The beautiful Euler spiral, defined by the linear relationship between curvature and arclength, was first proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, first by Augustin Fresnel to compute diffraction of light through a slit, and again by Arthur Talbot to produce an ideal shape for a rail...

The Riemann-Wirtinger integral is a function defined by a definite integral on a complex torus whose integrand is a power product of the exponential function and theta functions. It was found as a special solution of the system of differential equations which governs the monodromy-preserving deformation of Fuchsian differential equations on the complex torus [11], [12]. The main purpose of this...

In this paper, we aim at establishing certain integral transform and fractional integral formulas for the generalized Gauss hypergeometric functions, which was introduced and studied by Özergin et al. [J. Comput. Appl. Math. 235(2011), 4601-4610]. All the results derived here are of general character and can yield a number of (known and new) results in the theory of integral transforms and frac...

tions F(3), F(4), Lauricella’s quadruple hypergeometric function F A , Exton’s multiple hypergeometric functions X E:G;H , K10, K13, X8, (k)H 2 , (k)H 4 , Erdélyi’s multiple hypergeometric function Hn,k, Khan and Pathan’s triple hypergeometric function H (P) 4 , Kampé de Fériet’s double hypergeometric function F E:G;H , Appell’s double hypergeometric function of the second kind F2, and the Sriv...

A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of pr...

We determine generalised asymptotic solutions for the inflaton field, Hubble parameter, and equation-of-state parameter valid during oscillatory phase of reheating potentials that close to their global minima behave as even monomial potentials. For quadratic potential, we derive a expansion with respect scale set by inverse powers cosmic time. quartic an explicit, two-term solution in terms Jac...

Abstract. A one-parameter generalisation Rλ(X; b) of the symmetric Macdonald polynomials and interpolations Macdonald polynomials is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry, principal specialisation formula and q-difference equation for Rλ(X; b). We also prove a new multiple q-Gauss summation formula and several further results for sl...

The Orbital Angular Momentum (OAM) of light has been at the center several classical and quantum applications for imaging, information processing communication. However, complex structure inherent in OAM states makes their detection classification nontrivial many circumstances. Most current schemes are based on models built upon use Laguerre-Gauss modes. this may not general be sufficient to ca...

In this paper, we consider a class of fractional-order differential equations and investigate two aspects these equations. First, the existence unique solution, then, using new control functions, Gauss hypergeometric stability. We use Chebyshev Bielecki norms in order to prove by Picard method. Finally, give some examples illustrate our results.