© 1965 by British Computer Society
Evaluation of certain definite integrals frequently encountered in radiational and diffraction problems involving circular geometry*
1 Engineering Division, Midwest Research Institute, Kansas City, Missouri, USA, 2 Leiter der Rechengruppe, Institut für Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, Technische Hochschule, Stuttgart, Germany
Consideration is given to the evaluation of definite integrals of the form
[equation: see PDF]
where n is any integer and R = [r2 + a2 – 2ra sin 
cos 
]1/2. These integrals are encountered in a variety of electromagnetic radiation and diffraction problems involving circular geometry.
Various techniques that have been used to obtain approximate evaluations of these field integrals are discussed briefly. Subsequently, a rather straightforward analytical method is described which gives closed-form evaluation of the integrals without resorting to any simplifying approximations. The results of this method are presented in terms of near-zone or source-region functions, depending on whether r > a or vice versa. These functions are defined by infinite series which involves well-known mathematical functions.
Some interesting properties of the near-zone and source-region functions are considered, and it is shown that their infinite series representations are amenable to machine computation. Problems associated with the calculation of numerical values are discussed and generalized algorithms for the near-zone and source region functions are presented.
* The research reported herein was acomplished without sponsorship from any tax-supported agency of the U.S. Government.
Engineering Division, Midwest Research Institute, Kansas City, Missouri.
¶ Univac Division, Remington Rand Limited, 65, Holborn Viaduct, London, E.C. 1. Presently Leiter der Rechengruppe, Institut für Statik und Dynamik der Luft- und Raumfahrtkonstruktionen, Technische Hochschule, Stuttgart, Germany.