|
- 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
|
|