Here we compute some Fourier series to illustrate a few useful computational tricks and to illustrate why convergence of Fourier series can be subtle. Because the integral is over a symmetric interval, some symmetry can be exploited to simplify calculations.

Topics covered: Introduction to Fourier Series; Basic Formulas for Period 2 (pi) Instructor/speaker: Prof. Arthur Mattuck

The following examples are just meant to give you an idea of what sorts of computations are involved in nding a Fourier series. You're not meant to be able to carry out these computations yet.

To compute the Fourier series, use (4)-(6) with ` = 1. First, observe that f(x) is an even function, so f(x) cos n x is an even function; f(x) sin n x is an odd function (10) for all n (note that the product of an odd and even function is odd).

This section explains three Fourier series: sines, cosines, and exponentials eikx. Square waves (1 or 0 or −1) are great examples, with delta functions in the derivative.

We introduce general periodic functions and learn how to express them as Fourier series, which are sums of sines and cosines.

We can think of Fourier series as an orthogonal decomposition. Vector representation of 3-space: let r ̄ represent a vector with components {x, y, and z} in the {xˆ, yˆ, and zˆ} directions, respectively.

Fourier series are useful for periodic func-tions or functions on a fixed interval L (like a string). One can do a similar analysis for non-periodic functions or functions on an infinite interval (L → ∞) in which case the decomposition is known as a Fourier transform.

For today and the next two lectures, we are going to be studying Fourier series. Today will be an introduction explaining what they are.

Fourier Series From your di®erential equations course, 18.03, you know Fourier's expression representing a T -periodic time function x(t) as an in ̄nite sum of sines and cosines at the fundamental fre-quency and its harmonics, plus a constant term equal to the average value of the time function over a period: 1X