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The hyperbolic cosine function, defined as \(\cosh(x) = \frac{e^x + e^{-x}}{2}\).

The hyperbolic sine function, defined as \(\sinh(x) = \frac{e^x - e^{-x}}{2}\).

The hyperbolic tangent function, defined as \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\) or \(\frac{e^x - e^{-x}}{e^x + e^{-x}}\).

The hyperbolic cosecant function, defined as \(\csch(x) = \frac{1}{\sinh(x)}\) or \(\frac{2}{e^x - e^{-x}}\).

The hyperbolic cotangent function, defined as \(\coth(x) = \frac{\cosh(x)}{\sinh(x)}\) or \(\frac{e^x + e^{-x}}{e^x - e^{-x}}\).

The hyperbolic secant function, defined as \(\sech(x) = \frac{1}{\cosh(x)}\) or \(\frac{2}{e^x + e^{-x}}\).

The hyperbolic arcsine function or inverse hyperbolic sine, defined as \(\arsinh(x) = \ln(x + \sqrt{x^2 + 1})\).