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NASA Technical Note
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On the Theory of Boolean Formulas
The theory of substitutions for Boolean formulas developed in a previous report is applied here to the determination of those s-tuples of formulas that may be allowable replacements for s -tuples of sub-formulas of given Boolean formulas. The criteria of allowability are very general and flexible, yet may be expressed with great precision and ease. The results formalize and extend various known isolated instances. They may be used for the study of a large number of problems in the theory and application of Boolean formulas and functions.
Asymptotic Solutions of Dipoles in a Semi-infinite Medium
A theoretical analysis is made of the electromagnetic fields in two homogeneous media separated by a plane interface with a point source located in the denser medium. The solution is expressed in the form of integrals which cannot be evaluated explicitly. Asymptotic evaluations of the integrals have been made by many investigators using the saddlepoint technique. In the present work, all known asymptotic results are presented in one comprehensive form, using a modification of the method suggested by Lighthill for the asymptotic evaluation of the Fourier integrals. The regions of validity of the solutions are indicated wherever possible. The advantage of this method over others is its ease and simplicity. The present results agree term by term with the earlier ones of Banos and Wesley (1953-1954), and Paul (1959), who investigated the case of a source and receiver close to the interface, and an arbitrary location of source and receiver, respectively. The results obtained in the report are also compared with those of Stein (1955). (Author).
