EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 335-341 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Note on an Octuple Integral in terms of the Lerch Function Robert Reynolds1,∗, Allan Stauffer1 1 Department of Mathematics and Statistics, Faculty of Science, York University, Toronto, Ontario, Canada, M3J1P3 Abstract. The known exact expression for an octuple integral relating to research in the fields of mathematics and physics is summarized. A new closed form expression for this integral is given in terms of the Lerch function. 2020 Mathematics Subject Classifications: Primary 30E20, 33-01, 33-03, 33-04, 33-33B Key Words and Phrases: Octuple integral, Riemann zeta function, Cauchy integral, Lerch function 1. Significance Statement Octuple integrals are used and evaluated in may areas of mathematics and physics. Some areas of interest where these integrals are used are in multipupil in phase microscopy, where pairs of Fourier transforms are evaluated [8], the kinetic theory of simple and com- posite monatomic gases : viscosity, thermal conduction, and diffusion [1], statistical char- acteristics of the laser-radiation-intensity fluctuations in rainfall [6], the velocity distribu- tion function, and on the stresses in a non-uniform rarefied monatomic gas [5], and some applications of Marcel Riesz’s Integrals of Fractional Order [2]. In current literature octuple integrals expressed in terms of a closed form solution do not appear to be tabulated. In this work the authors derive and evaluate a octuple inte- gral in terms of the Lerch function and derive special cases of this integral transform in terms of special constants. It is our hope that researchers will find such evaluations useful for potential research requiring these formulae. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4154 Email addresses: milver@my.yorku.ca (R. Reynolds), stauffer@yorku.ca (A. Stauffer) https://www.ejpam.com 335 © 2022 EJPAM All rights reserved. R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 335-341 336 2. Introduction The octuple integral derived in this manuscript is given by ∫ R8 + (rs) m−1 2 (r + s)−m/2(tz)− m 2 −1(t+ z) m+1 2 (uv)− m 2 − 1 2 (u+ v)m/2(xy)m/2(x+ y) 1 2 (−m−1)e−p(r+u+x+z)−q(s+t+v+y) logk ( a √ rs √ t+ z √ u+ v √ xy √ r + s √ tz √ uv √ x+ y ) dxdydzdrdsdtdudv (1) where the parameters k, a ∈ C, Re(p, q) > 0 are general complex numbers with −1/2 ≥ Re(m) ≥ −1. The derivation of the definite integral follows the method used by us in [10] which involves Cauchy’s integral formula. The generalized Cauchy’s integral formula is given by yk Γ(k + 1) = 1 2πi ∫ C ewy wk+1 dw. (2) where C is in general an open contour in the complex plane where the bilinear concomitant [10] has the same value at the end points of the contour. The method in [10] involves using a form of equation (2) then multiply both sides by a function, then take a definite integral of both sides. This yields a definite integral in terms of a contour integral. A second contour integral is derived by multiplying equation (2) by a function and performing some substitutions and taking the infinite sum so that the contour integrals are the same. 3. Definite integral of the contour integral We use the method in [10]. The variable of integration in the contour integral is z = w+m. The cut and contour are in the second quadrant of the complex z-plane. The cut approaches the origin from the interior of the first or second quadrant and the contour goes round the origin with zero radius