# §4.38 Inverse Hyperbolic Functions: Further Properties

## §4.38(i) Power Series

 4.38.1 $\mathop{\mathrm{arcsinh}\/}\nolimits z=z-\frac{1}{2}\frac{z^{3}}{3}+\frac{1% \cdot 3}{2\cdot 4}\frac{z^{5}}{5}-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac% {z^{7}}{7}+\cdots,$ $|z|<1$. Symbols: $\mathop{\mathrm{arcsinh}\/}\nolimits\NVar{z}$: inverse hyperbolic sine function and $z$: complex variable A&S Ref: 4.6.31 Referenced by: §4.38(i) Permalink: http://dlmf.nist.gov/4.38.E1 Encodings: TeX, pMML, png See also: Annotations for 4.38(i)
 4.38.2 $\mathop{\mathrm{arcsinh}\/}\nolimits z=\mathop{\ln\/}\nolimits\!\left(2z\right% )+\frac{1}{2}\frac{1}{2z^{2}}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^{4}}+\frac{% 1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^{6}}-\cdots,$ $\Re{z}>0$, $|z|>1$. Symbols: $\mathop{\mathrm{arcsinh}\/}\nolimits\NVar{z}$: inverse hyperbolic sine function, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\Re{}$: real part and $z$: complex variable A&S Ref: 4.6.31 (misses a condition on $z$.) Referenced by: §4.38(i) Permalink: http://dlmf.nist.gov/4.38.E2 Encodings: TeX, pMML, png See also: Annotations for 4.38(i)
 4.38.3 $\mathop{\mathrm{arccosh}\/}\nolimits z=\mathop{\ln\/}\nolimits\!\left(2z\right% )-\frac{1}{2}\frac{1}{2z^{2}}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^{4}}-\frac{% 1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^{6}}-\cdots,$ $|z|>1$. Symbols: $\mathop{\mathrm{arccosh}\/}\nolimits\NVar{z}$: inverse hyperbolic cosine function, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 4.6.32 Referenced by: §4.38(i) Permalink: http://dlmf.nist.gov/4.38.E3 Encodings: TeX, pMML, png See also: Annotations for 4.38(i)
 4.38.4 $\mathop{\mathrm{arccosh}\/}\nolimits z=(2(z-1))^{1/2}\*{\left(1+\sum_{n=1}^{% \infty}(-1)^{n}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}n!(2n+1)}(z-1)^{n}% \right)},$ $\Re{z}>0$, $|z-1|\leq 2$.
 4.38.5 $\mathop{\mathrm{arctanh}\/}\nolimits z=z+\frac{z^{3}}{3}+\frac{z^{5}}{5}+\frac% {z^{7}}{7}+\cdots,$ $\left|z\right|\leq 1$, $z\neq\pm 1$. Symbols: $\mathop{\mathrm{arctanh}\/}\nolimits\NVar{z}$: inverse hyperbolic tangent function and $z$: complex variable A&S Ref: 4.6.33 Permalink: http://dlmf.nist.gov/4.38.E5 Encodings: TeX, pMML, png See also: Annotations for 4.38(i)
 4.38.6 $\mathop{\mathrm{arctanh}\/}\nolimits z=\pm\mathrm{i}\frac{\pi}{2}+\frac{1}{z}+% \frac{1}{3z^{3}}+\frac{1}{5z^{5}}+\cdots,$ $\Im{z}\gtrless 0$, $\left|z\right|\geq 1$. Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{arctanh}\/}\nolimits\NVar{z}$: inverse hyperbolic tangent function, $\Im{}$: imaginary part and $z$: complex variable A&S Ref: 4.6.33 Permalink: http://dlmf.nist.gov/4.38.E6 Encodings: TeX, pMML, png See also: Annotations for 4.38(i)
 4.38.7 $\mathop{\mathrm{arctanh}\/}\nolimits z=\frac{z}{1-z^{2}}\*{\left(1+\frac{2}{3}% \frac{z^{2}}{z^{2}-1}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}{z^{2}-1}% \right)^{2}+\cdots\right)},$ $\Re{(z^{2})}<\tfrac{1}{2}$, Symbols: $\mathop{\mathrm{arctanh}\/}\nolimits\NVar{z}$: inverse hyperbolic tangent function, $\Re{}$: real part and $z$: complex variable Referenced by: §4.38(i) Permalink: http://dlmf.nist.gov/4.38.E7 Encodings: TeX, pMML, png See also: Annotations for 4.38(i)

which requires $z$ $(=x+iy)$ to lie between the two rectangular hyperbolas given by

 4.38.8 $x^{2}-y^{2}=\tfrac{1}{2}.$ Symbols: $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/4.38.E8 Encodings: TeX, pMML, png See also: Annotations for 4.38(i)

## §4.38(ii) Derivatives

In the following equations square roots have their principal values.

