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Infinitely Many Stokes Smoothings in the Gamma Function

M. V. Berry
Proceedings: Mathematical and Physical Sciences
Vol. 434, No. 1891 (Aug. 8, 1991), pp. 465-472
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/51844
Page Count: 8
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Infinitely Many Stokes Smoothings in the Gamma Function
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Abstract

The Stokes lines for Γ (z) are the positive and negative imaginary axes, where all terms in the divergent asymptotic expansion for ln Γ (z) have the same phase. On crossing these lines from the right to the left half-plane, infinitely many subdominant exponentials appear, rather than the usual one. The exponentials increase in magnitude towards the negative real axis (anti-Stokes line), where they add to produce the poles of Γ (z). Corresponding to each small exponential is a separate component asymptotic series in the expansion for ln Γ (z). If each is truncated near its least term, its exponential switches on smoothly across the Stokes lines according to the universal error-function law. By appropriate subtractions from ln Γ (z), the switching-on of successively smaller exponentials can be revealed. The procedure is illustrated by numerical computations.

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