Math

Table of contents

Relations

(1)/(a)±(1)/(b) = (b±a)/(ab)

Aexp( − τ) ≈ (1 − τ)A, if τ≪1.

f(x)≃f(x₀) + (f’(x₀))/(1!)(x − x₀) + ... + (fⁿ(x₀))/(n!)(x − x0)ⁿ

tanθ ≈ θ

sinθ ≈ θ

cosθ ≈ 1 − (θ²)/(2) ≈ 1

sinα + sinβ = 2sin⎛⎝(α + β)/(2)⎞⎠cos⎛⎝(α − β)/(2)⎞⎠ = 2sin(α)/(2)cos(α)/(2) + 2sin(β)/(2)cos(β)/(2)

sinα + sinβ = 2⎛⎝cos²(β)/(2) + sin²(β)/(2)⎞⎠ × ⎛⎝sin(α)/(2)cos(α)/(2)⎞⎠ + 2⎛⎝cos²(α)/(2) + sin²(α)/(2)⎞⎠ × ⎛⎝sin(β)/(2)cos(β)/(2)⎞⎠

(x²)/(a²) + (y²)/(b²) = 1, where a > b and the focus on x axis.

x = acosθ and y = bsinθ.

The focus are ±c = ±a × e, where the eccentricity is e = √(1 − (b²)/(a²)).

Or Trammel of Archimedes is the definition of a ≡ p + q and b = q.

If q ≡ 1, then p = √(1 − e²) − 1.

R_x = ⎡⎢⎢⎢⎣ 1 0 0 0 cosα − sinα 0 sinα cosα ⎤⎥⎥⎥⎦

R_y = ⎡⎢⎢⎢⎣ cosβ 0 sinβ 0 1 0 − sinβ 0 cosβ ⎤⎥⎥⎥⎦

R_z = ⎡⎢⎢⎢⎣ cosγ − sinγ 0 sinγ cosγ 0 0 0 1 ⎤⎥⎥⎥⎦

R = R_x⋅R_y⋅R_z

In python:

R = np.dot(Rz,(np.dot(R_x,R_y)))

f(x) = ae^{− ((x − b)²)/(2c²)} + d

∫^∞_− ∞f(x) dx = ac√(2π), since ∫^∞_− ∞e^− x²dx = √(π).

The scale height H is P = P₀e^{− h ⁄ H}.