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.
The aim in this note is to provide a generalization of an interesting entry in Ramanujans Notebooks that relate sums involving the derivatives of a function Phi(t) evaluated at 0 and 1. The generalization obtained is derived with the help of expressi
ons for the sum of terminating 3F2 hypergeometric functions of argument equal to 2, recently obtained in Kim et al. [Two results for the terminating 3F2(2) with applications, Bull. Korean Math. Soc. 49 (2012) pp. 621{633]. Several special cases are given. In addition we generalize a summation formula to include integral parameter differences.
Expressions for the summation of a new series involving the Laguerre polynomials are obtained in terms of generalized hypergeometric functions. These results provide alternative, and in some cases simpler, expressions to those recently obtained in the literature.
