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# Inverse Hyperbolic Functions

Calculation of inverse hyperbolic functions of given argument

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See also Hyperbolic functions

This calculator shows values of inverse hyperbolic functions of given argument

#### Inverse Hyperbolic Functions

Digits after the decimal point: 2

Areasine or inverse hyperbolic sine
$\operatorname{Arsh}x=\ln(x+\sqrt{x^2+1})$
Odd, continuously increasing function.

Areacosine or inverse hyperbolic cosine
$\operatorname{Arch}x=\ln \left( x+\sqrt{x^{2}-1} \right)$
Increasing function. Function is defined only for x greater or equal 1.

Areatangent or inverse hyperbolic tangent
$\operatorname{Arth}x=\ln\left(\frac{\sqrt{1-x^2}}{1-x}\right)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$
Odd, continuously increasing function. Function is defined only for x greater then -1 and less then +1.

Areacotangent or inverse hyperbolic cotangent
$\operatorname{Arcth}x=\ln\left(\frac{\sqrt{x^2-1}}{x-1}\right)=\frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)$
Odd, continuously decreasing function.

Areasecant or inverse hyperbolic secant
$\operatorname{Arsch}x=\pm\ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)$
Multivalued function

Areacosecant or inverse hyperbolic cosecant
$\operatorname{Arcsch}x=\left\{\begin{array}{l}\ln\left(\frac{1-\sqrt{1+x^2}}{x}\right),\quad x<0 \\ \ln\left(\frac{1+\sqrt{1+x^2}}{x}\right),\quad x>0\end{array}\right$
Odd decreasing function. Function is not defined for x = 0.

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