Real Area Hyperbolic Tangent of Reciprocal equals Real Area Hyperbolic Cotangent

Theorem

Everywhere that the function is defined:

$\map \artanh {\dfrac 1 x} = \arcoth x$

where $\artanh$ and $\arcoth$ denote real area hyperbolic tangent and real area hyperbolic cotangent respectively.

Proof

\(\ds \map \artanh {\dfrac 1 x}\)

\(=\)

\(\ds y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \frac 1 x\)

\(=\)

\(\ds \tanh y\)

Definition of Real Area Hyperbolic Tangent

\(\ds \leadstoandfrom \ \ \)

\(\ds \frac 1 x\)

\(=\)

\(\ds \frac {\cosh y} {\sinh y}\)

Definition 2 of Hyperbolic Tangent

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds \frac {\sinh y} {\cosh y}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds \coth y\)

Definition 2 of Hyperbolic Cotangent

\(\ds \leadstoandfrom \ \ \)

\(\ds \arcoth x\)

\(=\)

\(\ds y\)

Definition of Real Area Hyperbolic Cotangent

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.63$: Relations Between Inverse Hyperbolic Functions