Power Series Expansion for Real Area Hyperbolic Sine/Mistake

Source Work

1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists

$8$. Taylor Series and Power Series:

Appendix: Table $8.2$: Power Series of Important Functions

Mistake

$\sinh^{-1} x = \paren {\ln 2} x + \dfrac 1 {\paren {2 \times 2} x^2} - \dfrac {1 \times 3} {\paren {2 \times 4 \times 4} x^4} + \dfrac {1 \times 3 \times 5} {\paren {2 \times 4 \times 6 \times 6} x^6} + \cdots \quad \paren {\size x \ge 1}$

Correction

There are two mistakes here:

$(1): \quad$ The first term should read $\map \ln {2 x}$

$(2): \quad$ The formula for $x \le -1$ should be:

$-\map \ln {-2 x} - \paren {\dfrac 1 {\paren {2 \times 2} x^2} - \dfrac {1 \times 3} {\paren {2 \times 4 \times 4} x^4} + \dfrac {1 \times 3 \times 5} {\paren {2 \times 4 \times 6 \times 6} x^6} + \cdots}$

Sources

• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions