The asymptotic expansion of Gordeyev's integral

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

Received: March 11, 1997