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XXI—The Evaluation of a Definite Integral Which Occurs in Asymptotic Partition Theory*

Published online by Cambridge University Press:  14 February 2012

M. M. Robertson
M. M. Robertson
Affiliation:
Department of Mathematics, University of Aberdeen.
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Synopsis

In order to estimate the number of partitions of a multi-partite number, the components of which are all large and of approximately the same order of magnitude, it is necessary to evaluate for ℛ(z1) > o(l=1, 2, …,j — 1) the integral

where

for o < u < 2π min (1, |z1|−1, …, |zj−1|−1) and Asymptotic expansions are obtained for I when the z1 are small. Simple expressions give an approximate value of I when every zl is real and exact formulæ are derived when every Zl is real and rational.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 65 , Issue 4 , 1961 , pp. 283 - 309
Copyright
Copyright © Royal Society of Edinburgh 1961

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References

REFERENCES TO LITERATURE

Knopp, K., 1928. Theory and Application of Infinite Series, 520–539. London and Glasgow.Google Scholar
Legendre, A. M., 1921. “Tables of the Logarithms of the Complete Γ-function to Twelve Figures”, Tracts for Computers, IV. Cambridge.Google Scholar
Whittaker, E. T., and Watson, G. N., 1920. Modern Analysis, 265–280. Cambridge.Google Scholar
Wright, E. M., 1958. “A Definite Integral in the Asymptotic Theory of Partitions”, Proc. Lond. Math. Soc., 8, 312–320.CrossRefGoogle Scholar