Identities, relations, and methods Flashcards by Ezra Carlin

Trig addition formulae

sin(A+B) = sinAcosB + cosAsinB

sin(A-B) = sinAcosB - cosAsinB

cos(A+B) = cosAcosB - sinAsinB

cos(A-B) = cosAcosB + sinAsinB

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Double angle formulae

sin(2x) = 2sinxcosx

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

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Fourier series (sin/cos)

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

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Orthogonality relations

∫±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

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Fourier series (complex)

f(x) = Σ±∞ cₙexp(inπx/L)

cₙ = 1/2L ∫±L f(x)exp(−inπx/L) dx

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Fourier transform

F(k) = ∫±∞ f(x) exp(−ikx) dx

f(x) = 1/2π ∫±∞ F(k) exp(ikx) dk

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Laplace equation

∇²φ = 0

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Wave equation

∇²φ = 1/c² ∂²φ/∂t²

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Schrödinger equation

−ħ²/2m ∇²ψ + Vψ = iħ ∂ψ/∂t

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

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

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Solution to
df/dx = −αf

f = Aexp(−αx)

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Solution to
d²f/dx² = −α²f

f = A cos(αx) + B sin(αx)

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Solution to
d²f/dx² = α²f

f = Aexp(αx) + Bexp(-αx)

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Trig to exponential

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

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

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Hyperbolic definitions

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

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General solution to wave equation

φ = f(x − ct) + g(x + ct)

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Method for solving wave equations

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

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sinA sinB as a sum

1/2 (cos(A−B) − cos(A+B))

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sinA cosB as a sum

1/2 (sin(A−B) + sin(A+B))

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cosA cosB as a sum

1/2 (cos(A−B) + cos(A+B))

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When is the Fourier series expansion valid

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.

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Define orthogonality

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

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Mean of a function

f̄ ≡ 1/2L ∫±L f(x) dx = a₀/2

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RMS of a function

∫±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

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Diffusion equation

∇²T = 1/α² ∂T/∂t T can refer to the temperature in a region or the density of particles as they diffuse.

Functional forms of Dirac delta

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)

Properties of Dirac delta (7)

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)

Bandwidth product relation

∆k∆x ≥ 1/2

Heisenberg uncertainty relation from bandwidth product relation

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

Practical check for Fourier transform analysis

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

Parseval's theorem

∫±∞|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

Define convolution

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

Fourier transform link to convolution

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

Fowards and backwards travelling waves

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)

Solution to the 3-d wave equation

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|.

Phase velocity

vₚ(k) = ω(k)/k

Group velocity

v₉ = dω/dk|ₖ₀ = ω'₀

Wave equation including dispersion

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

Shifting property of the Fourier transform

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

Transform of f(ax)

1/|a| F(k/a)

Transform of f(x + a)

exp(ika) F(k)

Transform of f(x) exp(−ixa)

F(k + a)

Transform of d/dx f(x)

ikF(k)

Transform of ∫(-∞tox) f(x′) dx′

F(k)/ik + πF(0)δ(k)

Transform of F(x)

2π f(−k)

Transform of −ixf(x)

d/dk F(k)

Transform of δ(x − a)

exp(−ika)

Transform of exp(iax)

2πδ(k − a)

Transform of H(x)

πδ(k) + 1/ik

Transform of 1/iπx

sgn(k)

Transform of sgn(x)

2/ik

Transform of a/2π sinc(ka/2)

Π(k/a)

Transform of Π(ax)

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

Transform of Λ(x)

sinc²(k/2)

Transform of cos(ax)

π[δ(k−a) + δ(k+a)]

Transform of sin(ax)

iπ[δ(k+a) − δ(k−a)]

Transform of H(x)cos(ax)

π/2 [δ(k−a) + δ(k+a)] + ik/a²−k²

Transform of H(x)sin(ax)

π/2i [δ(k−a) − δ(k+a)] + a/a²−k²

Transform of Π(x/a)cos(πx/a)

2π/a cos(ka/2)/(π/a)²−k²

Transform of exp(−x²/2σ²)

√2π' σ exp(−1/2 k²σ²)

Transform of 1/2a exp(−a|x|)

(a² + k²)⁻¹

Transform of F(−x)

2πf(k)

Rectangular, triangular, and sgn functions in terms of Heaviside

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

Taylor expansion

f(x) = f(x₀) + f'(x₀) (x − x₀) + f''(x₀) (x − x₀)²/2! + ... = Σ(0to∞) 1/n! f⁽ⁿ⁾ (x₀) (x − x₀)ⁿ

Hermite's equation

d²y/dx² − 2x dy/dx + 2ny = 0 for n >0