The asymptotic expansion of Gordeyev's integral

  • R. B. Paris

DOI: 10.1007/PL00001486

Cite this article as:
Paris, R. Z. angew. Math. Phys. (1998) 49: 322. doi:10.1007/PL00001486


We obtain asymptotic expansions for the integral¶¶\( G_\nu(\omega,\lambda)=\omega\int_0^\infty \exp [i\omega t-\lambda (1-\cos t)- {1\over2}\nu t^2] dt, \)¶for large values of \(\omega\) and \(\lambda\) and \(\nu\rightarrow 0+\). For positive real parameters, the real part of the integral is associated with an exponentially small expansion in which the leading term involves a Jacobian theta function as an approximant. The asymptotic expansions are compared with numerically computed values of \(G_\nu(\omega,\lambda)\).

Key words. Gordeyev's integral, asymptotic expansions, steepest descent, Stokes phenomenon.

Copyright information

© Birkhäuser Verlag, Basel, 1998

Authors and Affiliations

  • R. B. Paris
    • 1
  1. 1.Division of Mathematical Sciences, University of Abertay Dundee, Dundee DD1 1HG, UK, Fax: 01382 308877UK