  The velocity potential in the Kelvin ship-wave source can be partly expressed
in terms of space derivatives of the single integral
\[F(x,\rho,\alpha)=\int_{-\infty}^\infty \exp\,[-\frac{1}{2}\rho \cosh
(2u-i\alpha)] \cos (x\cosh u)\,du,\] where $(x, \rho, \alpha)$ are cylindrical
polar coordinates with origin based at the source and
$-\pi/2\leq\alpha\leq\pi/2$. An asymptotic expansion of $F(x,\rho,\alpha)$ when
$x$ and $\rho$ are small, but such that $M\equiv x^2/(4\rho)$ is large, was
given using a non-rigorous approach by Bessho in 1964 as a sum involving
products of Bessel functions. This expansion, together with an additional
integral term, was subsequently proved by Ursell in 1988.
  Our aim here is to present an alternative asymptotic procedure for the case
of large $M$. The resulting expansion consists of three distinct parts: a
convergent sum involving the Struve functions, an asymptotic series and an
exponentially small saddle-point contribution. Numerical computations are
carried out to verify the accuracy of our expansion.
