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Exponentially small expansions associated with a generalised Mathieu series R. B. Paris ∗ 6 Division of Computing and Mathematics, 1 0 University of Abertay Dundee, Dundee DD1 1HG, UK 2 n a J Abstract 8 2 We consider the generalised Mathieu series ] ∞ A nγ (µ>0) C (nλ+aλ)µ nX=1 . h whentheparametersλ(>0)andγ areevenintegersforlargecomplexainthesector t a arg a < π/λ. The asymptotics in this case consist of a finite algebraic expansion m | | together with an infinite sequence of increasingly subdominant exponentially small [ expansions. When µ is also a positive integer it is possible to give closed-form evaluations of this series. Numerical results are given to illustrate the accuracy of 1 v the expansion obtained. 1 5 Mathematics Subject Classification: 30E15,30E20, 34E05 7 Keywords: asymptotic expansions, exponentially small expansions, generalised 7 0 Mathieu series, Mellin transform method . 1 0 6 1. Introduction 1 : v This paper is a sequel to the asymptotic study of a generalised Mathieu series carried i X out by the author in [4]. The functional series r a ∞ n (1.1) (n2+a2)µ nX=1 in the case µ = 2 was introduced by Mathieu in his 1890 book [2] dealing with the elasticity of solid bodies. Considerable effort has been devoted to the determination of upper and low bounds for the series with µ = 2 when the parameter a > 0. Several integral representations for (1.1), and its alternating variant, have been obtained; see [8] and the references therein. ∗E-mail address: [email protected] 1 The asymptotic expansion of the more general functional series ∞ nγ S (a;λ) := (µ > 0, λ > 0, µλ γ > 1) (1.2) µ,γ (nλ+aλ)µ − nX=1 was considered by Zvastavnyi [11] for a + and by Paris [4] for a in the sector → ∞ | | → ∞ arg a < π/λ. In [4], the additional factor e := exp[ aλb/(nλ +aλ)] (with b > 0) was n | | − included in the summand which, although not affecting the rate of convergence of the series (since e 1 as n ), can modify the large-a growth, particularly with the n → → ∞ alternating variant of (1.2). Both these authors adopted a Mellin transform method and obtained the result1, when γ > 1, − S (a;λ) Γ(γ+λ1)Γ(µ− γ+λ1) 1 ∞ (−)kΓ(µ+k) ζ( γ λk) (1.3) µ,γ − λΓ(µ)aλµ γ 1 ∼ Γ(µ) k!aλ(k+µ) − − − − kX=0 as a in the sector arg a < π/λ, where ζ(s) denotes the Riemann zeta function. | | → ∞ | | In this paper we also employ the Mellin transform approach used in [4, 11], where our interest will be concerned with the parameter values µ > 0 and even integer values of λ (> 0) and γ. In this case, the asymptotic series on the right-hand side of (1.3) (the algebraic expansion) is either a finite series or vacuous on account of the trivial zeros of ζ(s). We shall find that the asymptotic expansion of S (a;λ) for large complex a in the µ,γ sector arg a < π/λ for these parameter values consists of a finite algebraic expansion | | together with an infinitesequence (whenµis notan integer) of increasingly subdominant exponentially small contributions. In the case of positive integer µ it is possible to give a closed-form evaluation of S (a;λ). µ,γ It is perhaps rather surprising that such an innocent-looking series should possess such an intricate asymptotic structure. A similar phenomenon has been recently ob- served in the expansion of the generalised Euler-Jacobi series n wexp[ anp] as ∞n=1 − − the parameter a 0 when p and w are even integers; see [5]P. The leading terms in → the expansion of S (a;λ) when γ = 0 and λ = 2,4 have been given in [10] using the µ,γ Poisson-Jacobi formula. In the application of the Mellin transform method to the series in (1.2) and its alternating variant we shall require the following estimates for the gamma function and the Riemann zeta function. For real σ and t, we have the estimates 1 Γ(σ it) = O(tσ−2), ζ(σ it) = O(tΩ(σ)logαt) (t + ), (1.4) ± | ± | → ∞ where Ω(σ) = 0 (σ > 1), 1 1σ (0 σ 1), 1 σ (σ < 0) and α = 1 (0 σ 1), 2 − 2 ≤ ≤ 2 − ≤ ≤ α = 0 otherwise [9, p. 95]. The zeta function ζ(s) has a simple pole of unit residue at s = 1 and the evaluations for positive integer k (2π)2k ζ(0) = 1, ζ( 2k) = 0, ζ(2k) = B (k 1), −2 − 2(2k)!| 2k| ≥ 1The restriction γ >−1 was imposed in [4] to avoid the formation of a double pole for odd negative integer values of γ; in [11], theparameter γ was allowed to assume arbitrary real values. 2 B = 1, B = 1, B = 1 , B = 1 ,..., (1.5) 0 2 6 4 −30 6 42 where B are the Bernoulli numbers. Finally, we have the well-known functional relation k satisfied by ζ(s) given by [3, p. 603] ζ(s)= 2sπs 1ζ(1 s)Γ(1 s)sin 1πs. (1.6) − − − 2 2. An integral representation The generalised Mathieu series defined in (1.2) can be written as S (a;λ) = a δ ∞ h(n/a), h(x) := xγ , δ := λµ γ (2.1) µ,γ − (1+xλ)µ − nX=1 where the parameter δ > 1 for convergence. We employ a Mellin transform approach as discussed in [6, Section 4.1.1]. The Mellin transform of h(x) is H(s) = 0∞xs−1h(x)dx, where R xγ+s 1 1 τ(γ+s)/λ 1 (s) = ∞ − dx = ∞ − dτ H Z (1+xλ)µ λ Z (1+τ)µ 0 0 Γ(γ+s)Γ(µ γ+s) = λ − λ λΓ(µ) in the strip γ < (s) < δ. Using the Mellin inversion theorem (see, for example, [6, − ℜ p. 118]), we find ∞ h(n/a) = 1 ∞ c+∞i (s)(n/a) sds = 1 c+∞i (s)ζ(s)asds, − 2πi Z H 2πi Z H nX=1 nX=1 c−∞i c−∞i where 1< c < δ. The inversion of the order of summation and integration is justified by absolute convergence provided 1 < c< δ. Then, from (2.1), we have [4, 11] a δ 1 c+ i γ +s γ+s S (a;λ) = − ∞ Γ Γ µ ζ(s)asds, (2.2) µ,γ λΓ(µ) 2πi Z (cid:18) λ (cid:19) (cid:18) − λ (cid:19) c i −∞ where 1 <c < δ. From the estimates in (1.4), the integral in (2.2) then defines S (a;λ) µ,γ for complex a in the sector arg a < π/λ. The asymptotic expansion of S (a;λ) for µ,γ | | large a and real parameters λ, µ and γ, such that δ > 1 is given in (1.4); see [4, Theorem 3] for complex a and [11] for positive a and unrestricted γ. We now suppose in the remainder of this paper that µ > 0, with λ (> 0) and γ both chosen to be even integers. More specifically, we write λ = 2p (p =1,2,...), γ = 2m (m = 0, 1, 2,...). (2.3) ± ± 3 The integration path in (2.2) lies to the right of the simple pole of ζ(s) at s= 1 and the poles of Γ((γ+s)/λ) at s = γ λk (k = 0,1,2,...), but to the left of the poles of the − − second gamma