk ln #1 ðp½y þ a; qÞ: nþ1 n þ 1 k¼0 k 0

ð4:8Þ

The integrals that appear in (4.7) and (4.8) can be computed from the results of the following lemma. Lemma 2. For 0 < Iz < pIs with q ¼ eips , one has a Fourier-type representation

ln #1 ðz; qÞ ¼

1 X 1 q2n cosð2nzÞ ln q þ ln gðsÞ þ lnð2sinzÞ  2 : 6 n 1  q2n n¼1

ð4:9Þ

Proof. The logarithmic derivative of the first Jacobi theta function can be represented in terms of an infinite series for 0 < Iz < pIs as [5,6] 1 1 X X d q2n q2n sinð2zÞ ln #1 ðz; qÞ ¼ cot z þ 4 sinð2nzÞ ¼ cot z þ 4 : 2n 2n cosð2zÞ þ q4n dz 1  2q 1  q n¼1 n¼1

ð4:10Þ

606

G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

Anti-differentiation of (4.10) yields 1 X ln #1 ðz; qÞ ¼ f ðqÞ þ ln sinz  2

1 X

q2n cosð2nzÞ ¼ f ðqÞ þ ln sinz þ ln 1  2q2n cosð2zÞ þ q4n ; 2n n 1  q n¼1 n¼1

ð4:11Þ

where f ðqÞ is an arbitrary function. In order to determine the unknown f ðqÞ we use the infinite product representation [5,6] 1 Y

1 #1 ðz; qÞ ¼ 2GðqÞq4 sinz 1  2q2n cosð2zÞ þ q4n ;

ð4:12Þ

n¼1

where

GðqÞ ¼

1 Y

1  q2n :

ð4:13Þ

n¼1

From Eq. (4.12) and by recalling that the Dedekind eta function is defined for Is > 0 and q ¼ eips as ips

gðsÞ ¼ e 12

1 Y

1  q2n ;

ð4:14Þ

n¼1

one can easily find

ln #1 ðz; qÞ ¼ ln 2 þ ln gðsÞ þ

1 X

1 ln q þ ln sinz þ ln 1  2q2n cosð2zÞ þ q4n : 6 n¼1

ð4:15Þ

By comparing (4.11) with (4.15) we finally find that

f ðqÞ ¼ ln 2 þ ln gðsÞ þ

1 ln q: 6

ð4:16Þ

Substitution of the explicit expression (4.16) for the function f ðqÞ in (4.11) yields the claim (4.9).

h

The use of (4.9) in the expression (4.7) allows us to obtain the following result for the derivative of the doubly-periodic Barnes zeta function at s ¼ 0,

f0 ð0; a; sÞ ¼  ln #1 ðpa; qÞ þ

Z

ip 1 þ ln q þ ln gðsÞ þ 2 6

1

ln ð2sin½pðy þ aÞÞ:

ð4:17Þ

0

When n P 1, we use once again the result (4.9) in (4.8). By performing this substitution and by recalling (4.6) one obtains

" #  n  1 X nþ1 ð1Þn X q2l 1 ð1Þn þ i ðn  k þ 1ÞBk I nk ðaÞ  2 f ðn; a; sÞ ¼ J ð a Þ p Bnþ1 ; nk;l n þ 1 k¼0 nþ1 1  q2l l k l¼1 0

ð4:18Þ

where we have introduced for typographical convenience the functions

I nk ðaÞ ¼

Z

1

dy ynk ln ð2sin½pðy þ aÞÞ;

ð4:19Þ

0

J nk;l ðaÞ ¼

Z

1

dy ynk cos ð2pl½y þ aÞ:

ð4:20Þ

0

Note that the integral appearing in (4.17) reduces, according to the definition (4.19), to I 0 ðaÞ. The integrals (4.19) and (4.20) can actually be explicitly computed in terms of polylogarithmic functions and trigonometric functions, respectively. The following lemma provides an expression for the function I nk ðaÞ. Lemma 3. For n 2 N0 and for A 2 C n Z,

I n ð AÞ ¼ 

  l X  ! nþ1  k kl  X

nþ1 k  l j ipn ipA ðAÞnþ1 X k! i Lilþ1 e2piA : ð1Þk A  þ 2ðn þ 1Þðn þ 2Þ n þ 1 ðk  lÞ! 2 n þ 1 k¼1 p A k j j¼1 l¼1 ð4:21Þ

Proof. Performing an integration by parts leads to the expression

I n ðAÞ ¼

ln 2 ip 1 p ln sinðpAÞ  þ þ nþ1 nþ1 nþ1 nþ1

Z 0

1

dy ynþ1 cot ðp½y þ AÞ:

ð4:22Þ

G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

607

The integral in (4.22) containing the cotangent can be explicitly computed in terms of polylogarithmic functions. First, by exploiting the change of variables y þ A ! y, we obtain

Z

1

dy ynþ1 cot ðp½y þ AÞ ¼

0

 Z Aþ1 nþ1  X nþ1 dy yk cot ðpyÞ; ðAÞnþ1k k A k¼0

ð4:23Þ

and by rewriting the cotangent in terms of complex exponentials the integral on the right-hand side of (4.23) takes the form

