Fourier Transforms Part 2

Question 1

How would you compute the Fourier Transform of 𝛿(x-y)?

Answer 1

Substitute 𝛿(x-y) for f(x) in the forward Fourier Transform equation.

Question 2

What is the equation we get from computing the fourier transform of 𝛿(x-y)?

Answer 2

𝛿(x-y) = 1/2π * integral from -inf to inf of exp(ik(x-y)) dk

Question 3

What is reciprocal space and what is it denoted by?

Answer 3

The space which is made by the fourier transform, denoted by k.

Question 4

What do we do to prove that the forward and reverse transforms are properly defined?

Answer 4

f(x) is equal to the inverse fourier transform of the fourier transform of f(x).

Question 5

What equation do we get after taking the inverse fourier transform of the fourier transform of f(x)? What does this prove?

Answer 5

f(x) = integral from -inf to inf of dx’ 𝛿(x-x’) * f(x’)

This shows that the fourier transform and inverse fourier transforms precisely invert eachother.

Question 6

What is one wya we can define the forward and inverse fourier transforms differently?

Answer 6

f(x) = 1/2π * a * integral from -inf to inf of f(k)(hat) * exp(ikx) dk

f(k)(hat) = 1/a * integral from -inf to inf of f(x) * exp(ikx) dx

Question 7

How do we solve the transform of f(x) = 1?

Answer 7

Recall that 1/2π * integral from -inf to inf pf exp(ik(x-y)) dk = 𝛿(x-y), and rearrange this so that it fits our equation for the transform with f(x) = 1.

Question 8

What is the fourier transform of f(x) = 1?

Answer 8

f(k) = 2π*𝛿(k)

Question 9

How do we solve the transform of f(x) = exp(-|x|)?

Answer 9

• Sub in f(x), get rid of the moduls by dividing the integral into two equal parts
• Do these integrals to find the answer

Question 10

What is the fourier transform of f(x) = exp(-|x|) equal to?

Answer 10

f(k) = 2/(1+k^2)

Question 11

How do we solve the transform of f(x) = a*g(x)?

Answer 11

• Sub it in and take a out as it is a constant

- Find that it is just the constant multiplied by the transform of that function

Question 12

What is the fourier transform of f(x) = a*g(x) equal to?

Answer 12

a*g(k)

Question 13

How do we solve the transform of f(x+a)?

Answer 13

• Sub in f(x) to the equation
• Use dummy variables to get rid of x+a
• Rearrange

Question 14

What is the fourier transform of f(x+a) equal to?

Answer 14

The transform of the untranslated function multiplied by a complex exponential with phase ka

transform of (f(x+a)) = f(k)*exp(ika)

Question 15

How do we solve the inverse fourier transform of 𝛿(k-q)

Answer 15

• Sub it into the inverse equation

- 𝛿 in reciprocal space picks out only the k=q mode, hence the real space function is a “pure” complex exponential

Question 16

What is the inverse fourier transform of 𝛿(k-q) equal to?

Answer 16

1/2π * exp(iqx)

Question 17

How do we solve the transform of sin(qx)?

Answer 17

• Sub it in
• Put sin(qx) into its exponential form
• Do the integral after taking out the 1/2i

Question 18

What is the fourier transform of sin(qx) equal to?

Answer 18

2π/2i * [𝛿(k-q) (-) 𝛿(k+q)]

Question 19

What is the orthogonality relation for fourier series?

Answer 19

𝛿(mn) = 𝛿(nm) = 1/L * integral from -L/2 to L/2 of exp(+/- i(n-m) * 2πx/L) dx, because either (n-m) or (m-n)=-(n-m)

Question 20

What is the orthogonality relation for fourier transforms?

Answer 20

𝛿(x-y) = 𝛿(y-x) = 1/2π * integral from -inf to inf of exp(+/- ik(x-y)) dk, because either x-y or y-x=-(x-y)

Question 21

What function f(x) can we use for a Gaussian example?

Answer 21

f(x) = Nexp(-αx^2) = Nexp(-x^2/(2σ^2))

Question 22

How do we normalise the Gaussian function?

Answer 22

Set the integral equal to 1 (normalisation)

Question 23

How do we find the normalised gaussian function?

Answer 23

• Make integral 2 parts with x part and y part
• Change variables from (x,y) to (r,θ)
• Change the integral accordingly
• Find the integral and set equal to 1/N, hence finding N
• Substitute N into the original equation

Question 24

What is the equation for the normalised gaussian function?

Answer 24

f(x) = sqrt(α/π)*exp(-αx^2)

Question 25

What is the first step in doing the transform of the normalised gaussian?

Answer 25

Identify that the integrand is of the form exp(ax^2+bx), and complete the square

Question 26

How do you complete the square for ax^2 + bx?

Answer 26

• ax^2 + bx = a(x+c)^2 + d

- Balance powers of x and balance units of 1 (units with no x in) to find c and d in terms of b and a

Question 27

What is the completed square version of ax^2 + bx?

Answer 27

ax^2 + bx = a(x+b/2a)^2 - b^2/4a

Question 28

How do we use the completed square to solve the transform of the gaussian?

Answer 28

Substitute it into the equation and rearrange, taking out exponential factors etc and set part inside the brackets to x’ to make simpler

Question 29

What is the final answer for the transform of a gaussian and why?

Answer 29

f(k) = exp(-k^2/4α)

This is because the integral part is equal to sqrt(π/α), so these cancel out.

Question 30

What is the transform of df/dx (partial) equal to?

Answer 30

f(k) = ik * f(k)

Question 31

What is a general solution to the transform of d^n f/dx^n?

Answer 31

f(k) = (ik)^n * f(k)