An integral transform and its application in the propagation of Lorentz-Gaussian beams

The aim of the present note is to derive an integral transform I = ∫ 0 ∞ x s + 1 e - β x 2 + γ x M k , v ( 2 ζ x 2 ) J μ ( χ x ) d x , I = \int_0^\infty {{x^{s + 1}}{e^{ - \beta x}}^{2 + \gamma x}{M_{k,v}}} \left( {2\zeta {x^2}} \right)J\mu \left( {\chi x} \right)dx, involving the product of the Whittaker function Mk, ν and the Bessel function of the first kind Jµ of order µ. As a by-product, we also derive certain new integral transforms as particular cases for some special values of the parameters k and ν of the Whittaker function. Eventually, we show the application of the integral in the propagation of hollow higher-order circular Lorentz-cosh-Gaussian beams through an ABCD optical system (see, for details [13], [3]).