Chapter 6 - Fourier Transforms

Question 1

Motivation

Fo

Answer 1

Fo = ao/2

Question 2

Motivation

an

Answer 2

an = Fn+ + Fn-

Question 3

Motivation

bn

Answer 3

bn = i(Fn+ - Fn-)

Question 4

Motivation

Fn+

Answer 4

Fn+ = 1/2(an - i*bn)
= 1/2L ∫ f(x) e^(-inπx/k) dx
-where the integral is taken from -L to L

Question 5

Motivation

Fn-

Answer 5

Fn- = 1/2(an + i*bn)
= 1/2L ∫ f(x) e^(inπx/k) dx
-where the integral is taken from -L to L

Question 6

The Fourier Transform

Definition

Answer 6

-the Fourier Transform of a function f(x) for which:
∫|f(x)| dx < ∞
-i.e. the integral (between -∞ and +∞) is finite, is defined as:
F(k) = 1/√(2π) ∫f(x)*e^(-ikx) dx
-where -∞

Question 7

The Fourier Transform Inverse

Definition

Answer 7

f(x) = 1/√(2π) ∫F(k)*e^(ikx) dk

-where -∞

Question 8

Differences Between Fourier Series and the Fourier Transform

Answer 8

1) the discrete index n has been replaces by a continuous parameter k, and the summation over n has been replaced by integration over k
2) the range of integration of x has changed from (-L,L) to (-∞,∞)
3) the integral has been normalised differently

Question 9

Useful Properties

Linearity

Answer 9

-Let Ƒ(f) and Ƒ(g) denote the Fourier transforms of the functions f(x) and g(x)
-then for constants a and b:
Ƒ(af+bg) = aƑ(f) + bƑ(g)

Question 10

Useful Properties

Dilation of x

Answer 10

-for a>0, we find:

Ƒ(f(ax)) = 1/a F(k/a)

Question 11

Useful Properties

The First Shift Theorem

Answer 11

-for constant a,

Ƒ(f(x-a)) = e^(-iak) Ƒ(f(x))

Question 12

Useful Properties

The Second Shift Theorem

Answer 12

Ƒ[e^(iax) f(x)] = F(k-a)
-in terms of the Inverse Fourier Transform, the second shift theorem takes the form:
Ƒ^(-1)[F(k-1)] = e^(iax) Ƒ^(-1)[F(k)]
-which is of the form of the first shift theorem

Question 13

Useful Properties

Fourier Transform of the Derivative

Answer 13

-suppose f(x) is piecewise smooth, f(x) and f’(x) are integrable and f(x)->0 as |x|->∞
-THEN:
Ƒ[f’(x)] = ikƑ[f(x)]

Question 14

Useful Properties

The Derivative of the Fourier Transform

Answer 14

-suppose f(x) and xf(x) are integrable
-THEN:
Ƒ[xf(x)] = i d/dk Ƒ[f(x)] = i F’(k)

Question 15

The Gaussian Function

Answer 15

-the Gaussian Function is defined as:
f(x) = e^(-ax²)
-for a positive constant a

Question 16

Integrate the Gaussian Function

Answer 16

I = ∫ e^(-ax²) dx = √[π/a]

-where the integral is taken between -∞ and +∞

Question 17

Fourier Transform of the Gaussian Function

Answer 17

a=1/2
f(x) = e^(-1/2 x²)

F(k) = e^(-1/2 k²)
-with F(0)=1

Question 18

Fourier Transform of u(x,t)

Status of x and t

Answer 18

• the variables x and t have a different status
• the variable x is the variable in the definition of the Fourier Transform
• the variable t is playing the role of an evolution parameter