## Abstract

Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sums *S*_{2} (for conductivity) and *T*_{2} (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between *S*_{2} and *T*_{2} for different lattices are established. In particular, the value *S*_{2}=*π* for the square and hexagonal arrays is discussed and *T*_{2}=*π*/2 for the hexagonal is deduced.

## 1. Introduction

The mathematical questions of convergence, numerically effective algorithm and closed-form evaluation of the lattice sums were discussed in the fundamental book Borwein *et al.* [1] and works cited therein. In particular, as it is noted in Stremler’s paper [2], Glasser [3] evaluated lattice sums, using Jacobi imaginary transformation, integral representation for the modified Bessel function, and relations for the Jacobi theta-functions. This paper is devoted to closed-form evaluation of the conditionally convergent two-dimensional (2D) lattice sums. One of them, *S*_{2}, defined by (2.2), was considered by Lord Rayleigh [4]. Numerically effective series for Rayleigh’s sum based on the elliptic functions are outlined in Borwein *et al.* [1], §3.2. McPhedran *et al.* [5], Movchan *et al.* [6] and Greengard *et al.* [7] developed the Rayleigh method to elastostatic (see lattice sum (2.3)) and elastodynamic problems having paid the main attention to computationally convenient and accurate expressions for the lattice sums constructed for the square array.

The effective properties of unidirectional fibrous composites can be expressed in terms of the series in concentration *f*. The famous Clausius–Mossotti approximation also known as the Maxwell formula [8], ch. 10 is valid in the first-order approximation. The second-order approximation includes the value of *S*_{2}. In order to shortly describe these approximations following Rayleigh, we consider a doubly periodic rectangular array of discs of conductivity *λ*_{1} embedded in matrix of conductivity *λ*. Let *λ*_{ij} (*i*,*j*=1,2) be the components of the effective conductivity tensor, then [4,9,10,11]
*ρ*=(*λ*_{1}−*λ*)/(*λ*_{1}+*λ*) denotes the contrast parameter. For the square and hexagonal arrays, the medium becomes macroscopically isotropic, i.e. *λ*_{e}=*λ*_{11}=*λ*_{22}, *λ*_{12}=*λ*_{21}=0, and the above formulae becomes the Clausius–Mossotti approximation
*S*_{2}=*π* for the square and hexagonal arrays. Actually, the approximation in the right part of (1.3) holds up to *O*((|*ρ*|*f*)^{5}) (see formula (28) from [12] where the correction in the fifth-order term should be

The same rule holds for elastostatic problems [13]. An analytic formula for the macroscopic elastic constants must include lattice sums (2.2) and (2.3) in the second-order term *O*( *f*^{2}). Such formulae for elastic problems are similar to (1.1)–(1.3). For simplicity, consider macroscopically isotropic composites where elastic fibres with the shear modulus *μ*_{1} and the Poisson’s ratio *ν*_{1} are imbedded in the matrix with the constants *μ* and *ν*. Let *κ*=3−4*ν* be the corresponding Muskhelishvili’s constants for the plane strain. The effective shear modulus *μ*_{e} can be estimated by the asymptotic formula deduced in [13]
*T*_{2} stands for the lattice sum (2.3).^{1} Therefore, analytical formulae for the lattice sums (2.2) and (2.3) have the fundamental applications in 2D composites.

In this paper, we deduce closed-form formulae for the lattice sums (2.2), (2.3) and other new formulae. Our method, involving complete elliptic integrals, arithmetic summation formulae and inverse Mellin transform for the product of Euler’s gamma- and Riemann’s zeta- functions drives to the closed-form formulae for these sums and numerous interesting particular cases. Explicit formulae deduced in §§3 and 4 yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions (1.1)–(1.4). The obtained fundamental values *S*_{2} and *T*_{2} give a possibility to pass through the approximation *O*( *f*^{2}) terms to get high order analytical formulae for the effective elastic constants of fibrous composites [14,13].

