Multiple-location matched approximation for Bessel function $J_0$ and its derivatives

Abstract

I present an approximation of Bessel function $J_0(r)$ of the first kind for small arguments near the origin. The approximation comprises a simple cosine function that is matched with $J_0(r)$ at $r=\pi /\text{e}$. A second matching is then carried out with the standard, but slightly modified, far-field approximation for $J_0(r)$, such that first and second derivatives are also considered. The approximation is practical when nonlinear dynamics come into play, in particular in the case of nonlinear interactions that involve second order differential equations as in acoustic--gravity wave theory. A demonstration of the proposed matching technique applied to three-dimensional acoustic--gravity wave triad resonance in cylindrical coordinates is provided.

10 Pages, 3 figures

References in corpus (8)

• Magnetars
• Transient pulsed radio emission from a magnetar
• Magnetar-like Emission from the Young Pulsar in Kes 75
• Discovery of luminous pulsed hard X-ray emission from anomalous X-ray pulsars 1RXS J1708-4009, 4U 0142+61 and 1E 2259+586 by INTEGRAL and RXTE
• A Magnetar-like Outburst from a High-B Radio Pulsar
• Magnetar Broadband X-ray Spectra Correlated with Magnetic Fields: Suzaku Archive of SGRs and AXPs Combined with NuSTAR, Swift, and RXTE
• Evidence for the magnetar nature of 1E 161348-5055 in RCW 103
• Chandra Observations of the High-Magnetic-Field Radio Pulsar J1718-3718

Cited by in corpus (1)

• On the role of magnetic fields into the dynamics and gravitational wave emission of binary neutron stars