Efficient Evaluation of Integrals of Analytic Functions by the Trapezoidal Rule

Quite often the problem of determining t h e value of a definite integral arises. When the integral cannot be readily evaluated by analysis, we m u s t resort to numerical methods. Here we discuss a method of numerical q u a d r a t u r e which gives promise of being useful in evaluating some types of integrals t h a t are difficult to handle by conventional numerical methods. In particular, we consider the problem ( M o r a n 1 and Schwartz 2 ) of transforming a given integral of an analytic function /(.r) into a rapidly converging one (with limits ± °c ) which can be efficiently evaluated by the trapezoidal rule, (1) T h e integral and series are assumed to converge. In addition to t h e trapezoidal error E, a second error is introduced when t h e series is t r u n c a t e d in the process of computation. It is supposed t h a t both errors are m a d e negligible, E by taking h small, and the truncation error by taking enough terms in the series. T h e feature which makes the use of (1) a t t r a c t i v e is t h a t E often decreases in proportion to exp ( -- C/h) as h decreases, C being a constant. T h u s if h gives threefigure accuracy, h/2 will give six-figure accuracy in m a n y cases. 707