Inverse Hyperbolic Sine is Odd Function

Theorem

Let $x \in \R$.

Then:

$\map \arsinh {-x} = -\arsinh x$

where $\arsinh$ denotes the inverse hyperbolic sine function.

Proof

\(\ds \map \arsinh {-x}\)

\(=\)

\(\ds y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds -x\)

\(=\)

\(\ds \sinh y\)

Definition 1 of Inverse Hyperbolic Sine

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds -\sinh y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds \map \sinh {-y}\)

Hyperbolic Sine Function is Odd

\(\ds \leadstoandfrom \ \ \)

\(\ds \arsinh x\)

\(=\)

\(\ds -y\)

Definition 1 of Inverse Hyperbolic Sine

\(\ds \leadstoandfrom \ \ \)

\(\ds -\arsinh x\)

\(=\)

\(\ds y\)

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.64$: Relations Between Inverse Hyperbolic Functions