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70+ years of the Watson integrals. (English) Zbl 1231.82026

Summary: G. N. Watson [Q. J. Math., Oxf. Ser. 10, 266–276 (1939; Zbl 0022.33202)] published in 1939 the evaluation of three integrals submitted to him, which had arisen from a problem in physics [W. F. Van Peype, “Zur Theorie der magnetischen Anisotropie kubischer Kristalle beim absoluten Nullpunkt”, Physica 5, No. 6, 465–482 (1938; doi:10.1016/S0031-8914(38)80158-0)]. 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.

MSC:

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
33E05 Elliptic functions and integrals
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C75 Elliptic integrals as hypergeometric functions
33-03 History of special functions
33E20 Other functions defined by series and integrals
60G50 Sums of independent random variables; random walks
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
01A60 History of mathematics in the 20th century
01A61 History of mathematics in the 21st century

Citations:

Zbl 0022.33202
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Full Text: DOI

References:

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