§4.38 Inverse Hyperbolic Functions: Further Properties

4.38.1		$$\mathrm{arcsinh}z=z-\frac{1}{2}\frac{z^3}{3}+\frac{1\cdot 3}{2\cdot 4}\frac{z^5}{5}-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{z^7}{7}+\mathrm{\cdots },$$
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ⓘ Symbols: $\mathrm{arcsinh}z$: 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 and Ch.4

4.38.2		$$\mathrm{arcsinh}z=\ln \left(2z\right)+\frac{1}{2}\frac{1}{2z^2}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^4}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^6}-\mathrm{\cdots },$$
$\mathrm{\Re }z>0$, $|z|>1$.
ⓘ Symbols: $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: 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 and Ch.4

4.38.3		$$\mathrm{arccosh}z=\ln \left(2z\right)-\frac{1}{2}\frac{1}{2z^2}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^4}-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^6}-\mathrm{\cdots },$$
$|z|>1$.
ⓘ Symbols: $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: 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 and Ch.4

4.38.4		$$\mathrm{arccosh}z={(2(z-1))}^{1/2}\left(1+\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{1\cdot 3\cdot 5\mathrm{\cdots }(2n-1)}{2^{2n}n!(2n+1)}{(z-1)}^n\right),$$
$\mathrm{\Re }z>0$, $|z-1|\le 2$.
ⓘ Symbols: $!$: factorial (as in $n!$), $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Re }$: real part, $n$: integer and $z$: complex variable Referenced by: §4.38(i) Permalink: http://dlmf.nist.gov/4.38.E4 Encodings: TeX, pMML, png See also: Annotations for §4.38(i), §4.38 and Ch.4

4.38.5		$$\mathrm{arctanh}z=z+\frac{z^3}{3}+\frac{z^5}{5}+\frac{z^7}{7}+\mathrm{\cdots },$$
$\left|z\right|\le 1$, $z\ne \pm 1$.
ⓘ Symbols: $\mathrm{arctanh}z$: 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 and Ch.4

4.38.6		$$\mathrm{arctanh}z=\pm \mathrm{i}\frac{\pi }{2}+\frac{1}{z}+\frac{1}{3z^3}+\frac{1}{5z^5}+\mathrm{\cdots },$$
$\mathrm{\Im }z\gtrless 0$, $\left|z\right|\ge 1$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit 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 and Ch.4

4.38.7		$$\mathrm{arctanh}z=\frac{z}{1-z^2}\left(1+\frac{2}{3}\frac{z^2}{z^2-1}+\frac{2\cdot 4}{3\cdot 5}{\left(\frac{z^2}{z^2-1}\right)}^2+\mathrm{\cdots }\right),$$
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ⓘ Symbols: $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\mathrm{\Re }$: 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), §4.38 and Ch.4

which requires $z$ $(=x+\mathrm{i}y)$ to lie between the two rectangular hyperbolas given by

4.38.8		$$x^2-y^2=\frac{1}{2}.$$
ⓘ 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 and Ch.4

In the following equations square roots have their principal values.

4.38.9		${\displaystyle \frac{d}{dz}}\mathrm{arcsinh}z$	$={(1+z^2)}^{-1/2}.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable A&S Ref: 4.6.37 Permalink: http://dlmf.nist.gov/4.38.E9 Encodings: TeX, pMML, png See also: Annotations for §4.38(ii), §4.38 and Ch.4
4.38.10		${\displaystyle \frac{d}{dz}}\mathrm{arccosh}z$	$=\pm {(z^2-1)}^{-1/2},$
$\mathrm{\Re }z\gtrless 0$.
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Re }$: real part and $z$: complex variable A&S Ref: 4.6.38 (has an error.) Permalink: http://dlmf.nist.gov/4.38.E10 Encodings: TeX, pMML, png See also: Annotations for §4.38(ii), §4.38 and Ch.4
4.38.11		${\displaystyle \frac{d}{dz}}\mathrm{arctanh}z$	$={\displaystyle \frac{1}{1-z^2}}.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable A&S Ref: 4.6.39 Permalink: http://dlmf.nist.gov/4.38.E11 Encodings: TeX, pMML, png See also: Annotations for §4.38(ii), §4.38 and Ch.4
4.38.12		${\displaystyle \frac{d}{dz}}\mathrm{arccsch}z$	$=\mp {\displaystyle \frac{1}{z{(1+z^2)}^{1/2}}},$
$\mathrm{\Re }z\gtrless 0$.
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{\Re }$: real part and $z$: complex variable A&S Ref: 4.6.40 Permalink: http://dlmf.nist.gov/4.38.E12 Encodings: TeX, pMML, png See also: Annotations for §4.38(ii), §4.38 and Ch.4
4.38.13		${\displaystyle \frac{d}{dz}}\mathrm{arcsech}z$	$=-{\displaystyle \frac{1}{z{(1-z^2)}^{1/2}}}.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{arcsech}z$: 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 and Ch.4
4.38.14		${\displaystyle \frac{d}{dz}}\mathrm{arccoth}z$	$={\displaystyle \frac{1}{1-z^2}}.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{arccoth}z$: inverse hyperbolic cotangent function and $z$: complex variable A&S Ref: 4.6.42 Permalink: http://dlmf.nist.gov/4.38.E14 Encodings: TeX, pMML, png See also: Annotations for §4.38(ii), §4.38 and Ch.4

4.38.15		$$\mathrm{Arcsinh}u\pm \mathrm{Arcsinh}v=\mathrm{Arcsinh}\left(u{(1+v^2)}^{1/2}\pm v{(1+u^2)}^{1/2}\right),$$
ⓘ Symbols: $\mathrm{Arcsinh}z$: 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 and Ch.4

4.38.16		$$\mathrm{Arccosh}u\pm \mathrm{Arccosh}v=\mathrm{Arccosh}\left(uv\pm {((u^2-1)(v^2-1))}^{1/2}\right),$$
ⓘ Symbols: $\mathrm{Arccosh}z$: 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 and Ch.4

4.38.17		$$\mathrm{Arctanh}u\pm \mathrm{Arctanh}v=\mathrm{Arctanh}\left(\frac{u\pm v}{1\pm uv}\right),$$
ⓘ Symbols: $\mathrm{Arctanh}z$: 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 and Ch.4

4.38.18		$$\begin{array}{l}\mathrm{Arcsinh}u\pm \mathrm{Arccosh}v=\mathrm{Arcsinh}\left(uv\pm {((1+u^2)(v^2-1))}^{1/2}\right)\\ =\mathrm{Arccosh}\left(v{(1+u^2)}^{1/2}\pm u{(v^2-1)}^{1/2}\right),\end{array}$$
ⓘ Symbols: $\mathrm{Arccosh}z$: general inverse hyperbolic cosine function and $\mathrm{Arcsinh}z$: 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 and Ch.4

4.38.19		$$\mathrm{Arctanh}u\pm \mathrm{Arccoth}v=\mathrm{Arctanh}\left(\frac{uv\pm 1}{v\pm u}\right)=\mathrm{Arccoth}\left(\frac{v\pm u}{uv\pm 1}\right).$$
ⓘ Symbols: $\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function and $\mathrm{Arctanh}z$: 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), §4.38 and Ch.4

The above equations are interpreted in the sense that every value of the left-hand side is a value of the right-hand side and vice-versa. All square roots have either possible value.