Journal of Statistical Physics

, Volume 145, Issue 3, pp 591–612 | Cite as

70+ Years of the Watson Integrals

  • I. J. ZuckerEmail author


Watson (Q. J. Math. Oxford 10:266–276, 1939) published in 1939 the evaluation of three integrals submitted to him, which had arisen from a problem in physics (van Pepye in Physica 5:465–482, 1938). Over the years these integrals have continued to occur in other aspects of physics such as random walk problems. This article reviews these integrals and generalisations over the past 70 years.


Cubic LN curve Convolution curve Minkowski sum Offsets Linear normal map G-1 Bezier approximation Error estimate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    van Pepye, W.F.: Zür Theorie der Magetischen Anisotropic Kubischer Kristalle beim Absoluten Nullpunkt. Physica 5, 465–482 (1938) ADSCrossRefGoogle Scholar
  2. 2.
    Watson, G.N.: Three triple integrals. Q. J. Math. Oxford 10, 266–276 (1939) CrossRefGoogle Scholar
  3. 3.
    Kummer, E.E.: Über die hypergeometrische Reihe. J. Reine Angew. Math. 15, 39–83 (1836) zbMATHCrossRefGoogle Scholar
  4. 4.
    Kummer, E.E.: Über die hypergeometrische Reihe. J. Reine Angew. Math. 15, 127–172 (1836) zbMATHCrossRefGoogle Scholar
  5. 5.
    Maradudin, A.A., Montroll, E.W., Weiss, G.H., Herman, R., Miles, W.H.: Green’s functions for monatomic cubic lattices. Acadèmie Royale de Belgique 5–15 (1960) Google Scholar
  6. 6.
    Watson, G.N.: The expansion of products of hypergeometric functions. Q. J. Math. 39, 27–51 (1908) Google Scholar
  7. 7.
    McCrea, W.H., Whipple, F.J.W.: Random paths in two and three dimensions. Proc. R. Soc. Edinb. 60, 281–298 (1940) MathSciNetGoogle Scholar
  8. 8.
    McCrea, W.H.: A problem on random paths. Math. Gazette 20, 311–317 (1936) CrossRefGoogle Scholar
  9. 9.
    Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichung der mathematischen Physik. Math. Ann. 100, 32–74 (1928) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Pòlya, G.: Über eine Aufgabe der Wahrscheinlichkeitstheorie betreffend die Irrfahrt im Strassennetz. Math. Ann. 84, 149–160 (1921) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Stewart, I.: How to Cut a Cake, Chap. 2. Oxford University Press, Oxford (2006) Google Scholar
  12. 12.
    Domb, C.: On multiple returns in the random walk problem. Proc. Camb. Philos. Soc. 50, 586–591 (1954) MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Duffin, R.J.: Discrete potential theory. Duke Math. J. 20, 233–251 (1953) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Courant, R.: Über partielle Differenzengleichung. Atti Congresso Internazionale Dei Matematici-Bolgna. 3, 83–89 (1929) Google Scholar
  15. 15.
    Davies, H.: Poisson’s partial differential equation. Q. J. Math. (2) 6, 232–240 (1955) zbMATHCrossRefGoogle Scholar
  16. 16.
    Doyle, P.G., Snell, L.J.: Random Walks and Electric Networks. Carus Mathematical Monographs, vol. 22. Mathematical Association of America, Washington (1984) zbMATHGoogle Scholar
  17. 17.
    Montroll, E.W.: Theory of the vibration of simple cubic lattices with nearest neighbor interaction. In Proc. Third Berkeley Symp. on Math. Statistics and Probability, vol. 3, pp. 209–246. University of California Press, Berkeley (1956) Google Scholar
  18. 18.
    Iwata, G.: Evaluation of the Watson integral of a face-centered lattice. Nat. Sci. Rep. Ochanomizu Univ. 20, 13–18 (1869) MathSciNetGoogle Scholar
  19. 19.
    Joyce, G.S.: Lattice Green function for the anisotropic face-centred cubic lattice. J. Phys. C, Solid State Phys. 4, L53–L56 (1971) ADSCrossRefGoogle Scholar
