On the accuracy of the approximation of the complex exponent by the first terms of its Taylor expansion with applications

A modification of the Taylor expansion for the complex exponential function eixeix, x∈Rx∈R, is proposed yielding precise moment-type estimates of the accuracy of the approximation of a Fourier transform by the first terms of its Taylor expansion. Moreover, a precise upper bound for the third moment of a probability distribution in terms of the absolute third moment is established. Based on these results, new precise bounds for Fourier–Stieltjes transforms of probability distribution functions and for their derivatives are obtained that are uniform in classes of distributions with prescribed first three moments.