Power Series Expansion for Real Area Hyperbolic Cosine/Also presented as

Power Series Expansion for Real Area Hyperbolic Cosine: Also presented as

Some sources present this result in the form:

\(\ds \cosh^{-1} x\)

\(=\)

\(\ds \map \pm {\map \ln {2 x} - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } } }\)

\(\ds \begin {cases} \text {$+$ if $x \ge 1$} \\ \text {$-$ if $x \le -1$} \end {cases}\)

\(\ds \)

\(=\)

\(\ds \map \pm {\map \ln {2 x} - \paren {\dfrac 1 {2 \times 2 x^2} + \dfrac {1 \times 3} {2 \times 4 \times 4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6 x^6} + \cdots} }\)

\(\ds \begin {cases} \text {$+$ if $x \ge 1$} \\ \text {$-$ if $x \le -1$} \end {cases}\)

This takes into account the interpretation that $\cosh^{-1} x$ is a multifunction arising from the fact that $\cosh x = \map \cosh {-1}$ for $\size x \ge 1$.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Series for Hyperbolic Functions: $20.40$