Generalization of the integral representation gamma function has been obtained, which shows that Hankel contour assumes rotation in complex plane. The range admissible values for angle is set. Using this representation, generalization Mittag-Leffler obtained expresses value terms integral.

Abstract We apply the Lorentzian path integral to decay of a false vacuum and estimate false-vacuum rate. To make convergent, deformation an integration contour is performed by following Picard-Lefschetz theory. show that nucleation rate critical bubble, for which corresponding bounce action extremized, has same exponent as Euclidean approach. also extend our computation bubble larger or smalle...

1 The Schwinger-Keldysh contour 2 1.1 States and observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The entropy principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Path integral formulation . . . . . . . . . . . . . . . . . . ...

In his third notebook, Ramanujan claims that ∫ ∞ 0 cos(nx) x2 + 1 log xdx+ π 2 ∫ ∞ 0 sin(nx) x2 + 1 dx = 0. In a following cryptic line, which only became visible in a recent reproduction of Ramanujan’s notebooks, Ramanujan indicates that a similar relation exists if log x were replaced by log x in the first integral and log x were inserted in the integrand of the second integral. One of the go...

Using a circular contour integral representation for the generalized Marcum-Q function, Qm(a, b ) , we derive a new closed-form formula for the moment generating function (MGF) of the output signal power of a dual-diversity selection combiner (SC) in bivariate Nakagami-m fading with positive integer fading severity index. This result involve% only elementary functions and holds for any value of...

Numerical evaluations of Feynman integrals often proceed via a deformation the integration contour into complex plane. While valid contours are easy to construct, numerical precision for multi-loop integral can depend critically on chosen contour. We present methods optimize this using combination optimized, global shifts and normalizing flow. They lead significant gain in precision.

We derive the fixed-$\mathrm{\ensuremath{\Lambda}}$ and unimodular propagators using path integral formalism as applied to Einstein-Cartan action. The simplicity of action (which is linear in lapse function) allows for an exact integration starting from function enforcement Hamiltonian constraint, leading a product Chern-Simons states if connection fixed at endpoints. No saddle point approximat...