On the use of asymptotic approximations of Green's type when the coefficient has zeros

Abstract

ABSTRACTAllowance is made for the error terms in the cases where χ has a simple zero at x = 0, or two simple zeros at a; = ± 1 (with χ either positive or negative for − 1 < x < 1). The errors of the approximations given by neglecting the error terms are found to be small in all cases, sometimes much smaller than there was any previous reason to expect.