Maths
- Just a few random, useful things, e.g.,
-
- iff : "if and only if," ⇔.
- wrt : "with respect to."
i2 = − 1 |x + yi| = √(x2+y2) (note, |z1 z2| = |z1| |z2|) ei x = cos x + i sin x sinh x = 1/2 (ex − e−x) = − i sin ix cosh x = 1/2 (ex + e−x) = cos ix tanh x = sinh x / cosh x cos2 x + sin2 x = 1 1 + tan2 x = 1/cos2x cosh2 x − sinh2 x = 1 - iff : "if and only if," ⇔.
- ∫[0..∞] 1/(1+x2) dx (letting x=tanθ, so dx/dθ=1/cos2θ)
- = ∫[0..π/2]
(1/(1+tan2θ))
(1/cos2θ) dθ
= ∫[0..π/2] 1 dθ
= [θ]0..π/2
= π/2 - This leads to the Cauchy(0,1) probability distribution with pdf(x) = 1/(π(1+x2)), for x∈(−∞,∞), and the half-Cauchy(0,1) with pdf(x) = 2/(π(1+x2)), for x∈[0,∞).
- ∫[0..∞] x/(1+x2)3/2 dx (θ as above)
- = ∫[0..π/2]
(sinθ/cosθ)
(1/(1+tan2θ)3/2)
(1/cos2θ) dθ
= ∫[0..π/2] sinθ dθ
= [−cosθ]0..π/2
= 1