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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
[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
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