and is on opposite sides of the cut. Using equation (2) we replace y by (3)log ( a √ rs √ t+ z √ u+ v √ xy √ r + s √ tz √ uv √ x+ y ) then multiply by both sides by (4)(rs) m−1 2 (r + s)−m/2(tz)− m 2 −1(t+ z) m+1 2 (uv)− m 2 − 1 2 (u + v)m/2(xy)m/2(x+ y) 1 2 (−m−1)e−p(r+u+x+z)−q(s+t+v+y) and take the definite octuple integral over x, y, z, r, s, t, u, v ∈ [0,∞) to get; R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 335-341 337∫ R8 + (rs) m−1 2 (r + s)−m/2(tz)− m 2 −1(t+ z) m+1 2 (uv)− m 2 − 1 2 (u+ v)m/2(xy)m/2(x+ y) 1 2 (−m−1)e−p(r+u+x+z)−q(s+t+v+y) logk ( a √ rs √ t+ z √ u+ v √ xy √ r + s √ tz √ uv √ x+ y ) dxdydzdrdsdtdudv Γ(k + 1) = 1 2πi ∫ C ∫ R8 + aww−k−1(rs) 1 2 (m+w−1)(r + s) 1 2 (−m−w) (tz) 1 2 (−m−w)−1(t+ z) 1 2 (m+w+1)(uv) 1 2 (−m−w)− 1 2 (u+ v) m+w 2 (xy) m+w 2 (x+ y) 1 2 (−m−w−1)e−p(r+u+x+z)−q(s+t+v+y)dwdxdydzdrdsdtdudv = 1 2πi ∫ R8 + ∫ C aww−k−1(rs) 1 2 (m+w−1)(r + s) 1 2 (−m−w) (tz) 1 2 (−m−w)−1(t+ z) 1 2 (m+w+1)(uv) 1 2 (−m−w)− 1 2 (u+ v) m+w 2 (xy) m+w 2 (x+ y) 1 2 (−m−w−1)e−p(r+u+x+z)−q(s+t+v+y)dxdydzdrdsdtdudvdw = − 1 2πi ∫ C 2π4aww−k−1 csc(π(m+ w)) p2q2 dw (5) from equation (3.1.3.9) in [9] where −1 < Re(w+m) < 1 and using the reflection formula for the gamma function. The logarithmic function is given for example in section (4.2) in [3]. We are able to switch the order of integration over w +m and x, y, z, r, s, t, u, v using Fubini’s theorem since the integrand is of bounded measure over the space C × [0,∞) × [0,∞)× [0,∞)× [0,∞)× [0,∞)× [0,∞)× [0,∞)× [0,∞). 4. The Lerch function We use section (25.14) in [3] where Φ(z, s, v) is the Lerch function which is a generaliza- tion of the Hurwitz zeta ζ(s, v) and Polylogarithm functions Lin(z). The Lerch function has a series representation given by Φ(z, s, v) = ∞∑ n=0 (v + n)−szn (6) where |z|< 1, v ̸= 0,−1, .. and is continued analytically by its integral representation given by Φ(z, s, v) = 1 Γ(s) ∫ ∞ 0 ts−1e−vt 1− ze−t dt = 1 Γ(s) ∫ ∞ 0 ts−1e−(v−1)t et − z dt, (7) R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 335-341 338 where Re(v) > 0, and either |z|≤ 1, z ̸= 1, Re(s) > 0, or z = 1, Re(s) > 1. 5. Infinite sum of the contour integral In this section we will again use Cauchy’s integral formula (2) and take the infinite sum to derive equivalent sum representations for the contour integrals. We proceed using equation (2) and replace y by log(a) + iπ(2y + 1) and multiply both sides by 4iπ4 p2q2 then take the infinite sum over y ∈ [0,∞) simplifying in terms of the Lerch function to get ik+12k+2πk+4eiπmΦ ( e2imπ,−k, π−i log(a) 2π ) p2q2Γ(k + 1) = 1 2πi ∞∑ y=0 ∫ C 4iπ4aww−k−1eiπ(2y+1)(m+w) p2q2 dw = 1 2πi ∫ C ∞∑ y=0 4iπ4aww−k−1eiπ(2y+1)(m+w) p2q2 dw = − 1 2πi ∫ C 2π4aww−k−1 csc(π(m+ w)) p2q2 dw (8) from equation(1.232.3) in [4] where Im(w +m) > 0 in order for the sum to converge. Theorem 1. For all k, a ∈ C, Re(p, q) > 0,−1/2 < Re(m) < −1,∫ R8 + (rs) m−1 2 (r + s)−m/2(tz)− m 2 −1(t+ z) m+1 2 (uv)− m 2 − 1 2 (u+ v)m/2(xy)m/2(x+ y) 1 2 (−m−1)e−p(r+u+x+z)−q(s+t+v+y) logk ( a √ rs √ t+ z √ u+ v √ xy √ r + s √ tz √ uv √ x+ y ) dxdydzdrdsdtdudv = ik+12k+2πk+4eiπmΦ ( e2imπ,−k, π−i log(a) 2π ) p2q2 (9) Proof. Observe the right-hand side of equation (5) is equal to the right-hand side of equation (8) so we may equate the left-hand sides and simplify the gamma function to yield the stated result. Example 1. The degenerate case.