Hyperbolic Functions


Useful Hyperbolic functions references:

Hyperbolic Identities:

$$\cosh^{2}{u}-\sinh^{2}{u} = 1$$

$$1-\text{tanh}^{2} u = \text{sech}^{2}u$$

$$\text{coth}^{2} u-1 = \text{cosech}^{2} u$$

$$\sinh{\left(x \pm y \right)} = \sinh{x}\cosh{y} \pm \cosh{x}\sinh{y}$$

$$\cosh{\left(x \pm y \right)} = \cosh{x}\cosh{y} \pm \sinh{x}\sinh{y}$$

$$\sinh{2u} = 2 \sinh{u}\cosh{u}$$

$$\begin{aligned} \cosh{2u} &= \cosh^{2}{u} + \sinh^{2}{u} \\ &= 2 \cosh^{2}{u}-1 \\ &= 1 + 2\sinh^{2}{u} \end{aligned}$$

Relation between inverse hyperbolic functions and natural logarithms:

$$\sinh^{-1}{u} = \text{ln} \left( u + \sqrt{u^{2} +1} \right) \,\,\,\, \text{for all } u$$

$$\cosh^{-1}{u} = \text{ln} \left( u + \sqrt{u^{2} -1} \right) \,\,\,\, \text{for all } u$$

$$\text{tanh}^{-1} u = \frac{1}{2}\text{ln} \left( \frac{1+u}{1-u} \right) \,\,\,\, \text{for } |u| < 1$$

$$\text{coth}^{-1} u = \frac{1}{2} \text{ln} \left( \frac{u+1}{u-1} \right) \,\,\,\, \text{for } |u| > 1$$

$$\text{sech}^{-1} u = \text{ln} \left( \frac{1+\sqrt{1-u^{2}}}{u} \right) \,\,\,\, \text{for } 0 < u \le 1$$

$$\text{cosech}^{-1} u = \text{ln} \left( \frac{1}{u} + \frac{\sqrt{1+u^{2}}}{|u|}\right) \,\,\,\, \text{for } u \neq 0$$

Some extra things:

$$\cosh{x} = \frac{e^{x} + e^{-x}}{2}$$

$$\sinh{x} = \frac{e^{x}-e^{-x}}{2}$$

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