Lecture 3.6: Real vs. complex Fourier series Matthew Macauley - - PowerPoint PPT Presentation

Lecture 3.6: Real vs. complex Fourier series

Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics

• M. Macauley (Clemson)

Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 1 / 6

Overview

Last time, we derived formulas for the complex Fourier series of a function.

Complex Fourier series

If f (x) is a piecewise continuous 2L-periodic function, then we can write f (x) =

∞

• n=−∞

cne

iπnx L

= c0 +

∞

• n=1
• cne

iπnx L

+ c−ne− iπnx

L

where c0 =

• f , 1
• = 1

2L L

−L

f (x) dx, cn =

• f , e

iπnx L

= 1 2L L

−L

f (x)e− iπnx

L

dx. Here, we will see how to go between the real and complex versions of a Fourier series. It’s just a simple application of the following identities that we’ve already seen:

Euler’s formula (and consequences)

eiθ = cos θ + i sin θ, e−iθ = cos θ − i sin θ, cos θ = eiθ + e−iθ 2 , sin θ = eiθ − e−iθ 2i .

• M. Macauley (Clemson)

Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 2 / 6

From the real to the complex Fourier series

Proposition

The complex Fourier coefficients of f (x) = a0 2 +

∞

• n=1

an cos nπx

L

+ bn sin nπx

L

are cn = an − ibn 2 , c−n = an + ibn 2 .

• M. Macauley (Clemson)

Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 3 / 6

From the complex to the real Fourier series

Proposition

The real Fourier coefficients of f (x) =

∞

• n=−∞

cne

iπnx L

are an = cn + c−n, bn = i(cn − c−n) .

• M. Macauley (Clemson)

Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 4 / 6

Computations

Example 1: square wave

Find the complex Fourier series of f (x) =

• 1,

0 < x < π −1, π < x < 2π.

• M. Macauley (Clemson)

Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 5 / 6

Computations

Example 2

Compute the real Fourier series of the 2π-periodic extension of the function ex defined on −π < 0 < π.

• M. Macauley (Clemson)

Lecture 3.6: Real vs. complex Fourier series Advanced Engineering Mathematics 6 / 6