Computer visualization of the approximations of functions by several terms of their Fourier series

Abstract

Fourier trigonometric series are a constant component of the basic course of calculus at Universities. If we teach the topic about the Fourier series in the classical way it is often necessary to give a lot of time to a tedious mechanical performing of some operations. For a given function and a given interval, a student must count the coefficients of the Fourier series by definite integrals, to investigate the convergence of this series, to draw graphs and so on. Then a teacher with his students cannot sufficiently deal with such an important problem as the approximation of functions in a given interval by the Fourier polynomial, it means by the sum of several initial terms of the Fourier series. The reasons for it are the lack of time and the fact that these problems require more complex calculating and graphics tools for their solution. Fourier series occur very frequently in the problems of mathematical physics. The expansions of the functions can help to solve them in various physical problems. Our aim is to show on concrete simple examples that the graphical and numerical investigations Fourier series expansions by Mathematica commands and programs may be effectively used.