## 2. Eisenstein summation method and Rayleigh’s integral

Let *ω*_{1}, *ω*_{2}. Without loss of generality we assume that *ω*_{1}>0 and *Im* *τ*>0, where *τ*=*ω*_{2}/*ω*_{1}. Let the area of the fundamental parallelogram be normalized to unity, hence,

The Eisenstein summation method [15] is defined as the limit of iterated sums

Moreover, Rayleigh [4] found the beautiful formula *S*_{2}(*i*)=*π*, where *τ*=*i* corresponds to the square array. The method of calculation was based on the reduction of the sum *S*_{2}(*i*) to the integral
*v*,*v*)×(−*v*,*v*) is eliminated in (2.5) as it vanishes. Further, the equality *S*_{2}=*π* was proved in [17] for the hexagonal array (under the normalization of the area of the fundamental cell to unity).

The following formula was independently deduced in [19]
*ζ*_{W} denotes the *ζ*-Weierstrass function. The equality *S*_{2}(*i*)=*π* was proved by Legendre’s identity. Although formulae (2.4) and (2.6) seem to be similar, a reduction of one to other can be justified only through the Eisenstein summation method applied to the *ζ*-Weierstrass function.

We now proceed to discuss the lattice sum (2.3) beginning from the relation [15]
*m* and application of (2.7) yields
*n* in accordance with (2.2) and (2.3) gives the computationally effective formula

Surprisingly, that Rayleigh’s integral gives a wrong result
*S*(*i*)=*π* despite in the same paper Rayleigh gave a rigorous proof based on the elliptic integrals (see remark 4.2). Another curious fact that Rayleigh’s integral for the three-dimensional (3D) cubic lattice [4] gives also the correct result

## 3. Exact relations between lattice sums

Let *x*>0 be given by the formula
*K*(*k*) is the complete elliptic integral of the first kind [21], [22], vol. II, [23]
*k* is called the elliptic modulus and *k*^{′} is the complementary modulus. As we see the function *x* as a function of *k*∈(0,1) is monotone decreasing and continuously differentiable bijective map *x*>0 is uniquely defined by the corresponding modulus *k*, which, in turn, can be computed by Jacobi’s inversion formula
*θ*_{2} and *θ*_{3} are Jacobi theta functions. The complete elliptic integral *K*(*k*) satisfies the Legendre relation
*E*(*k*) is the complete elliptic integral of the second kind
*K*(*k*),*K*(*k*^{′}) satisfy the differential equation
*E*(*k*), *E*(*k*^{′})−*K*(*k*^{′}) are, in turn, solutions of the differential equation
*K*(*k*) can be calculated by the formula

Let *k*_{r} be an elliptic modulus such that *r* and the corresponding singular values of the elliptic integral *K*(*k*_{r}) (see [24,25]), namely
*Γ*(*z*) is Euler’s gamma-function [22], vol. I. According to Borwein & Zucker [25], the so-called elliptic alpha function for the integral singular values can be calculated as follows:
*ζ*(*s*) is the Riemann’s zeta-function [22], vol. I, which satisfies the familiar functional equation

We will use the Rayleigh formula (2.4) and formula (2.10) obtained in §2 in order to deduce functional equations for *S*_{2}(*τ*) and *T*_{2}(*τ*) on the imaginary positive half-axis *τ*=*ix* and positive half-lines *x*>0).

### Theorem 3.1

*Let x>0. Then,
**and
*

### Proof.

Indeed, employing integral representation (3.14), we substitute it into (2.4) to write for *τ*=*ix*
*γ*>2 and the zeta-function is bounded on the vertical line *ζ*(*s*−1)|≤*ζ*(*γ*−1), the interchange of the order of summation and integration is allowed for each *x*>0 via the absolute and uniform convergence by virtue of the estimate

On the other hand, the product of zeta-functions *ζ*(*s*)*ζ*(*s*−1) can be represented by the Ramanujan identity [28]
*σ*(*m*) is the arithmetic function [21], denoting the sum of divisors of *m*. Hence, substituting in (3.20) and inverting the order of integration and summation owing to the same motivation, we obtain

