Identities, relations, and methods

Question 1

Trig addition formulae

Answer 1

sin(A+B) = sinAcosB + cosAsinB

sin(A-B) = sinAcosB - cosAsinB

cos(A+B) = cosAcosB - sinAsinB

cos(A-B) = cosAcosB + sinAsinB

Question 2

Double angle formulae

Answer 2

sin(2x) = 2sinxcosx

cos(2x) = cos²x - sin²x
= 2cos²x - 1
= 1 - 2sin²x

Question 3

Fourier series (sin/cos)

Answer 3

For a function with period 2L in the range -L to +L

f(x) = a₀/2 + Σ(1to∞) (aₙcos(nπx/L) + bₙsin(nπx/L))

where
a₀ = 1/L ∫±L f(x) dx
aₙ = 1/L ∫±L f(x)cos(nπx/L) dx
bₙ = 1/L ∫±L f(x)sin(nπx/L) dx

Question 4

Orthogonality relations

Answer 4

∫±L sin(nπx/L) cos(mπx/L) dx = 0

∫±L sin(nπx/L) sin(mπx/L) dx = Lδₙₘ
where
δₙₘ =
1 if m = n
0 if m ̸= n

∫±L cos(nπx/L) cos(mπx/L) dx =
0 if m ̸= n
L if m = n
2L if m = n = 0

Question 5

Fourier series (complex)

Answer 5

f(x) = Σ±∞ cₙexp(inπx/L)

cₙ = 1/2L ∫±L f(x)exp(−inπx/L) dx

Question 6

Fourier transform

Answer 6

F(k) = ∫±∞ f(x) exp(−ikx) dx

f(x) = 1/2π ∫±∞ F(k) exp(ikx) dk

Question 7

Laplace equation

Answer 7

∇²φ = 0

Question 8

Wave equation

Answer 8

∇²φ = 1/c² ∂²φ/∂t²

Question 9

Schrödinger equation

Answer 9

−ħ²/2m ∇²ψ + Vψ = iħ ∂ψ/∂t

Question 10

Solution to
a d²f/dx² + b df/dx + cf = 0

Answer 10

f = Aexp(λ₁x) + Bexp(λ₂x)
where aλᵢ² + bλᵢ + c = 0

Question 11

Solution to
df/dx = −αf

Answer 11

f = Aexp(−αx)

Question 12

Solution to
d²f/dx² = −α²f

Answer 12

f = A cos(αx) + B sin(αx)

Question 13

Solution to
d²f/dx² = α²f

Answer 13

f = Aexp(αx) + Bexp(-αx)

Question 14

Trig to exponential

Answer 14

exp(ix) = cosx + isinx
exp(-ix) = cos-x - isinx

cosx = (exp(ix) + exp(-ix)) / 2
sinx = (exp(ix) - exp(-ix)) / 2i

Question 15

Hyperbolic definitions

Answer 15

sinhx = (exp(x) - exp(-x)) / 2
coshx = (exp(x) + exp(-x)) / 2

Question 16

General solution to wave equation

Answer 16

φ = f(x − ct) + g(x + ct)

Question 17

Method for solving wave equations

Answer 17

1. Investigate solutions in the separable form φ = X(x)T(t).
2. Build the general solution as a superposition of these.
3. Fix the undetermined constants by applying the initial conditions.

Question 18

sinA sinB as a sum

Answer 18

1/2 (cos(A−B) − cos(A+B))

Question 19

sinA cosB as a sum

Answer 19

1/2 (sin(A−B) + sin(A+B))

Question 20

cosA cosB as a sum

Answer 20

1/2 (cos(A−B) + cos(A+B))

Question 21

When is the Fourier series expansion valid

Answer 21

Valid for any function satisfying Dirichlet’s conditions:
* Single valued
* Finite number of discontinuities
* ∫±L |f(x)| dx must be finite

Note that sin(nπx/L) and cos(nπx/L) are orthogonal basis vectors.

Question 22

Define orthogonality

Answer 22

Two functions u(x) and v(x) are orthogonal on an interval a ≤ x ≤ b if
∫ₐᵇ u(x) v(x) dx = 0

Question 23

Mean of a function

Answer 23

f̄ ≡ 1/2L ∫±L f(x) dx = a₀/2

Question 24

RMS of a function

Answer 24

∫±L [f(x)]² dx
= 1/2 a₀ ∫±L f(x) dx
+ ∫±L Σ(1to∞) f(x) aₙ cos(nπx/L) dx
+ ∫±L Σ(1to∞) f(x) bₙ sin(nπx/L) dx

Evaluate each term
f̄² ≡ 1/2L ∫±L [f(x)]² dx = (a₀/2)² + 1/2 Σ(1to∞) aₙ² + bₙ²

rms is given by √f̄²

Question 25

Diffusion equation

Answer 25

∇²T = 1/α² ∂T/∂t

T can refer to the temperature in a region or the density of particles as they diffuse.

