We argue that the Lorentzian path integral is a better starting point for quantum cosmology than its Euclidean counterpart. In particular, we revisit the minisuperspace calculation of the Feynman path integral for quantum gravity with a positive cosmological constant. Instead of rotating to Euclidean time, we deform the contour of integration over metrics into the complex plane, exploiting Pica...

Some finite integrals are difficult to evaluate numerically because the integrand has a high peak or contains a rapidly oscillating function as a factor. If the integrand is an analytic function Cauchy's theorem may be applied to replace the integral by a contour integral, the path being chosen to avoid singularities of the integrand, together with a possible residue contribution. If the integr...

The result was that, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a ...

Integral representations are derived for the parabolic cylinder functions U(a, x), V (a, x) andW (a, x) and their derivatives. The new integrals will be used in numerical algorithms based on quadrature. They follow from contour integrals in the complex plane, by using methods from asymptotic analysis (saddle point and steepest descent methods), and are stable starting points for evaluating the ...

In this paper, a new framework for one-dimensional contour extraction from discrete two-dimensional data sets is presented. Contour extraction is important in many scientific fields such as digital image processing, computer vision, pattern recognition, etc. This novel framework includes (but is not limited to) algorithms for dilated contour extraction, contour displacement, shape skeleton extr...

We present an iterative Monte Carlo path integral methodology for evaluating thermally averaged real-time correlation functions. Standard path integral Monte Carlo methods are used to sample paths along the imaginary time contour. Propagation of the density matrix is performed iteratively on a grid composed of the end points of the sampled paths. Minimally oscillatory propagators are constructe...

This article is concerned with a new method for the approximate evaluation of Fourier sine and cosine transforms. We develope and analyse a new quadrature rule for Fourier sine and cosine transforms involving transforming the integral to one over the entire real line and then using the trapezoidal rule in order to approximate the transformed integral. This method follows on from the work of Oou...

In this work we demonstrate how to derive the Kohn-Sham equations of time-dependent current-density functional theory from a generating action functional defined on a Keldysh time contour. These Kohn-Sham equations contain an exchange-correlation contribution to the vector potential. For this quantity we derive an integral equation. We further derive an integral equation for its functional deri...

The steady-state waiting time in a stable GI/G/1 queue is equivalent to the maximum of a general random walk with negative drift. Thus, the distribution of the steady-state waiting time in the GI/G/1 queue is characterized by Spitzer’s (1956) formula. However, earlier, Pollaczek (1952) derived an equivalent contour-integral expression for the Laplace transform of the GI/G/1 steady-state waiting...