∫ R8 + (rs) m−1 2 (r + s)−m/2(tz)− m 2 −1(t+ z) m+1 2 (uv)− m 2 − 1 2 (u+ v)m/2(xy)m/2(x+ y) 1 2 (−m−1) e−p(r+u+x+z)−q(s+t+v+y)dxdydzdrdsdtdudv R. Reynolds, A. Stauffer / Eur. J. Pure Appl. Math, 15 (2) (2022), 335-341 339 = −2π4 csc(πm) p2q2 (10) Proof. Use equation (9) and set k = 0 and simplify using entry (2) in Table below (64:12:7) in [7]. Example 2.∫ R8 + 4 √ r + s 4 √ t+ ze−r−2s−2t−u−2v−x−2y−z (rs)3/4(tz)3/4 4 √ uv 4 √ u+ v 4 √ xy 4 √ x+ y ( log2 (√ rs √ t+z √ u+v √ xy√ r+s √ tz √ uv √ x+y ) + π2 )dxdydzdrdsdtdudv = 1 2 π2 log(2) (11) and (12 ) ∫ R8 + 4 √ rs 4 √ r + s 4 √ tz 4 √ t+ z(uv)3/4(xy)3/4e−r−2s−2t−u−2v−x−2y−z rstuvxyz 4 √ u+ v 4 √ x+ y ( log2 (√ rs √ t+z √ u+v √ xy√ r+s √ tz √ uv √ x+y ) + π2 ) log (√ rs √ t+ z √ u+ v √ xy √ r + s √ tz √ uv √ x+ y ) dxdydzdrdsdtdudv = 0 Proof. Use equation (9) and set a = −1, p = 1, q = 2,m = −1/2 and simplify in terms of the Riemann zeta function using entry (2) in Table below (64:7) and entry (4) in Table below (64:12:7) in [7]. Next apply l’Hopital’s rule to the right-hand side as k → −1 rationalize the denominator equate real and imaginary parts and simplify. Example 3. (13 ) ∫ R8 + (r + s)3/8 8 √ t+ ze−r−2s−2t−u−2v−x−2y−z (rs)7/8(tz)5/8 8 √ uv(u+ v)3/8(xy)3/8 8 √ x+ y ( log2 ( − √ rs √ t+z √ u+v √ xy√ r+s √ tz √ uv √ x+y ) + π2 ) dxdydzdrdsdtdudv = π2(π + log(4)) 8 √ 2 and (14 ) ∫ R8 + (r + s)3/8 8 √ t+ ze−r−2s−2t−u−2v−x−2y−z log (√ rs √ t+z √ u+v √ xy√ r+s √ tz √ uv √ x+y ) (rs)7/8(tz)5/8 8 √ uv(u+ v)3/8(xy)3/8 8 √ x+ y ( log2 (√ rs √ t+z √ u+v √ xy√ r+s √ tz √ uv √ x+y ) + π2 ) dxdydzdrdsdtdudv = π3(log(4)− π) 8 √ 2 REFERENCES 340 Proof. Use equation (9) and set k = −1, a = −1, p = 1, q = 2,m = −1/2 and simplify in terms of the polylogarithm function Lin(z) function using entry (2) in in Table below (64:12:7) in [7] and equation (25.12.10) in [3] and rationalize the denominator equate real and imaginary parts and simplify. Example 4.∫ R8 + (r + s)3/8 8 √ t+ ze−r−s−t−u−v−x−y−z (rs)7/8(tz)5/8 8 √ uv(u+ v)3/8(xy)3/8 8 √ x+ y √ log ( − √ rs √ t+z √ u+v √ xy√ r+s √ tz √ uv √ x+y ) dxdydzdrdsdtdudv = (−1− i) √ 2π7/2 ( ζ ( 1 2 , 1 4 ) − iζ ( 1 2 , 3 4 )) (15) Proof. Use equation (9) and set k = −1/2, a = −1, p = q = 1,m = −3/4 and simplify in terms of the Hurwitz zeta function using entry (4) in Table below (64:12:7) in [7]. 6. Discussion In this paper, we have presented a novel method for deriving a new octuple integral along with some interesting definite integrals using contour integration. The results pre- sented were numerically verified for both real and imaginary and complex values of the parameters in the integrals using Mathematica by Wolfram. Some of the challenges en- countered were in the numerical evaluation of the integrals. We know from our method the definite integral is equal to the Lerch function so this is a new way of computing this octuple integral. We tried various numerical methods in the Mathematica software to achieve the best possible result relative to the Lerch function. Acknowledgements This research is supported by NSERC Canada under grant 504070. References [1] S Chapman. The kinetic theory of simple and composite monatomic gases : viscosity, thermal conduction, and diffusion. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 93:1–20, 12 1916. [2] E. T. Copson. Xxii.—some applications of marcel riesz’s integrals of fractional order. Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences, 61:260–272, 1943. REFERENCES 341 [3] Nist digital library of mathematical functions. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds. [4] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. 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