Cheat Sheet

\[ \begin{aligned} \coth (1/n) &=& [n; 3n, 5n,...] \\ \tan(1) &=& [1; 1, 1, 3, 1, 5, 1, 7, 1, ...] & \\ \tan(1/n) &=& [0; n-1, 1, 3n - 2, 1, 5n -2, 1, 7n -2, 1, ...] & (n \gt 1) \\ \end{aligned} \]

Invert (insert or remove a 0) to derive expansions for $\tanh$ and $\cot$. We can compute the other trigonometric and hyperbolic trigonometric functions by solving quadratic equations involving continued fractions via the $\tan x/2$ and $\tanh x/2$ identities.

\[ \begin{aligned} \arctan z &=& \frac{z}{1 + \frac{z^2}{3 + \frac{(2 z)^2}{5 + \frac{(3 z)^2}{...}}}} \\ \tanh z &=& \frac{z}{1 + \frac{z^2}{3 + \frac{z^2}{5 + \frac{z^2}{...}}}} \\ \tan z &=& \frac{z}{1 - \frac{z^2}{3 - \frac{z^2}{5 - \frac{z^2}{...}}}} \end{aligned} \]

The inverse tangent is useful for computing other inverse trigonometric functions. Its expansion also gives an expansion for $\pi$ by setting $z = 1$. Also,

\[ \pi = 3 + \frac{1}{6 + \frac{9}{6 + \frac{25}{6 + ...}}} \]

but this converges far too slowly for practical purposes.

\[ \begin{aligned} \log(1+x) &=& \frac{x}{1 + \frac{1^2 x}{2-x +\frac{2^2 x}{3-2x + ...}}} \\ e^x &=& 1+\frac{x}{1-\frac{x}{x+2-\frac{2x}{x+3-\frac{3x}{x+4-...}}}} \end{aligned} \]

Other well-known expansions [scoured from web searches; I wish I knew how these were derived]:

\[ \begin{aligned} e &=& [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ...] & \\ \exp(1/n) &=& [1; n - 1, 1, 1, 3n - 1, 1 ,1 , 5n - 1, 1 ,1, ...] & (n \gt 1) \\ \exp(2m/n) &=& 1 + \frac{2m}{(n-m)+\frac{m^2}{3n + \frac{m^2}{5n +\frac{m^2}{7n+...}}}} \\ \exp(z) &=& 1+\frac{z}{1-\frac{z}{2+\frac{z}{3-\frac{z}{2+\frac{z}{5-\frac{z}{2+...}}}}}} \\ \end{aligned} \]