The normal distribution:
$ python equajson.py normal
Normal distribution
1 ⎛ (x−μ)²⎞
P(x) = ―――――― exp⎜− ――――――⎟
σ√(2π) ⎝ 2σ² ⎠
where:
P = probability
x = independent variable
μ = mean
σ = standard deviation
exp = exponential function
π = 3.14159…
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$ equajson.py schro
Schrödinger equation
⎛ ℏ² ⎞
E Ψ(r) = ⎜‒ ‒‒‒ ∇² + V(r)⎟ Ψ(r)
⎝ 2μ ⎠
where:
E = energy
Ψ = wave function
r = radius
ℏ = reduced Planck constant
μ = reduced mass
∇² = Laplacian
V = potential energy function
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$ python equajson.py lapla
Laplacian in spherical coordinates
1 ∂ ⎛ ∂ƒ⎞ 1 ∂ ⎛ ∂ƒ⎞ 1 ∂²ƒ
∇²ƒ = ― ―― ⎜r ――⎟ + ―――――― ―― ⎜sinθ ――⎟ + ――――――― ―――
r ∂r ⎝ ∂r⎠ r²sinθ ∂θ ⎝ ∂θ⎠ r²sin²θ ∂φ²
where:
∇² = Laplacian
ƒ = function in spherical coordinates
r = radius
∂ = partial derivative
θ = zenith angle, spans π radian
φ = azimuthal angle, spans 2π radian
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Some approximations:
$ equajson.py approximation
Stirling's approximation
n! ≈ nⁿ e⁻ⁿ √(2πn)
where:
n = integer of interest
e = 2.71828…
π = 3.14159…
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Linear approximation
f(x) ≈ f(a) + f'(a)(x-a)
where:
x = independent variable
a = point of tangency
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Small angle approximation
x²
cos(x) ≈ 1 − ‒‒‒
2
where:
cos = cosine function
x = angle
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Binomial approximation
(1+x)ⁿ ≈ 1 + nx
where:
x = small real number (|x| ≪ 1)
n = exponent
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