Z

Aþ1

dy yk cot ðpyÞ ¼ i

A

Z

Aþ1

dy yk þ 2i

A

Z

Aþ1

A

dy yk

e2piy : 1

e2piy

ð4:24Þ

This can be rewritten using the polylogarithmic function defined for jzj < 1 and s 2 C by the sum

Lis ðzÞ ¼

1 X zn n¼1

ns

ð4:25Þ

;

and by analytic continuation in the entire complex z-plane [7]. Of particular importance to our analysis is the following property satisfied by the polylogarithmic function [7]





1 d Lin e2piy ¼ Linþ1 e2piy ; 2pi dy

ð4:26Þ

and since



Li0 e2piy ¼ 

e2piy ; e2piy  1

ð4:27Þ

we have that





1 d e2piy Li1 e2piy ¼ 2piy : 2pi dy e 1

ð4:28Þ

The result (4.28) employed in (4.24) yields

2i

Z

Aþ1

dy yk A

e2piy 1 ¼ e2piy  1 p

Z

Aþ1

dy yk

A



d Li1 e2piy : dy

ð4:29Þ

Integrating by parts k times and using, at each step, the relation (4.26) leads to



1

Z

p

Aþ1

dy yk

A

k



d 1X k! ð1Þn ykn Li1 e2piy ¼  Linþ1 e2piy jAþ1 n A dy p n¼0 ðk  nÞ! ð2piÞ   k kn

k  n knl 1X k! ð1Þn X A ¼ Linþ1 e2piA : n p n¼0 ðk  nÞ! ð2piÞ l¼1 l

ð4:30Þ

In light of the previous result and after computing the elementary integral on the right-hand side of (4.24) we can conclude that

Z

Aþ1

dy yk cot ðpyÞ ¼ 

A

  kþ1  k kn 

k þ 1 kþ1l 1 X k  n knl i X k! ð1Þn X A A  Linþ1 e2piA : n k þ 1 l¼1 p n¼0 ðk  nÞ! ð2piÞ l¼1 l l

ð4:31Þ

By substituting expression (4.31) in (4.23) we obtain, for (4.22),

I n ðAÞ ¼

  nþ1  kþ1  n þ 1 ð1Þkþ1 X k þ 1 l ln 2 ip 1 ipðAÞnþ2 X A ln sinðpAÞ þ þ þ nþ1 nþ1 nþ1 n þ 1 k¼0 k þ 1 l¼1 k l   l X  ! nþ1  k kl  X

nþ1 k  l j ðAÞnþ1 X k! i k Lilþ1 e2piA : ð1Þ A þ ðk  lÞ! 2 n þ 1 k¼0 p A k j j¼1 l¼0

ð4:32Þ

The above result can be simplified further, in fact

" #    kþ1 nþ1  kþ1  nþ1  n þ 1 ð1Þkþ1 X n þ 1 ð1Þkþ1 k þ 1 l ipðAÞnþ2 X ipðAÞnþ2 X 1 A ¼ 1 ; 1þ A n þ 1 k¼0 k þ 1 l¼1 n þ 1 k¼0 kþ1 k k l

ð4:33Þ

and by using the relation, valid for a 2 C [6],

 m  X m akþ1 ða þ 1Þmþ1  1 ¼ ; mþ1 k kþ1 k¼0

ð4:34Þ

608

G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

one obtains

  nþ1  kþ1  n þ 1 ð1Þkþ1 X k þ 1 l ipðAÞnþ2 X ip A ¼ : n þ 1 k¼0 k þ 1 l¼1 ðn þ 1Þðn þ 2Þ k l

ð4:35Þ

The l ¼ 0 contribution from the last term in (4.32) can be simplified by following an argument similar to the one leading from (4.33)–(4.35). By noticing that [7]

Li1 ðzÞ ¼  lnð1  zÞ;

ð4:36Þ

one obtains

 nþ1  k   X nþ1 k j 2piA

ðAÞnþ1 X ipA ln 2 1 3pi ð1Þk A Li1 e ln sinðpAÞ    : ¼ n þ 1 n þ 1 n þ 1 2ðn þ 1Þ n þ 1 k¼0 k j j¼1

ð4:37Þ

By substituting (4.35) and (4.37) in the expression (4.32) the claim (4.21) immediately follows. h Let us now focus on the analysis of the function J nk;l ðaÞ in (4.20), which can be rewritten, after a suitable change of variables, as

J nk;l ðaÞ ¼

1 ð2plÞ

kþ1

cosð2plaÞ

Z 2p l

dx xk cos x þ

0

1

sinð2plaÞ kþ1 ð2plÞ

Z 2p l

dx xk sinx:

ð4:38Þ

0

The use of known reduction formulae for the trigonometric integrals on the right-hand side of (4.38) yields

Z

dx xk



cos x sin x



 ¼

sin

 ½2k x X

cos x

l¼0

k! ð1Þl xk2l þ ðk  2lÞ!



 ½k1  2 cos x X k! ð1Þl xk2l1 ; sin x l¼0 ðk  2l  1Þ!