  20. 20.
    Joyce, G.S.: On the simple cubic lattice Green function. Philos. Trans. R. Soc. Lond. A 273, 583–610 (1973) MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, London and New York (1946) Google Scholar
  22. 22.
    Bailey, W.N.: A reducible case of the fourth type of Appell’s hypergeometric functions of two variables. Q. J. Math. Oxford 4, 305–308 (1933) CrossRefGoogle Scholar
  23. 23.
    Bailey, W.N.: Some infinite integrals involving Bessel functions. Proc. Lond. Math. Soc. 40, 37–48 (1935) CrossRefGoogle Scholar
  24. 24.
    Tikson, M.: Tabulation of an integral arising in the theory of cooperative phenomena. J. Res. Natl. Bur. Stds. 50, 177–178 (1953) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Appell, M.: Sur les transformations des equations differentielles lineaires. C. R. Acad. Sci. Paris 91, 211–214 (1880) Google Scholar
  26. 26.
    Glasser, M.L.: A Watson sum for a cubic lattice. J. Math. Phys. 13, 1145–1146 (1972) MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Hioe, F.T.: A Green’s function for a cubic lattice. J. Math. Phys. 19, 1064–1067 (1978) MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. 28.
    Rashid, M.A.: Lattice Green’s functions for cubic lattices. J. Math. Phys. 21, 2549–2552 (1980) MathSciNetADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Montaldi, E.: The evaluation of Green’s functions for Cubic lattices, revisited. Lett. Nuovo Cimento 30, 403–409 (1981) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Glasser, M.L., Wood, V.E.: A closed form evaluation of the elliptical integral. Math. Comput. 25, 535–536 (1971) zbMATHCrossRefGoogle Scholar
  31. 31.
    Ramanujan, S.: Modular equations and approximations to π. Q. J. Math. XLV, 350–372 (1914) Google Scholar
  32. 32.
    Zucker, I.J.: The evaluation in terms of Γ-functions of the periods of elliptic curves admitting complex multiplication. Math. Proc. Camb. Philos. Soc. 82, 111–118 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Selberg, A., Chowla, S.: On Epsrein’s zeta-function. J. Reine Angew. Math. 227, 86–110 (1967) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Glasser, M.L., Zucker, I.J.: Extended Watson integrals for the cubic lattices. Proc. Natl. Acad. Sci. USA 74, 1800–1801 (1977) MathSciNetADSzbMATHCrossRefGoogle Scholar
  35. 35.
    Borwein, J.M., Zucker, I.J.: Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal. 12, 519–526 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Joyce, G.S.: On the cubic lattice Green functions. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 455, 463–477 (1994) MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Joyce, G.S.: On the cubic modular transformation and the cubic lattice Green functions. J. Phys. A, Math. Gen. 31, 5105–5115 (1998) MathSciNetADSzbMATHCrossRefGoogle Scholar
  38. 38.
    Delves, R.T., Joyce, G.S.: On the Green function for the anisotropic simple cubic lattice. Ann. Phys. 291, 71–133 (2001) MathSciNetADSzbMATHCrossRefGoogle Scholar
  39. 39.
    Clausen, T.: Ueber die Falle wenn die Reihe ein quadrat von der Form hat. J. Math. 3, 89–95 (1828) zbMATHGoogle Scholar
  40. 40.
    Bornemann, F., Laurie, D., Wagon, S., Waldvogel, J.: The SIAM 100-Digit Challenge. A Study in High- Accuracy Numerical Computing. SIAM, Philadelphia (2004), 146 zbMATHGoogle Scholar
  41. 41.
    Joyce, G.S., Delves, R.T., Zucker, I.J.: Exact evaluation for the anisotropic face-centred and simple cubic lattices. J. Phys. A, Math. Gen. 36, 8661–8672 (2003) MathSciNetADSzbMATHCrossRefGoogle Scholar
