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Inverse Hyperbolic Functions

Calculation of inverse hyperbolic functions of given argument

See also Hyperbolic functions

This calculator shows values of inverse hyperbolic functions of given argument

Created on PLANETCALC

Inverse Hyperbolic Functions

Digits after the decimal point: 2
Inverse Hyperbolic Functions

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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