ABSTRACT
If we have a way of assessing C(z) for large lzl, then the magnitude of the optimum remainder term in the expansion (2.01) can be estimated. In some cases it is actually possible to construct an asymptotic expansion for C(z) in descending powers of z. In these fortunate circumstances C(z) can be calculated to several significant figures, considerably increasing the attainable accuracy in the computed value of f(z). For this reason, C(z) is called a converging factor. 2.2 In the next three sections procedures for expanding C(z) are introduced by means of examples. Instead of using z as asymptotic variable it is convenient to change to n. Suppose, for example, that n(z) = [lzl]. Then we write
and seek an asymptotic representation of Cn{(n+C)eiB) for large n, which is uniformly valid with respect to 8 E [el, 8,] and C E [0, I] (or some larger region). To apply the result for an assigned value of z, we take 8 = phz, n = [lzl], and r = IZI-CIZII.