function at s = δ+λk. When γ = 2,4,..., the poles at s = γ λk are − − cancelled by the trivial zeros of ζ(s) at s = 2, 4,.... When γ = 2m, m = 0,1,2,..., − − − however, there is a finite set of poles of this sequence on the left of the integration path situated in (s) 0 with 0 k k , where k is the index that satisfies ∗ ∗ ℜ ≥ ≤ ≤ m k p 0, m (k +1)p < 0. (2.4) ∗ ∗ − ≥ − The remaining poles of this sequence correspondingto k > k are cancelled by the trivial ∗ zeros of ζ(s) with the result that there are again no poles in (s)< 0. ℜ We consider theintegral in (2.2) taken roundthe rectangular contour with vertices at c iT and c iT, where c > 0 and T > 0. The contribution from the upper and lower ′ ′ ± − ± sidesoftherectangles = σ iT, c σ c,vanishesasT provided arg a < π/λ, ′ ± − ≤ ≤ → ∞ 1 |3 | sincefrom(1.4),themodulusoftheintegrandiscontrolledbyO(TΩ(σ)+2µ−4 log T e−∆T), where ∆ = π/λ arg a . Evaluation of the residues then yields −| | Γ(γ+1)Γ(µ γ+1) S (a;λ) = λ − λ +H (a;λ)+J(a), (2.5) µ,γ λΓ(µ)aδ 1 µ,γ − where the finite algebraic expansion H (a;λ) (with γ =2m) is given by µ,γ 0 (m = 1,2,...) H (a;λ) = a λµ k∗ ( )kΓ(µ+k) (2.6) µ,γ −  − ζ(2m 2kp) (m = 0, 1, 2,...), Γ(µ) k!aλk − − − kX=0  with the index k being defined in (2.4), and ∗ a δ 1 c′+ i γ+s γ+s J(a) = − − ∞ Γ Γ µ ζ(s)asds (c > 0). (2.7) ′ λΓ(µ) 2πi Z c′ i (cid:18) λ (cid:19) (cid:18) − λ (cid:19) − −∞ The values of ζ(s) at s = 0,2,4,... can be expressed in terms of the Bernoulli numbers, if so desired, by (1.5). The integrand in J(a) is holomorphic in (s) < 0, so that further displacement of ℜ the contour to the left can produce no additional terms in the algebraic expansion of S (a;λ). We shall see in the next section that J(a) possesses an infinite sequence of µ,γ increasingly exponentially small terms in the large-a limit. 3. The exponentially small expansion of J(a) In the integral (2.7), we make the change of variable s s γ to find → − − a λµ 1 d+ i s s J(a) = − ∞ Γ − Γ µ+ ζ( s γ)a−sds, d = c′ γ. λΓ(µ) 2πi Z (cid:18) λ (cid:19) (cid:18) λ(cid:19) − − − d i −∞ 4 We now employ (1.4) to convert the zeta function into one with real part greater than unity. With the parameters λ and γ in (2.3), the above integral can then be written in the form ( )ma λµ 1 d+ i sin(1πs) J(a) = (2−π)γλΓ−(µ) 2πi Z ∞ G(s)ζ(1+s+γ)(2πa)−s sin(2πs) ds, d i λ −∞ where Γ(s+γ+1)Γ(µ+s) G(s) := λ . (3.1) Γ(1+s) λ Making use of the expansion sin(1πs) sin(πps) p−1 π 2 λ = e iωrs, ω := (p 1 2r) , (3.2) sin(πs) ≡ sin(πs) − r − − λ λ λ rX=0 we then obtain ( )ma−λµ p−1 J(a) = − (a), (3.3) (2π)γλΓ(µ) Er rX=0 where 1 d+ i (a) = ∞ G(s)ζ(s+γ+1)(2πaeiωr) sds (3.4) r − E 2πiZ d i −∞ and d+γ = c >0. ′ The integrals (a) (0 r p 1) have no poles to the right of the integration path, r E ≤ ≤ − so that we can displace the path as far to the right as we please. On such a displaced path, which we denote by L, s is everywhere large. Let M denote an arbitrary positive | | integer. The quotient of gamma functions in G(s) may be expanded by appealing to the inverse-factorial expansion given in [6, p. 53] to obtain M 1 G(s) = λ1 µ − ( )jc Γ(s+ϑ j)+ρ (s)Γ(s+ϑ M) , ϑ := µ+γ, (3.5) − j M (cid:26) − − − (cid:27) jX=0 where c = 1 and ρ (s) = O(1) as s in arg s < π. An algorithm for the 0 M | | → ∞ | | evaluation of the coefficients c is discussed in Section 4. Substitution of the expansion j (3.5) into (3.4) then produces M 1 (a) = λ1 µ − ( )jc 1 Γ(s+ϑ j)ζ(s+γ+1)(2πaeiωr) sds+R , (3.6) r − j − M,r E (cid:26) − 2πi Z − (cid:27) jX=0 L where the remainders R are given by M,r 1 R = ρ (s)Γ(s+ϑ M)ζ(s+γ+1)(2πaeiωr) sds. (3.7) M,r M − 2πi Z − L The integrals appearing in (3.6) can be evaluated by making use of the well-known result 1 Γ(s+α)z sds = zαe z ( arg z < 1π), 2πi ZL′ − − | | 2 5 where L is a path parallel to the imaginary s-axis lying to the right of all the poles of ′ Γ(s + α); see, for example, [6, Section 3.3.1]. Upon expansion of the zeta function in (3.6) (since on L its argument satisfies (s)+γ +1> 1) we find ℜ 1 Γ(s+ϑ j)ζ(s+γ+1)(2πaeiωr) sds = ∞ (2πnaeiωr)ϑ−j exp[ 2πnaeiωr] − 2πi Z − n1+γ − L nX=1 = Xϑ je XrK (X ;µ), X := Xeiωr, X := 2πa, (3.8) r− − j r r where we have defined the exponential sum ∞ e−(n−1)Xr K (X ;µ) := . (3.9) j r n1 µ+j nX=1 − Thisevaluation isvalid providedthatthevariableX satisfies theconvergence conditions r arg a+ω < 1π (0 r p 1). | r| 2 ≤ ≤ − From the definition of ω in (3.2), it is easily verified that these conditions are met when r arg a < π/λ. It is then evident that K (X ;µ) 1 as a in arg a < π/λ. j r | | ∼ | | → ∞ | | Thus we find M 1 (a) = λ1 µe Xr − ( )jc Xϑ jK (X ;µ)+R . (3.10) Er − − − j r− j r M,r jX=0 Bounds for the remainders of the type R have been considered in [6, p. 71]; see also M,r [1, 10.1]. The integration path in (3.7) is such that (s)+γ +1 > 1, so that we may § ℜ employ the bound ζ(x+iy) ζ(x) for real x, y with x > 1. A slight modification of | | ≤ Lemma 2.7 in [6, p. 71] then shows that R = O(Xϑ Me Xr) (3.11) M,r r− − as a in the sector arg a < π/λ. | | → ∞ | | The expansion of S (a;λ) then follows from (2.5), (3.3), (3.10) and (3.11) and is µ,γ given in the following theorem. Theorem 1. Let µ > 0, γ = 2m, λ = 2p, where m = 0, 1, 2,... and p = 1,2,.... ± ± Further, let M denote a positive integer, ω = π(p 1 2r)/(2p) for 0 r p 1 and r − − ≤ ≤ − δ = λµ γ, ϑ = µ+γ. Then − Γ(γ+1)Γ(µ γ+1) ( )m π µ p−1 Sµ,γ(a;λ) = λλΓ(µ)aδ−1 λ +Hµ,γ(a;λ)+ Γ−(µ)(cid:18)p(cid:19) aµ−δ Er(a) (3.12) − rX=0 as a in the sector arg a < π/λ. The finite algebraic expansion H (a;λ) is µ,γ | | → ∞ | | defined in (2.6) and the exponentially small expansions E (a) are given by r M 1 E (a) = e Xr+iϑωr − ( )jc X jK (X ;µ)+O(X M) (0 r p 1), (3.13) r − (cid:26) − j r− j r r− (cid:27) ≤ ≤ − jX=0 where the leading coefficient c = 1 and X = 2πaeiωr. The infinite exponential sums 0 r K (X ;µ) are defined in (3.9). j r 6 When a is a real variable, the expansion in Theorem 1 can be expressed in a different form. We have the following theorem. Theorem 2. Let the parameters µ, γ, λ and δ, ϑ, ω be as in Theorem 1. Then with r N = 1p and X = 2πa, the expansion for S (a;λ) becomes ⌊2 ⌋ µ,γ Γ(γ+1)Γ(µ γ+1) S (a;λ) = λ − λ +H (a;λ) µ,γ λΓ(µ)aδ 1 µ,γ − +(Γ−(µ)m)(cid:18)πp(cid:19)µaµ−δ(cid:26)N−1Er∗(a)+(cid:18)1E0(a)(cid:19)(cid:27) (cid:26)pp