  42. 42.
    Joyce, G.S., Delves, R.T.: Exact product forms for the simple cubic lattice Green functions: I. J. Phys. A, Math. Gen. 37, 3645–3671 (2004a) MathSciNetADSzbMATHCrossRefGoogle Scholar
  43. 43.
    Joyce, G.S., Delves, R.T.: Exact product forms for the simple cubic lattice Green functions: II. J. Phys. A, Math. Gen. 37, 5417–5447 (2004b) MathSciNetADSzbMATHCrossRefGoogle Scholar
  44. 44.
    Delves, R.T., Joyce, G.S.: Exact product form for the anisotropic simple cubic lattice Green function. J. Phys. A, Math. Theor. 39, 4119–4145 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  45. 45.
    Delves, R.T., Joyce, G.S.: Derivation of exact product forms for the simple cubic lattice Green function using Fourier generating functions and Lie group identities. J. Phys. A, Math. Theor. 40, 8329–8343 (2007) MathSciNetADSzbMATHCrossRefGoogle Scholar
  46. 46.
    Miller, J. Jr: Lie Theory and Special Functions. Academic Press, New York (1968) zbMATHGoogle Scholar
  47. 47.
    Jacobi, C.G.J.: Gesammelte Werke, vol 3. Chelsea, New York (1969) Google Scholar
  48. 48.
    Cayley, A.: The Collected Mathematical Papers, vol. 1, pp. 224–227. Cambridge University Press, Cambridge (1889) CrossRefGoogle Scholar
  49. 49.
    Cayley, A.: An Elementary Treatise on Elliptic Functions, 2nd edn., pp. 319–323. Deighton, Bell and Co, Cambridge (1895) Google Scholar
  50. 50.
    Goursat, E.: Sur l’équation différentielle linéaire qui admet pour intégrale la serie hypergéometrique. Ann. Sci. École Norm. Sup. (2) 10, S3–S142 (1881) MathSciNetGoogle Scholar
  51. 51.
    Guttmann, A.J.: Lattice Green functions in all dimensions. J. Phys. A, Math. Theor. 43, 305205 (2010) MathSciNetCrossRefGoogle Scholar
  52. 52.
    Guttmann, A.J., Prellberg, T.: Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47, R2233–R2236 (1993) ADSCrossRefGoogle Scholar
  53. 53.
    Glasser, M.L., Montaldi, E.: Staircase polygons and recurrent lattice walks. Phys. Rev. E 48, 2339–2342 (1993) ADSCrossRefGoogle Scholar
  54. 54.
    Joyce, G.S., Zucker, I.J.: On the evaluation of generalized Watson integrals. Proc. Am. Math. Soc. 133, 77–81 (2004) MathSciNetGoogle Scholar
  55. 55.
    Bailey, D.H., Borwein, J.M., Broadhurst, D., Glasser, M.L.: Elliptic integral evaluations of Bessel moments and applications. J. Phys. A, Math. Theor. 41, 205203 (2008) MathSciNetADSCrossRefGoogle Scholar
  56. 56.
    Broadhurst, D.: Elliptic integral evaluation of a Bessel moment by contour integration of a lattice Green function. arXiv:0801.0891v1
  57. 57.
    Rosengren, A.: On the number of spanning trees for the 3D simple cubic lattice. J. Phys. A, Math. Gen. 20, L923–L927 (1987) MathSciNetADSCrossRefGoogle Scholar
  58. 58.
    Madras, N., Soteros, C.E., Whittington, S.G., Martim, J.L., Sykes, M.F., Flesia, S., Gaunt, D.S.: The free energy of a collapsing branched polymer. J. Phys. A, Math. Gen. 23, 5327–5350 (1990) ADSCrossRefGoogle Scholar
  59. 59.
    Joyce, G.S., Zucker, I.J.: Evaluation of the Watson integral and associated logarithmic integral for the d-dimensional hypercubic lattice. J. Phys. A, Math. Gen. 34, 7349–7354 (2001) MathSciNetADSzbMATHCrossRefGoogle Scholar
  60. 60.
    Rogers, M.D.: New 5 F 4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π. Ramanujan J. 18, 327–349 (2009) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of PhysicsKing’s College LondonLondonUK

Personalised recommendations