In discussions of the type undertaken in this monograph it is common to deal with functions represented by integrals that are resistant to exact evaluation, yet whose integrands contain one or more parameters that approach specific values in the problem of interest. In such cases it is often possible to find asymptotic representations of the function in a series of terms rapidly decreasing in value as z → z0, say. Even if such series do not converge, they can provide representations of the function for those parameter values to any desired degree of accuracy. With sufficient attention to detail, such expansions can often be differentiated and integrated term by term.

Many asymptotic developments pursued here arise after continuation into the complex plane, in which case additional difficulties emerge because we must insist that asymptotic relations be unique and independent of the path of approach of z to z0.