Hyperbolic / inverse hyperbolic functions
What Are Hyperbolic Functions?
Hyperbolic functions are defined based on exponential functions and have properties similar to trigonometric functions.
They are used in a wide range of fields such as control engineering, signal processing, machine learning and numerical analysis.
In particular, tanh is widely used as an activation function in neural networks.
What Are Inverse Hyperbolic Functions?
Inverse hyperbolic functions are the inverses of hyperbolic functions.
They include asinh, acosh and atanh, and are used in coordinate transformation, numerical computation and physics simulation.
Graphs of hyperbolic/inverse hyperbolic functions
Function List
Ladder example
Formulas
\begin{align} sinh(x)=&\frac{e^x-e^{-x}}{2} \\[1em] cosh(x)=&\frac{e^x+e^{-x}}{2} \\[1em] tanh(x)=&\frac{e^x-e^{-x}}{e^x+e^{-x}} \\[1em] sech(x)=&\frac{2}{e^x+e^{-x}} \\[1em] csch(x)=&\frac{2}{e^x-e^{-x}} \qquad (x\neq0) \\[1em] coth(x)=&\frac{e^x+e^{-x}}{e^x-e^{-x}} \qquad (x\neq0) \\[0.5em] \end{align}
\begin{align} asinh(x)=&\ln{\left( x+\sqrt{x^2+1}\right) } \\[0.5em] acosh(x)=&\ln{\left( x+\sqrt{x^2-1}\right) } \qquad (x\geq1) \\[0.5em] atanh(x)=&\frac{1}{2}\ln{\left( \frac{1+x}{1-x}\right) } \qquad (-1 \lt x \lt1) \\[0.5em] asech(x)=&\ln{\left( \frac{1}{x}+\sqrt{\left( \frac{1}{x}\right) ^2-1}\right) } \qquad (0 \lt x \leq1) \\[0.5em] acsch(x)=&\ln{\left( \frac{1}{x}+\sqrt{\frac{1}{x^2}+1}\right) } \qquad (x\neq0) \\[0.5em] acoth(x)=&\frac{1}{2}\ln{\left( \frac{x+1}{x-1}\right) } \qquad (x\gt1) \\[0.5em] \end{align}