In the meantime, Nasim’s identity [29] says that

In order to prove equations (3.18), we invoke representation (3.15), motivating all passages analogously to the previous case. Moreover, as we will see it is sufficient to prove (3.18) for positive real parts. So, we have (see (2.4))
*γ*>1 via the definition of the Riemann’s zeta-function and we obtain
*a*=*x*/2, *b*=*x*)
*x*, which is justified by the absolute and uniform convergence, and multiply by *x* the obtained equality. Thus, we get
*x* by 1/*x* in (3.28) and adding these two equalities with the use of (3.17), we obtain (3.18). ▪

### Remark 3.2

The identities (3.17)–(3.18) were proved in [18] by means of the Weierstrass elliptic functions. The proof of theorem 3.1 proposed here actually contains advanced formulae which will be used in the next section for the elastic lattice sums.

### Theorem 3.3

*Let x>0. Then,
**and
*

### Proof.

Put *τ*=*ix*, *x*>0 in (2.10) and write it in the form
*x* of the series *x*≥*x*_{0}>0 due to the absolute and uniform convergence. Hence,
*σ*(*m*) in (3.22) by the same reasons, we obtain
*x* and then multiplying both sides of the obtained equality by −*x*^{2}/(2*π*), we find

Therefore, with the use of (3.33), equality (3.35) becomes

## 4. Closed-form formulae for the lattice sums

In the previous section, we established exact relations between lattice sums and the corresponding functional equations for *T*_{2}(*τ*),*S*_{2}(*τ*), involving the Mellin transform technique and integral representations related to the Riemann’s zeta-function. Moreover, such stunning relations were possible due to our manipulations with the Nasim summation formula and Ramanujan’s identity for the product of Riemann’s zeta-functions. In this section, we obtain closed-form formulae for these sums, employing for the first time a new method of termwise differentiation of the series with respect to the elliptic modulus. As a result closed-form values for important constants are also obtained.

The explicit expressions of *S*_{2}(*τ*) on the imaginary axis and on the lines

### Theorem 4.1

*Let* *. Then the following formulae hold
**and
**where
**and k*^{′} *is defined by (*3.1*).*

### Proof.

First, consider a positive *x* being defined by (3.1). Fortunately, the series in (2.4) for *τ*=*ix* is calculated in [26], see eqn (5.3.4.5) as follows:
*x* and it can be easily extended to negative *x* via (2.4).

In order to prove (4.2), we employ relation (5.3.6.6) in [26]
*x*, we find from (2.4), (4.3), (4.4)
*x*, we get (4.2). ▪

### Remark 4.2

Formula (4.1) was first proved by Rayleigh [4], formula (41). Rayleigh also deduced formula (4.13) below having used the Legendre identity (3.3).

### Remark 4.3

Some particular results of the exact formulae for *S*_{2},*T*_{2} can be found in [1], table 1.6, which corresponds to 2D lattice sums in [1], (1.4.6) when *s*=1. In fact, it will give interesting analytic relations between theta functions and number theory functions.

### Corollary 4.4

*Formula* (3.30) *can be written in the form*

Now, we are able to calculate interesting particular values of (4.1) and (4.2).

### Corollary 4.5

*The following exact formulae take place*

### Proof.

As we observe from (4.1) and (4.2), it is sufficient to establish the above constants for a positive imaginary part of the corresponding *τ*. To do this, we employ particular cases (3.9) of the modulus *k*_{r} and the corresponding singular values *K*(*k*_{r}) (*r*=1,2,3,4) given by (3.10) and (3.11). In fact, letting

A more technically difficult task is to find explicit expressions for *T*_{2}(*τ*) on the same lines in the complex plane. To achieve our goal, we will involve the method of termwise differentiation of the series in (2.4) with respect to the elliptic modulus (for *τ*=*ix*(*k*) or *τ*=(±1+*ix*(*k*))/2). Indeed, as we mentioned above, *x*(*k*) by formula (3.1) is continuously differentiable. When *k*∈[*a*_{0},*b*_{0}], 0<*a*_{0}<*b*_{0}<1, the corresponding series (2.4) is absolutely and uniformly convergent. Moreover, it is not difficult to show that the series of the derivatives with respect to *k* converges absolutely and uniformly on the segment [*a*_{0},*b*_{0}]. Thus, the known theorem from calculus says that the termwise differentiation of the series is allowed. This leads us to

