The Numerical Evaluation of a Certain Integral with a Rapidly Oscillating Integrand

摘要

This memorandum considers the numerical evaluation of integrals of the form 'over the range from O to T, the integral: E(To the alpha times S power), Times sin 2W(Sub O)S, times the-3/2 power of the quantity (D squared + the square of (Y(Sub O)-VS)), times DS' by two methods. The first is conventional. The second is unconventional and exploits the possibility of expanding the expression 'D(squared) + (Y(sub O)-VS)(squared)' in terms of legendre polynomials. The integrals whose evaluation is discussed, arose in the study of the following problem. A sinusoidal source, with strength that falls off as the inverse cube of the distance from the source, moves in a straight line past a receiver with velocity much smaller than the field propagation velocity.