4 Elementary Functions Hyperbolic Functions 4.37 Inverse Hyperbolic Functions 4.39 Continued Fractions

§4.38 Inverse Hyperbolic Functions: Further Properties

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Referenced by:: §4.2(i)
Permalink:: http://dlmf.nist.gov/4.38
See also:: Annotations for Ch.4

§4.38(i) Power Series
§4.38(ii) Derivatives
§4.38(iii) Addition Formulas

§4.38(i) Power Series

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Keywords:: inverse hyperbolic functions, power series
Notes:: For (4.38.1) expand $(1+z^{2})^{-1/2}$ by the binomial theorem and integrate term by term. For (4.38.2), write $(1+z^{2})^{-1/2}=z^{-1}(1+(1/z^{2}))^{-1/2}$ , $\Re z>0$ , and then expand and integrate. For the constant of integration note that for large $z$ , $\operatorname{arcsinh}z$ behaves like $\ln\left(2z\right)$ and the constant is $\ln 2$ . (4.38.3) is proved similarly. (4.38.4)–(4.38.7) follow from the corresponding series for inverse trigonometric functions in §4.24.
Permalink:: http://dlmf.nist.gov/4.38.i
See also:: Annotations for §4.38 and Ch.4

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

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

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

4.38.4		$\operatorname{arccosh}z=(2(z-1))^{1/2}\*{\left(1+\sum_{n=1}^{\infty}(-1)^{n}% \frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}n!(2n+1)}(z-1)^{n}\right)},$
$\Re z>0$ , $\|z-1\|\leq 2$ .
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\Re$ : 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		$\operatorname{arctanh}z=z+\frac{z^{3}}{3}+\frac{z^{5}}{5}+\frac{z^{7}}{7}+\cdots,$
$\left\|z\right\|\leq 1$ , $z\neq\pm 1$ .
ⓘ Symbols: $\operatorname{arctanh}\NVar{z}$ : 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 and Ch.4

4.38.6		$\operatorname{arctanh}z=\pm\mathrm{i}\frac{\pi}{2}+\frac{1}{z}+\frac{1}{3z^{3}% }+\frac{1}{5z^{5}}+\cdots,$
$\Im z\gtrless 0$ , $\left\|z\right\|\geq 1$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function, $\Im$ : 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		$\operatorname{arctanh}z=\frac{z}{1-z^{2}}\*{\left(1+\frac{2}{3}\frac{z^{2}}{z^% {2}-1}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}{z^{2}-1}\right)^{2}+\cdots% \right)},$
$\Re\left(z^{2}\right)<\tfrac{1}{2}$ ,
ⓘ Symbols: $\operatorname{arctanh}\NVar{z}$ : 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), §4.38 and Ch.4

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 and Ch.4

§4.38(ii) Derivatives

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Keywords:: derivatives, inverse hyperbolic functions
Notes:: Take the derivatives of the logarithmic forms of inverse hyperbolic functions in §4.37(iv).
Permalink:: http://dlmf.nist.gov/4.38.ii
See also:: Annotations for §4.38 and Ch.4

In the following equations square roots have their principal values.

4.38.9	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsinh}z$	$\displaystyle=(1+z^{2})^{-1/2}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arcsinh}\NVar{z}$ : 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	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccosh}z$	$\displaystyle=\pm(z^{2}-1)^{-1/2},$
$\Re z\gtrless 0$ .
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\Re$ : 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	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arctanh}z$	$\displaystyle=\frac{1}{1-z^{2}}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arctanh}\NVar{z}$ : 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	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsch}z$	$\displaystyle=\mp\frac{1}{z(1+z^{2})^{1/2}},$
$\Re z\gtrless 0$ .
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arccsch}\NVar{z}$ : inverse hyperbolic cosecant function, $\Re$ : 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	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsech}z$	$\displaystyle=-\frac{1}{z(1-z^{2})^{1/2}}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arcsech}\NVar{z}$ : 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 and Ch.4
4.38.14	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccoth}z$	$\displaystyle=\frac{1}{1-z^{2}}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arccoth}\NVar{z}$ : 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(iii) Addition Formulas

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Keywords:: addition formulas, inverse hyperbolic functions
Notes:: Similar analysis to that used in §4.24(iii) for inverse trigonometric functions applies here.
Permalink:: http://dlmf.nist.gov/4.38.iii
See also:: Annotations for §4.38 and Ch.4

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

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

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

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

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

4.37 Inverse Hyperbolic Functions 4.39 Continued Fractions

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§4.38 Inverse Hyperbolic Functions: Further Properties

Contents

§4.38(i) Power Series

§4.38(ii) Derivatives

§4.38(iii) Addition Formulas