eodvedn (3.14) rX=0 2 N∗ as a + , where for arbitrary positive integer M → ∞ M 1 E (a) = 2e Xcosωr − ( )jc X jK (X;ω )+O(X M) (0 r N) (3.15) r∗ − (cid:26) − j − j∗ r − (cid:27) ≤ ≤ jX=0 and the infinite exponential sums K (X;ω ) are defined by j∗ r ∞ e−(n−1)Xcosωr K (X;ω )= cos[nXsinω +(j ϑ)ω ]. j∗ r n1 µ+j r − r nX=1 − When p is odd, the quantity ω = 0. N 4. The coefficients c j We describe an algorithm for the computation of the coefficients c that appear in the j exponentially small expansions E (a) and E (a) in (3.13) and (3.15). The expression for r r∗ the ratio of gamma functions in G(s) in (3.5) may be written in the form M 1 G(s) = λ1 µ − cj + ρM(s) , − Γ(s+ϑ) (cid:26) (1 s ϑ) (1 s ϑ) (cid:27) jX=0 − − j − − M where (α) = Γ(α + j)/Γ(α) is the Pochhammer symbol. If we introduce the scaled j 1 gamma function Γ∗(z) = Γ(z)/(√2πzz−2e−z), then we have 1 1 Γ(βs+γ) = Γ∗(βs+γ)(2π)2e−βs(βs)βs+γ−2 e(βs;γ), where γ e(βs;γ) := exp (βs+γ 1)log(1+ ) γ . (cid:20) − 2 βs − (cid:21) The above ratio of gamma functions may therefore be expressed as M 1 − cj ρM(s) R(s)Υ(s)= + (4.1) (1 s ϑ) (1 s ϑ) jX=0 − − j − − M 7 as s in arg s < π, where | |→ ∞ | | e(s;γ+1)e(s/λ;µ) Γ (s+γ+1)Γ (µ+s/λ) ∗ ∗ R(s):= , Υ(s) := . e(s/λ;1)e(s;ϑ) Γ (1+s/λ)Γ (s+ϑ) ∗ ∗ We now let ξ := s 1 and follow the procedure described in [6, p.47]. We expand R(s) − and Υ(s) for ξ 0 making use of the well-known expansion (see, for example, [6, p. 71]) → Γ (z) ∞ ( )kγ z k (z ; arg z < π), ∗ k − ∼ − | | → ∞ | | kX=0 where γ are the Stirling coefficients, with k γ = 1, γ = 1 , γ = 1 , γ = 139 , γ = 571 , ... . 0 1 −12 2 288 3 51840 4 −2488320 After some straightforward algebra we find that R(s)= 1+ 1(µ 1) (λ 1)µ 2γ ξ +O(ξ2), 2 − { − − } Υ(s) =1 1 (µ 1)(λ2 1)ξ2+O(ξ3), − 12 − − so that upon equating coefficients of ξ in (4.1) we can obtain c . The higher coefficients 1 can be obtained by matching coefficients recursively with the aid of Mathematica to find c = 1, c = 1(µ 1) 2γ (λ 1)µ , 0 1 2 − { − − } c = 1 (µ 1)(µ 2) 12γ(γ (λ 1)µ 1)+(λ 1)µ(5 3µ+λ(3µ 1)) , 2 24 − − { − − − − − − } c = 1 (µ 1)(µ 2)(µ 3) 2 2γ +(λ 1)µ 4γ(γ (λ 1)µ 2) 3 −48 − − − { − − }{ − − − +(λ 1)µ(3+λ(µ 1) µ) ,... . (4.2) − − − } The rapidly increasing complexity of the coefficients with j 4 prevents their presenta- ≥ tion. However, this procedure is found to work well in specific cases when the various parameters have numerical values, where many coefficients have been so calculated. In Table 1 we present some values2 of the coefficients c for 1 j 10, which are used in j ≤ ≤ the specific examples considered in Section 5. 4.1 The coefficients c when λ = 2, γ = 0 j When λ = 2 and γ = 0, it is possible to express the coefficients c in closed form for j arbitrary µ > 0. From (3.1), we have Γ(1+s)Γ(µ+ 1s) 2s G(s) = 2 = Γ(1 + 1s)Γ(µ+ 1s) Γ(1+ 1s) √π 2 2 2 2 2In the tables we write the valuesas x(y)instead of x×10y. 8 Table 1: The coefficients cj (1≤j ≤10) for different γ when µ=5/4 and λ=4. j γ = 0 γ = 2 γ = 2 − 1 4.6875000000( 1) +3.1250000000( 2) 2.1250000000(+0) − − − − 2 3.5888671875( 1) +1.5673828125( 1) 2.5546875000(+0) − − − − 3 