 4.38.9 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arcsinh}\/}\nolimits z$ $\displaystyle=(1+z^{2})^{-1/2}.$ 4.38.10 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arccosh}\/}\nolimits z$ $\displaystyle=\pm(z^{2}-1)^{-1/2},$ $\Re{z}\gtrless 0$. 4.38.11 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arctanh}\/}\nolimits z$ $\displaystyle=\frac{1}{1-z^{2}}.$ 4.38.12 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arccsch}\/}\nolimits z$ $\displaystyle=\mp\frac{1}{z(1+z^{2})^{1/2}},$ $\Re{z}\gtrless 0$. 4.38.13 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arcsech}\/}\nolimits z$ $\displaystyle=-\frac{1}{z(1-z^{2})^{1/2}}.$ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $\mathop{\mathrm{arcsech}\/}\nolimits\NVar{z}$: inverse hyperbolic secant function and $z$: complex variable A&S Ref: 4.6.41 (has an error.) Permalink: http://dlmf.nist.gov/4.38.E13 Encodings: TeX, pMML, png See also: Annotations for 4.38(ii) 4.38.14 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arccoth}\/}\nolimits z$ $\displaystyle=\frac{1}{1-z^{2}}.$

 4.38.15 $\mathop{\mathrm{Arcsinh}\/}\nolimits u\pm\mathop{\mathrm{Arcsinh}\/}\nolimits v% =\mathop{\mathrm{Arcsinh}\/}\nolimits\!\left(u(1+v^{2})^{1/2}\pm v(1+u^{2})^{1% /2}\right),$ Symbols: $\mathop{\mathrm{Arcsinh}\/}\nolimits\NVar{z}$: general inverse hyperbolic sine function A&S Ref: 4.6.26 Permalink: http://dlmf.nist.gov/4.38.E15 Encodings: TeX, pMML, png See also: Annotations for 4.38(iii)
 4.38.16 $\mathop{\mathrm{Arccosh}\/}\nolimits u\pm\mathop{\mathrm{Arccosh}\/}\nolimits v% =\mathop{\mathrm{Arccosh}\/}\nolimits\!\left(uv\pm((u^{2}-1)(v^{2}-1))^{1/2}% \right),$ Symbols: $\mathop{\mathrm{Arccosh}\/}\nolimits\NVar{z}$: general inverse hyperbolic cosine function A&S Ref: 4.6.27 Permalink: http://dlmf.nist.gov/4.38.E16 Encodings: TeX, pMML, png See also: Annotations for 4.38(iii)
 4.38.17 $\mathop{\mathrm{Arctanh}\/}\nolimits u\pm\mathop{\mathrm{Arctanh}\/}\nolimits v% =\mathop{\mathrm{Arctanh}\/}\nolimits\!\left(\frac{u\pm v}{1\pm uv}\right),$ Symbols: $\mathop{\mathrm{Arctanh}\/}\nolimits\NVar{z}$: general inverse hyperbolic tangent function A&S Ref: 4.6.28 Permalink: http://dlmf.nist.gov/4.38.E17 Encodings: TeX, pMML, png See also: Annotations for 4.38(iii)
 4.38.18 $\mathop{\mathrm{Arcsinh}\/}\nolimits u\pm\mathop{\mathrm{Arccosh}\/}\nolimits v% =\mathop{\mathrm{Arcsinh}\/}\nolimits\!\left(uv\pm((1+u^{2})(v^{2}-1))^{1/2}% \right)=\mathop{\mathrm{Arccosh}\/}\nolimits\!\left(v(1+u^{2})^{1/2}\pm u(v^{2% }-1)^{1/2}\right),$ Symbols: $\mathop{\mathrm{Arccosh}\/}\nolimits\NVar{z}$: general inverse hyperbolic cosine function and $\mathop{\mathrm{Arcsinh}\/}\nolimits\NVar{z}$: general inverse hyperbolic sine function A&S Ref: 4.6.29 Permalink: http://dlmf.nist.gov/4.38.E18 Encodings: TeX, pMML, png See also: Annotations for 4.38(iii)
 4.38.19 $\mathop{\mathrm{Arctanh}\/}\nolimits u\pm\mathop{\mathrm{Arccoth}\/}\nolimits v% =\mathop{\mathrm{Arctanh}\/}\nolimits\!\left(\frac{uv\pm 1}{v\pm u}\right)=% \mathop{\mathrm{Arccoth}\/}\nolimits\!\left(\frac{v\pm u}{uv\pm 1}\right).$ Symbols: $\mathop{\mathrm{Arccoth}\/}\nolimits\NVar{z}$: general inverse hyperbolic cotangent function and $\mathop{\mathrm{Arctanh}\/}\nolimits\NVar{z}$: general inverse hyperbolic tangent function A&S Ref: 4.6.30 Permalink: http://dlmf.nist.gov/4.38.E19 Encodings: TeX, pMML, png See also: Annotations for 4.38(iii)