ABSTRACT
The higher transcendental functions of mathematical physics either satisfy second-order differential equations with analytic coefficients, or they have integral representations, or both. A comparatively simple method is described for finding error bounds for integrals involving one or more small parameters as well as a large positive parameter N. This method makes use of the maximum-modulus theorem of complex-variable theory and has been applied in at least three previous instances, but in each instance so much supporting analytical detail was required that the simple idea underlying the argument was perhaps not readily perceived. Uniform bounds for the coefficient functions can therefore be found from simpler non-uniform bounds by applying the maximum-modulus theorem. The chapter shows how the inequalities can be established in a simple manner by use of the maximum-modulus principle.