Question 26

Functional forms of Dirac delta

Answer 26

Rectangular function
δ(x) = lim(k→∞) kΠ(kx)

Triangle function
Λ(x) = (1−|x| for −1 ≤ x ≤ 1, 0 otherwise)
δ(x) = lim(k→∞) kΛ(kx)

Gaussian
δ(x) = lim(k→∞) √(k/π)’ exp(−kx²)

Sinc
δ(x) = lim(k→∞) k/π sin(kx)/kx

Exponential (derived from sinc)
δ(x) = 1/2π ∫±∞ exp(−ikx) dk = 1/2π ∫±∞ exp(ikx) dk = δ(−x)

Question 27

Properties of Dirac delta (7)

Answer 27

lim(x→0) δ(x) = ∞

∫±∞ δ(x) dx = 1

δ(x) = δ(−x)

xδ(x) = 0

x d/dx δ(x) = −δ(x)

∫±∞ δ(x−a) f(x) dx = f(a)

δ(ax) = 1/|a| δ(x)

Question 28

Bandwidth product relation

Answer 28

∆k∆x ≥ 1/2

Question 29

Heisenberg uncertainty relation from bandwidth product relation

Answer 29

Bandwidth product relation
∆k∆x ≥ 1/2

k is essentially the wavenumber
de Broglie’s theorem states
λp = h
⇒ k = 2π/λ = p/ħ

Substitute in
∆x∆p ≥ ħ/2

Question 30

Practical check for Fourier transform analysis

Answer 30

F(0) = ∫±∞ f(x) dx
The total area under f(x) is given by the value of the Fourier transform at the origin.

Question 31

Parseval’s theorem

Answer 31

∫±∞|f(x)|² dx = a ∫±∞|F(k)|² dk
where a = 1 for the mathematical (symmetric) transform
and a = 1/2π for the physics transform

Question 32

Define convolution

Answer 32

h(x) = f(x) ∗ g(x)
= ∫±∞ f(x−x′) g(x′) dx′ = ∫±∞ f(x′) g(x−x′) dx′

Question 33

Fourier transform link to convolution

Answer 33

H(k) = F(k)G(k)

Question 34

Fowards and backwards travelling waves

Answer 34

Forwards: ϕ(x, t) = exp(−i(kx−ωt))
Backwards: ϕ(x, t) = exp(i(kx+ωt))
where ω = ck
(can be found by substituting into wave equation)

Question 35

Solution to the 3-d wave equation

Answer 35

ϕ(x, t) = exp(i(k.x−ωt))
ω/|k| = c, plane wave traveling along k
Note that k.x = α is the equation of a plane perpendicular to k and whose distance from the origin is α/|k|.

Question 36

Phase velocity

Answer 36

vₚ(k) = ω(k)/k

Question 37

Group velocity

Answer 37

v₉ = dω/dk|ₖ₀ = ω’₀

Question 38

Wave equation including dispersion

Answer 38

ϕ(x, t) = 1/√2π’ ∫±∞ G(k) exp(i(kx−ω(k)t)) dk

where G(k) is the Fourier transform of ϕ(x, 0)
and ω(k) is determined after solving the wave equation for ϕ

Question 39

Shifting property of the Fourier transform

Answer 39

∫±∞ g(x−x₀) exp(−ikx) dx = exp(−ikx₀) ∫±∞ g(x) exp(−ikx) dx
= exp(−ikx₀) G(k)

Question 40

Transform of f(ax)

Answer 40

1/|a| F(k/a)

Question 41

Transform of f(x + a)

Answer 41

exp(ika) F(k)

Question 42

Transform of f(x) exp(−ixa)

Answer 42

F(k + a)

Question 43

Transform of d/dx f(x)

Answer 43

ikF(k)

Question 44

Transform of ∫(-∞tox) f(x′) dx′

Answer 44

F(k)/ik + πF(0)δ(k)

Question 45

Transform of F(x)

Answer 45

2π f(−k)

Question 46

Transform of −ixf(x)

Answer 46

d/dk F(k)

Question 47

Transform of δ(x − a)

Answer 47

exp(−ika)

Question 48

Transform of exp(iax)

Answer 48

2πδ(k − a)

Question 49

Transform of H(x)

Answer 49

πδ(k) + 1/ik

Question 50

Transform of 1/iπx

Answer 50

sgn(k)

Question 50

Transform of sgn(x)

Answer 50

2/ik

Question 51

Transform of a/2π sinc(ka/2)

Answer 51

Π(k/a)

Question 52

Transform of Π(ax)

Answer 52

1/a sinc(k/2a) = 1/a sin(k/2a)/(k/2a)

Question 53

Transform of Λ(x)

Answer 53

sinc²(k/2)

Question 54

Transform of cos(ax)

Answer 54

π[δ(k−a) + δ(k+a)]

Question 55

Transform of sin(ax)

Answer 55

iπ[δ(k+a) − δ(k−a)]

Question 56

Transform of H(x)cos(ax)

Answer 56

π/2 [δ(k−a) + δ(k+a)] + ik/a²−k²

Question 57

Transform of H(x)sin(ax)

Answer 57

π/2i [δ(k−a) − δ(k+a)] + a/a²−k²

Question 58

Transform of Π(x/a)cos(πx/a)

Answer 58

2π/a cos(ka/2)/(π/a)²−k²

Question 59

Transform of exp(−x²/2σ²)

Answer 59

√2π’ σ exp(−1/2 k²σ²)

Question 60

Transform of 1/2a exp(−a|x|)

Answer 60

(a² + k²)⁻¹

Question 61

Transform of F(−x)

Answer 61

2πf(k)

Question 62

Rectangular, triangular, and sgn functions in terms of Heaviside

Answer 62

H(x) = 1 for x≥0, 0 for x<0

Π(x) = H(x+1/2) − H(x−1/2)
= 1 for |x|<1/2, 0 otherwise

Λ(x) = Π(x) ∗ Π(x)
= 1 − |x|

sgn(x) = H(x) − H(−x)
= 1 for x > 0, −1 for x < 0

Question 63

Taylor expansion

Answer 63

f(x) = f(x₀) + f’(x₀) (x − x₀) + f’‘(x₀) (x − x₀)²/2! + …
= Σ(0to∞) 1/n! f⁽ⁿ⁾ (x₀) (x − x₀)ⁿ

Question 64

Hermite’s equation

Answer 64

d²y/dx² − 2x dy/dx + 2ny = 0
for n >0