Reliable operations on oscillatory functions (Q1973552)

To differentiate or integrate a function of the form \(F(x)=f(x)w(\omega x+\delta)\) where \(w\) is \(sin\) or \(cos\) or \(sinh\) or \(cosh\), a method is proposed where only the \(f\)-part of \(F\) is approximated, while the \(w\)-factor is treated exactly. For the derivatives, the Leibniz formula is used to express \(F^{(n)}\) in terms of derivatives of \(f\) and the latter are approximated by classical finite differences. For the integrals, a second degree interpolating polynomial is used for the \(f\)-part and the resulting approximation is integrated exactly. These techniques give much better results than by the method of exponential fitting [cf. \textit{L. Ixaru}, Comput. Phys. Commun. 105, No. 1, 1-19 (1997; Zbl 0930.65150)]. Moreover the error estimates do not depend on \(\omega\).