Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA note on an integral associated with the Kelvin ship-wave pattern
95
0
0.0
(
0
)
Authors
R. B. Paris
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 with one pair of parameters differing by unity expressed as a linear combination of two ${}_3F_2(1)$ series.
In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potential $leftlangle p,qrightrangle_{M}=int_{mathbb{R}}p(x)q(x)e^{-x^{4}+2tx^{2}}dx+Mp(0)q(0).$ We analyze some properties of these polynomials, such as the ladder operators and the holonomic equation that they satisfy and, as an application, we give an electrostatic interpretation of their zero distribution in terms of a logarithmic potential interaction under the action of an external field. It is also shown that the coefficients of their three term recurrence relation satisfy a nonlinear difference string equation. Finally, an equation of motion for their zeros in terms of their dependence on $t$ is given.
We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified $p$-adic group $G$ and generalize the formulation to give a description of the Hecke algebra $H_G(V)$ of weight $V$, where $V$ is a lattice in an irreducible algebraic representation of $G$.
We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then f = g up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $pi$. We also prove that if f and g are functions in the Nevanlinna class, and if |f | = |g| on the unit circle and on a circle inside the unit disc, then f = g up to the multiplication of a unimodular constant.
Using observations of the INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL), we put upper limits on the gamma-ray and hard X-ray prompt emission associated with the gravitational wave event GW150914, discovered by the LIGO/Virgo collaboration. The omni-directional view of the INTEGRAL/SPI-ACS has allowed us to constrain the fraction of energy emitted in the hard X-ray electromagnetic component for the full high-probability sky region of LIGO trigger. Our upper limits on the hard X-ray fluence at the time of the event range from $F_{gamma}=2 times 10^{-8}$ erg cm$^{-2}$ to $F_{gamma}=10^{-6}$ erg cm$^{-2}$ in the 75 keV - 2 MeV energy range for typical spectral models. Our results constrain the ratio of the energy promptly released in gamma-rays in the direction of the observer to the gravitational wave energy E$_gamma/$E$_mathrm{GW}<10^{-6}$. We discuss the implication of gamma-ray limits on the characteristics of the gravitational wave source, based on the available predictions for prompt electromagnetic emission.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
