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-ialpha)] cos (xcosh u),du,] where $(x, rho, alpha)$ are
cylindrical polar coordinates with origin based at the source and $-pi/2leqalphaleqpi/2$. An asymptotic expansion of $F(x,rho,alpha)$ when $x$ and $rho$ are small, but such that $Mequiv x^2/(4rho)$ 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.
We examine a class of exact solutions for the eigenvalues and eigenfunctions of a doubly anharmonic oscillator defined by the potential $V(x)=omega^2/2 x^2+lambda x^4/4+eta x^6/6$, $eta>0$. These solutions hold provided certain constraints on the cou
pling parameters $omega^2$, $lambda$ and $eta$ are satisfied.
We consider the generalised Beta function introduced by Chaudhry {it et al./} [J. Comp. Appl. Math. {bf 78} (1997) 19--32] defined by [B(x,y;p)=int_0^1 t^{x-1} (1-t)^{y-1} exp left[frac{-p}{4t(1-t)}right],dt,] where $Re (p)>0$ and the parameters $x$
and $y$ are arbitrary complex numbers. The asymptotic behaviour of $B(x,y;p)$ is obtained when (i) $p$ large, with $x$ and $y$ fixed, (ii) $x$ and $p$ large, (iii) $x$, $y$ and $p$ large and (iv) either $x$ or $y$ large, with $p$ finite. Numerical results are given to illustrate the accuracy of the formulas obtained.
The purpose of this note is to provide an alternative proof of two transformation formulas contiguous to that of Kummers second transformation for the confluent hypergeometric function ${}_1F_1$ using a differential equation approach.
In this note we state (with minor corrections) and give an alternative proof of a very general hypergeometric transformation formula due to Slater. As an application, we obtain a new hypergeometric transformation formula for a ${}_5F_4(-1)$ series wi
th one pair of parameters differing by unity expressed as a linear combination of two ${}_3F_2(1)$ series.
