Identities, relations, and methods Flashcards
Trig addition formulae
sin(A+B) = sinAcosB + cosAsinB
sin(A-B) = sinAcosB - cosAsinB
cos(A+B) = cosAcosB - sinAsinB
cos(A-B) = cosAcosB + sinAsinB
Double angle formulae
sin(2x) = 2sinxcosx
cos(2x) = cos²x - sin²x
= 2cos²x - 1
= 1 - 2sin²x
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
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
Fourier series (complex)
f(x) = Σ±∞ cₙexp(inπx/L)
cₙ = 1/2L ∫±L f(x)exp(−inπx/L) dx
Fourier transform
F(k) = ∫±∞ f(x) exp(−ikx) dx
f(x) = 1/2π ∫±∞ F(k) exp(ikx) dk
Laplace equation
∇²φ = 0
Wave equation
∇²φ = 1/c² ∂²φ/∂t²
Schrödinger equation
−ħ²/2m ∇²ψ + Vψ = iħ ∂ψ/∂t
Solution to
a d²f/dx² + b df/dx + cf = 0
f = Aexp(λ₁x) + Bexp(λ₂x)
where aλᵢ² + bλᵢ + c = 0
Solution to
df/dx = −αf
f = Aexp(−αx)
Solution to
d²f/dx² = −α²f
f = A cos(αx) + B sin(αx)
Solution to
d²f/dx² = α²f
f = Aexp(αx) + Bexp(-αx)
Trig to exponential
exp(ix) = cosx + isinx
exp(-ix) = cos-x - isinx
cosx = (exp(ix) + exp(-ix)) / 2
sinx = (exp(ix) - exp(-ix)) / 2i
Hyperbolic definitions
sinhx = (exp(x) - exp(-x)) / 2
coshx = (exp(x) + exp(-x)) / 2
General solution to wave equation
φ = f(x − ct) + g(x + ct)
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.
sinA sinB as a sum
1/2 (cos(A−B) − cos(A+B))
sinA cosB as a sum
1/2 (sin(A−B) + sin(A+B))
cosA cosB as a sum
1/2 (cos(A−B) + cos(A+B))
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.
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
Mean of a function
f̄ ≡ 1/2L ∫±L f(x) dx = a₀/2
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
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)