The asymptotic expansion of Gordeyev's integral

Abstract.

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 $G_\nu(\omega,\lambda)$.