Inverse Hyperbolic Function - Logarithmic Representation

Logarithmic Representation

The operators are defined in the complex plane by:

\begin{align} \operatorname{arsinh}\, z &= \ln(z + \sqrt{z^2 + 1} \,) \\ \operatorname{arcosh}\, z &= \ln(z + \sqrt{z+1} \sqrt{z-1} \,) \\ \operatorname{artanh}\, z &= \tfrac12\ln\left(\frac{1+z}{1-z}\right) \\ \operatorname{arcoth}\, z &= \tfrac12\ln\left(\frac{z+1}{z-1}\right) \\ \operatorname{arcsch}\, z &= \ln\left( \frac{1}{z} + \sqrt{ \frac{1}{z^2} +1 } \,\right) \\ \operatorname{arsech}\, z &= \ln\left( \frac{1}{z} + \sqrt{ \frac{1}{z} + 1 } \, \sqrt{ \frac{1}{z} -1 } \,\right) \end{align}

The above square roots are principal square roots, and the logarithm function is the complex logarithm. For real arguments, i.e., z = x, which return real values, certain simplifications can be made e.g., which are not generally true when using principal square roots.

Inverse hyperbolic functions in the complex z-plane: the colour at each point in the plane represents the complex value of the respective function at that point

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