Astroparticle Physics

382 A Mathematical Appendixexamplesfor binomial expansionstrigonometric functionsexponential functionnatural logarithmA few examples for most commonly used binomial expansions:(1 ± x) 1/2 = 1 ± 1 2 x − 1 8 x2 ± 116 x3 − 5128 x4 ±··· ,(1 ± x) −1/2 = 1 ∓ 1 2 x + 3 8 x2 ∓ 516 x3 + 35128 x4 ∓··· ,(1 ± x) −1 = 1 ∓ x + x 2 ∓ x 3 + x 4 ∓··· ,(1 ± x) 4 = 1 ± 4x + 6x 2 ± 4x 3 + x 4 finite .Trigonometric functions:sin x = x − x33! + x55! −···+(−1)n x 2n+1(2n + 1)! ±··· ,cos x = 1 − x22! + x4x2n−···+(−1)n4! (2n)! ±··· ,tan x = x + 1 3 x3 + 215 x5 + 17315 x7 +··· , |x| < π 2 ,cot x = 1 x − x 3 − x345 − 2x5 −··· , 0 < |x|