Section 10.5 Fourier Series

Taylor Polynomials are a good approximation locally, but not necessarily globally We can use Fourier approximations –May not be as close as Taylor locally –Can be better globally –They are also useful for functions that are not continuous Approximates using trig functions –They are periodic

The Fourier polynomial of degree n is given by

As with Taylor polynomials, the higher the order, the better the approximation Therefore we introduce the Fourier series

Harmonics Given a periodic function (on 2π) expanded as a Fourier series The function is know as the k th harmonic What is the fifth harmonic of the square wave function?

Energy and the Energy Theorem The quantity is called the amplitude of the k th harmonic Now, we define the energy E of a periodic function f of period 2π to be the number Now for all positive k

That shows that the k th harmonic has energy We define the energy of the constant term to be Therefore we get

Example Let’s look at the energy from the first three harmonics (and the constant term) in the wave function approximation –We will compare that energy to the amount in the whole function

What if our function does not have a period of 2π? We can adapt by changing variables Suppose f(x) is periodic with period b We can let So t varies over the interval [-π, π]

The Fourier polynomial of degree n on [-b, b] is given by