| tan x = | x | x² | x² | x² | tanh x = | x | x² | x² | x² | ||||||||
| 1 | - | 3 | - | 5 | - | 7 | - ... | 1 | + | 3 | + | 5 | + | 7 | + ... |
These results were given by Lambert, and used by him to prove that π and π² incommensurable, and also any commensurable power of e.
Gauss in his famous memoir on the hypergeometric series
| F(α, β, γ, x) = 1 + | α · β | x + | α(α + 1)β(β + 1) | x² + ... |
| 1 · γ | 1 · 2 · γ · (γ + 1) |
gave the expression for F(α, β + 1, γ + 1, x) ÷ F(α, β, γ, x) as a continued fraction, from which if we put β = 0 and write γ - 1 for γ, we get the transformation
| 1 + | α | x + | α(α + 1) | x² + | α(α + 1)(α + 2) | x³ + ... = | 1 | β1x | β2x | |||
| γ | γ(γ + 1) | γ(γ + 1)(γ + 2) | 1 | - | 1 | - | 1 | - ... |
where
| β1 = | α | , β3 = | (α + 1)γ | , ..., β2n-1 = | (α + n - 1)(γ + n - 2) | , |
| γ | (γ + 1)(γ + 2) | (γ + 2n - 3)(γ + 2n - 2) |
| β2 = | γ - α | , β4 = | 2(γ + 1 - α) | , ..., β2n = | n(γ + n - 1 - α) | . |
| γ(γ + 1) | (γ + 2)(γ + 3) | (γ + 2n - 2)(γ + 2n - 1) |
From this we may express several of the elementary series as continued fractions; thus taking α = 1, γ = 2, and putting x for -x, we have