If you need an accessible version of this item please contact JSTOR User Support

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
  • Download PDF
  • Cite this Item

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support
Infinitely Many Stokes Smoothings in the Gamma Function
Preview not available

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.

Page Thumbnails

  • Thumbnail: Page 
465
    465
  • Thumbnail: Page 
466
    466
  • Thumbnail: Page 
467
    467
  • Thumbnail: Page 
468
    468
  • Thumbnail: Page 
469
    469
  • Thumbnail: Page 
470
    470
  • Thumbnail: Page 
471
    471
  • Thumbnail: Page 
472
    472