### Theorem 4.6

*Under conditions of theorem* 3.3 *the following formulae hold
**and
*

### Proof.

Substitute (4.3) with *τ*=*ix* (*x*>0) into (2.10) and write the result in the form
*x* is a function of *k* determined by (3.1). Using the termwise differentiation we obtain

Meanwhile, the derivative *x*^{′}(*k*) can be calculated explicitly, employing twice (3.8)
*x*. Further, we extend it to negative numbers as in theorem 3.3.

In order to establish identity (4.15), we write (2.10) for *τ*=(±1+*ix*)/2, *x*>0 in the same manner as in the proof of identity (3.30). Nevertheless, we will employ explicit expressions (4.3) and (4.4) and make the termwise differentiation with respect to the elliptic modulus. Hence, taking in mind (4.2), (4.18), we obtain
*k* with the aid of (3.5), (3.6), (3.8), we find
*x* as in theorem 3.3. ▪

As a corollary, we calculate particular values of *T*_{2} on the mentioned vertical lines, recalling *k*_{r} in (3.9) and *K*(*k*_{r}) in (3.10), (3.11) (*r*=1,2,3,4). In particular, the value *x*=1, corresponding to *T*_{2}(*i*)=4.078451 computed with (2.10).

We note that this numerical result *T*_{2}(*i*)=4.078451 coincides with the numerical value obtained by other approaches [5,7].

### Corollary 4.7

*Certain explicit constants related to* *T*_{2}(*τ*) *are the following values*

## 5. Discussion

In this section, we analyse the obtained exact formulae and apply them to investigation of the properties of various lattices. Attention is paid to the lattice sum *S*_{2}. Equation (3.17) (see its discussion in [12]) relates *S*_{2}(*ix*) and *S*_{2}(*ix*^{−1}). Within the normalization assumed at the beginning of §2, we consider here the rectangular array of discs when the fundamental rectangle has the sides *x*>0). Therefore, *S*_{2}(*ix*) and *S*_{2}(*ix*^{−1}) determine the effective conductivity up to *O*( *f*^{3}) in the direction *OX* and *OY* , respectively, of the same rectangular array. One can replace 2−(1/*π*)*S*_{2}(*ix*) in (1.2) by (1/*π*)*S*_{2}(*ix*^{−1}) in accordance with (3.17).

Consider equation (3.18) with *τ*=(1+*ix*)/2 and *τ*′=(1+*ix*^{−1})/2. The corresponding fundamental vectors have the form
*ω*′_{1} and *ω*′_{2} by rotation about 90^{°}
*k*, *m*, *k*′′ and *m*′′ run over integers. One can see that the sets (5.4) are isomorphic. Therefore, (3.18) relates *S*_{2} for the same lattice in two perpendicular directions.

Formulae (4.5) and (4.10) hold for the square and hexagonal arrays, respectively, which are macroscopically isotropic lattices for conductivity problems. The lattice with *τ*=(1+*i*)/2 is not macroscopically isotropic but in this case *S*_{2}=*π* in accordance with (4.6). Hence, the conductivity of this lattice is not distinguishable up to *O*( *f*^{3}) with the the square and hexagonal lattices.

In order to relate the properties of *T*_{2} with the macroscopic elastic properties one should construct general analytical formulae for the effective elastic constants up to *O*( *f*^{3}) for macroscopically anisotropic arrays.

## Authors' contributions

S.Y., P.D. and V.M. contributed to writing the paper.

## Competing interests

We have no competing interest.

## Funding

S.Y. was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. P.D. was partially supported by the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzeszow (grant no. WMP/GD-04/2016).

## Acknowledgements

The authors are sincerely indebted to anonymous referees for their invaluable comments and suggestions, which rather improved the presentation of the paper.

## Footnotes

↵1 We do not know arguments based on isotropy to justify that

*T*_{2}=*π*/2 for the hexagonal array as for the conductivity problem.

- Received June 24, 2016.
- Accepted September 27, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.