4.0534973145( 1) +2.3551940918( 1) 6.8701171875(+0) − − − − 4 3.3581793308( 1) +1.6646325588( 1) 2.5683746338(+1) − − − − 5 +7.5268601999( 1) 9.0884858742( 1) 1.1944799423(+2) − − − − 6 +6.4821335676(+0) 6.6501405553(+0) 6.6193037868(+2) − − 7 +2.6358910987(+1) 2.7627888119(+1) 4.2794038211(+3) − − 8 +4.5855530043(+1) 5.8401959193(+1) 3.1831413077(+4) − − 9 3.7955573596(+2) +2.8858407940(+2) 2.6901844936(+5) − − 10 5.1286970180(+3) +4.6231064924(+3) 2.5504879368(+6) − − upon use of the duplication formula for the gamma function. The inverse factorial expansion of a product of two gamma functions with equal coefficients of s is given in [6, pp. 51–52] in the form Γ(s+α)Γ(s+β) ∼ 223−α−β−2s√π ∞ djΓ(2s+α+β−12−j) jX=0 as s in arg s < π, where the coefficients satisfy d = 1 and 0 | |→ ∞ | | 2 j j d = − (α β)2 (r 1)2 (j 1). j j! { − − − 2 } ≥ rY=1 Putting α = 1 and β = µ, with s 1s, we therefore obtain the coefficients in the 2 → 2 inverse factorial expansion of G(s) when λ = 2, γ =0 and µ > 0 given by ( 2) j j − c = − (µ r)(µ+r 1) (j 1). (4.3) j j! − − ≥ rY=1 4.2 The coefficients c when µ is an integer j A study of the coefficients c with the aid of Mathematica enables us to conjecture that j they possess the general form c = (µ 1)(µ 2)...(µ j)P (µ,γ,λ) (j 1), j j − − − ≥ where P denotes a polynomial of degree j in the parameters µ, γ and λ. This implies j thatthesequenceofcoefficients isfiniteforinteger valuesofµ; thatis,forpositiveinteger q, we have c = 0 (j q; µ = q, q = 1,2,...). j ≥ 9 This can also be seen from (3.1) where, with µ = q, Γ(s+γ+1)Γ(q+s) s G(s) = λ = Γ(s+γ+1) 1+ . (4.4) Γ(1+s) (cid:18) λ(cid:19) λ q 1 − Whenq = 1,wehaveϑ = 1+γ andtheexpansion(3.5)issatisfiedtriviallybyterminating the series at the leading term with ρ (s) 0. When q 2, the right-hand side of (3.5) 1 ≡ ≥ must terminate at M = q with ρ (s) 0 to yield q ≡ q 1 − G(s) = λ1 qΓ(s+γ+1) ( )jc (s+γ +1) (4.5) − j g j 1 jX=0 − − − in order to have the polynomials in s in (4.4) and (4.5) of the same degree. Then the exponential expansions E (a) in (3.13) become the finite sums r q 1 − E (a) = e Xr+iϑωr ( )jc X jK (X ;q), ϑ = q+γ, r − − j r− j r jX=0 where the infinite sums K (X ;q) may be expressed exactly in terms of derivatives of an j r exponential by K (X ;q) = eXr ∞ nq j 1e nXr = ( 1)q j 1eXrDq j 1 ∞ e nXr j r − − − − − − − − − nX=1 nX=1 = ( )q j 1eXrDq j 1(eXr 1) 1 ( arg a < π/λ), (4.6) − − − − − − − | | with D d/dX . The coefficients c (0 j µ 1) are obtained by recursive solution r j ≡ ≤ ≤ − of (4.4) and (4.5) and are given below3 for µ = 1,2,...,5: µ = 1 : c = 1 0 µ = 2 : c = 1, c =γ λ 1 0 1 − − µ = 3 : c = 1, c =2γ 3λ+3, c = (1+γ λ)(1+γ 2λ) 0 1 2 − − − µ = 4 : c = 1, c =3γ 6λ+6, c = 3γ2+9γ +7 18λ 12λγ +11λ2, 0 1 2 − − − c = (1+γ λ)(1+γ 2λ)(1+γ 3λ) 3 − − − µ = 5 : c = 1, c =4γ 10λ+10, c = 6γ2+24γ +25 60λ 30λγ +35λ2, 0 1 2 − − − c = (2γ 5λ+3)(2γ2 +6γ +5 15λ 10λγ +10λ2), 3 − − − c = (1+γ λ)(1+γ 2λ)(1+γ 3λ)(1+γ 4λ). 4 − − − − (4.7) The generalised Mathieu series when µ is an integer can therefore be expressed by the following closed-form evaluation. 3The values of c1, c2 and c3 follow from (4.2). 10