16 - Different Forms of Fourier Series

Abstract

This chapter focuses on the different forms of Fourier series. The Fourier series discussed are for f(x) on –л ≤x ≤ л, for f(x) on –L ≤ x < L, and for f(x) on a ≤x ≤ b. The chapter also reviews the half-range Fourier cosine series for f(x) on 0 ≤x ≤л and for f(x) on 0 ≤x ≤L. The half-range Fourier sine series for f(x) on 0 ≤x ≤ л and for 0 ≤x ≤L is also reviewed. The chapter discusses the complex (exponential) Fourier series for f(x) on –л ≤x ≤ л and for f(x) on –L ≤ x ≤ L. Various representative examples of Fourier series are reviewed, and periodic extensions and convergence of Fourier series is discussed. If f(x) is defined on the interval –л ≤ x ≤ л, then since each function in the Fourier series of f(x) is periodic with a period that is a multiple of 2л, the Fourier series itself will be periodic with period 2л. Thus, irrespective of how f(x) is defined outside the fundamental interval –л ≤ x ≤ л, the Fourier series will replicate the behavior of f(x) in the intervals (2n – 1)л ≤x ≤ (2n + 1)л, for n= ±1, ±2, and so on. Each of these intervals is called a periodic extension of f(x).