Laplace's asymptotic methods (Q2713240)

The authors describe Laplace's method for the approximate evaluation of a definite integral of form NEWLINE\[NEWLINEI_n= \int_a^b u^n(x) f(x) dxNEWLINE\]NEWLINE as \(n\to \infty\). They examine the methodology in detail, as manifested in Laplace's (1812) Théorie analytique des probabilités, but staying free from specific probabilistic context. Such an approach, focused on the point \(c\) maximizing \(u(x)\), has manifested itself in recent decades in the method of saddle-point approximation in mathematical statistics. An additional reference useful to the reader is [\textit{N. G. De Bruijn}, Asymptotic methods in analysis. North Holland, Amsterdam (1958; Zbl 0082.04202); Reprint New York, Dover (1981; Zbl 0556.41021)].