The most usual area functions:

• The inverse function of the hyperbolic sine (in Latin sinus hyperbolicus) is $\arsinh$ (area sini hyperbolici): $$\arsinh{x} := \ln{(x+\sqrt{x^2+1})}$$
• The inverse function of the hyperbolic cosine (in Latin cosinus hyperbolicus) is $\arcosh$ (area cosini hyperbolici): $$\arcosh{x} := \ln(x+\sqrt{x^2-1})$$ It is defined for $x \geqq 1$
• The inverse function of the hyperbolic tangent (in Latin tangens hyperbolica) is $\artanh$ (area tangentis hyperbolicae): $$\artanh{x} := \frac{1}{2}\ln \frac{1+x}{1-x}$$ It is defined for $-1 < x < 1$
• The inverse function of the hyperbolic cotangent (in Latin cotangens hyperbolica) is $\arcoth$ (area cotangentis hyperbolicae): $$\arcoth{x} := \frac{1}{2}\ln \frac{x+1}{x-1}$$ It is defined for $|x| > 1$

These four functions are denoted also by $\sinh^{-1}x$ $\cosh^{-1}x$ $\tanh^{-1}x$ and $\coth^{-1}x$

Derivatives: $$\frac{d}{dx} \arsinh x = \frac{1}{\sqrt{x^2\!+\!1}}$$ $$\frac{d}{dx} \arcosh x = \frac{1}{\sqrt{x^2\!-\!1}}$$ $$\frac{d}{dx} \artanh x = \frac{1}{1\!-\!x^2}$$ $$\frac{d}{dx} \arcoth x = \frac{1}{1\!-\!x^2}$$

The functions $\arsinh$ , and $\artanh$ , have the simple Taylor series $$\arsinh{x} = x-\frac{1}{2}\!\cdot\!\frac{x^3}{3} +\frac{1\!\cdot\!3}{2\!\cdot\!4}\!\cdot\!\frac{x^5}{5} -\frac{1\!\cdot\!3\!\cdot\!5}{2\!\cdot\!4\cdot\!6}\!\cdot\!\frac{x^7}{7} +-\cdots\quad (|x|\leqq 1),$$ $$\artanh x = x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\cdots \quad (|x| < 1).$$ Because the inverse tangent function (see the cyclometric functions) has the expansion $\arctan x = x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+-\cdots\,\, (|x|\leqq 1)$ we see that $$\artanh x = \frac{1}{i}\arctan ix;$$ similarly we get $$\arsinh x = \frac{1}{i}\arcsin ix.$$ Some other formulae which may be obtained by means of the addition formulae of the hyperbolic functions: $$\arsinh x\pm\arsinh y = \arsinh(x\sqrt{y^2\!+\!1}\pm y\sqrt{x^2\!+\!1})$$ $$\arcosh x\pm\arcosh y = \arcosh(xy\pm\sqrt{x^2\!-\!1}\sqrt{y^2\!-\!1})$$ $$\artanh x\pm\artanh y = \artanh\frac{x\pm y}{1\pm xy}$$

The classic abbreviations ``$\arsinh$ ' and ``$\arcosh$ ' are explained as follows: The unit hyperbola $x^2\!-\!y^2 = 1$ ,(its right half) has the parametric representation $$\begin{cases} x = \cosh A,\\ y = \sinh A; \end{cases}$$ here $A$ means the area bounded by the hyperbola and the straight line segments $OP$ and $OQ$ where $O$ is the origin, $P$ is the point $(x,\,y)$ , of the hyperbola and $Q$ is the point $(x,\,-y)$ , of the hyperbola. Thus, conversely, $A$ is the area having hyperbolic cosine equal to $x$ (area cosini hyperbolici x), similarly $A$ is the area having hyperbolic sine equal to $y$ (area sini hyperbolici y).

Note. In some countries the abbreviation ``ar'' in the symbols arsinh etc. is replaced by ``a'', ``Ar'', ``arc'' or ``arg''.