The trapezoidal rule for analytic functions of rapid decrease

摘要

The evaluation of the integral of an analytic function f over the entire real line may be accurately approximated by the trapezoidal method of integration or by “mapped” trapezoidal rules. Typically the latter is employed to accelerate the convergence rate of the rule at the expense of a bit more complicated formula. In the case that the integrand f decreases rapidly and has its singularities sufficiently removed from the real line, the trapezoidal rule provides not only a simpler method of evaluation but also a more accurate method of approximation.