ð4:39Þ

where ½x denotes the integer part of x. The application of the expression (4.39) and (4.38) provides the result

3 ! k! 1 þ ð1Þkþ1 1 k2j1 j ½k1  5 2 ð1Þ k!  J nk;l ðaÞ ¼ ð1Þ ð2plÞ  kþ1 kþ1 ðk  2j  1Þ! 2 ð2plÞ ð2plÞ j¼0 2 k 3 ! ½2 X k k! 1 þ ð1Þk k2j ð1Þ½2 k!5:  sin ð2plaÞ4 ð1Þj ð2plÞ þ ðk  2jÞ! 2 j¼0 1

2 k1 ½ 2  X cos ð2plaÞ4

ð4:40Þ

We can conclude that Eq. (4.18), together with the explicit expressions (4.21) and (4.40), gives a formula for computing the derivative of the doubly-periodic Barnes zeta function at all negative integer points. Moreover, an expression for f0 ð0; a; sÞ is obtained from (4.17) by using (4.21) with n ¼ 0. The results obtained in this section allow for a very efficient way of computing f0 ðn; a; sÞ as the explicit formulae can be easily implemented in an algebraic computer program. For completeness, we display the expression for f0 ðn; a; sÞ when n ¼ 0; n ¼ 1, and n ¼ 2. For n ¼ 0, one has

f0 ð0; a; sÞ ¼  ln #1 ðpa; qÞ þ

  1 1 a ; ln q þ ln gðsÞ þ pi 6 2

ð4:41Þ

for n ¼ 1, we obtain

f0 ð1; a; sÞ ¼

1

1X 1 q2l 1 sin ð2plaÞ Li2 e2pia  2pi p l¼1 1  q2l l2

ð4:42Þ

and for n ¼ 2, one gets

f0 ð2; a; sÞ ¼ 

1

1 X 1 q2l 1 Li3 e2pia  2 cos ð2plaÞ: 2 2p p l¼1 1  q2l l3

ð4:43Þ

The above expressions are valid in the range of parameters stated below Eq. (1.3) and have to be periodically continued, as is clear from the fact that the integral representation (4.4) is periodic. 5. Concluding remarks In this work we have performed a detailed analysis of the meromorphic extension of the doubly-periodic Barnes zeta function in the entire complex plane. The contour integral representation (2.3) allowed us, after a suitable deformation of the integration contour, to obtain the meromorphic extension of the doubly-periodic Barnes zeta function fðs; a; sÞ. This analytically continued expression revealed to be particularly appropriate for the explicit computation of the values fðn; a; sÞ and of the derivative f0 ðn; a; sÞ for n 2 N0 .

G. Fucci, K. Kirsten / Applied Mathematics and Computation 221 (2013) 598–609

609

The process of analytic continuation delineated in this work is quite general and its applicability is not limited exclusively to the study of the doubly-periodic Barnes zeta function. In fact, the method developed here can be applied to more general elliptic functions. Although the representation of elliptic functions in terms of integrals over Rþ has been constructed, for instance in [3], our method would provide a contour integral representation for the class of elliptic functions. Since we have seen that such representation has some advantages, such as the double-periodicity of the results and the almost straightforward computation of the values and derivative at specific points, it could, perhaps, provide either new results or simplify already known ones regarding elliptic functions. This seems to be an interesting idea which deserves further investigation. Aside from their intrinsic mathematical interest, our results can find applications in problems related to physical systems. The Thirring model is used to describe simple interacting field theories [10]. The one-loop partition function for generalized Thirring models is proportional to the derivative at s ¼ 0 of the doubly-periodic Barnes zeta function [8]. The result (4.41) obtained here can then be directly applied to the analysis of these interacting field models. Acknowledgment KK acknowledges very fruitful discussions with Stuart Dowker on the subject. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

E.W. Barnes, The theory of the double gamma function, Philos. Trans. R. Soc. London, Ser. A 196 (1901) 265387. E.W. Barnes, On the theory of the multiple gamma function, Trans. Cambridge Philos. Soc. 19 (1904) 374–425. A. Dienstfrey, J. Huang, Integral representation for elliptic functions, J. Math. Anal. Appl. 316 (2006) 142–160. E. Elizalde, Some analytic continuations of the Barnes zeta function in two and higher dimensions, Appl. Math. Comput. 187 (2007) 141. A. Erdélyi, Higher Transcendental Functions, vol. II, McGraw-Hill, New York, 1953 (Bateman Project Staff ). I.S. Gradshtein, I.M. Ryzhik, in: A. Jeffrey, D. Zwillinger (Eds.), Table of Integrals, Series and Products, Academic, Oxford, 2007. L. Lewin, Polylogarithms and Associated Functions, Elsevier Science, North-Holland, New York, 1981. I. Sachs, A. Wipf, Generalized Thirring models, Ann. Phys. 249 (1996) 380. L.C. Shen, On the logarithmic derivative of a theta function and a fundamental identity of Ramanujan, J. Math. Anal. Appl. 177 (1993) 299–307. W.E. Thirring, A soluble relativistic field theory, Ann. Phys. 3 (1958) 91. E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press, 1927.