Simple analytical approximations to the integrals of the Bessel functions \(J_\nu\): Application to the transmittance of a circular aperture (Q2750765)

In this work by using a recently developed method known as quasi-fractional approximation [see \textit{P. Martín} and \textit{G. A. Baker jun.}, J. Math. Phys. 32, No. 6, 1470-1477 (1991; Zbl 0745.41016), two new and simple analytic approximations to the integral of the Bessel function \(J_0\), easily calculable and with good accuracy, are presented. The two approximants found are then applied to obtain first and second-order approximations to the coefficient of transmittance of a plane wave through a circular aperture of known radius. The advantaqe of the approximations attained, particularly the second order one, is about 10 times better than another published in a previous work [see the authors, J. Comput. Phys. 73, 481-489 (1987; Zbl 0631.65014)]. As an extension, a couple of first order approximations to the fractional order integral of the first kind Bessel function \(J_\nu (nu>-1)\), namely \(\int^x_0 J_\nu(t) dt\), are also derived.