§4.38 Inverse Hyperbolic Functions: Further Properties

Referenced by:: §4.2(i)
Permalink:: http://dlmf.nist.gov/4.38

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

§4.38(i) 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 $(1+z^{2})^{{-1/2}}=z^{{-1}}(1+(1/z^{2}))^{{-1/2}}$ , $\realpart{z}>0$ , and then expand and integrate. For the constant of integration note that for large $z$ , $\mathop{\mathrm{arcsinh}\/}\nolimits z$ behaves like $\mathop{\ln\/}\nolimits\!\left(2z\right)$ and the constant is $\mathop{\ln\/}\nolimits 2$ . (4.38.3) is proved similarly. (4.38.4)–(4.38.7) follow from the corresponding series for inverse trigonometric functions in §4.24.
Keywords:: inverse hyperbolic functions, power series
Permalink:: http://dlmf.nist.gov/4.38.i

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:: $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : 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

4.38.2 $\mathop{\mathrm{arcsinh}\/}\nolimits z=\mathop{\ln\/}\nolimits\!\left(2z\right)+\frac{1}{2}\frac{1}{2z^{2}}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^{4}}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^{6}}-\cdots,$ $\realpart{z}>0$ , $|z|>1$ .

Symbols:: $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : 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

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:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\ln\/}\nolimits z$ : 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

4.38.4 $\mathop{\mathrm{arccosh}\/}\nolimits z=(2(z-1))^{{1/2}}\*{\left(1+\sum _{{n=1}}^{\infty}(-1)^{n}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{{2n}}n!(2n+1)}(z-1)^{n}\right)},$ $\realpart{z}>0$ , $|z-1|\leq 2$ .

Symbols:: $!$ : $n!$ : factorial, $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\realpart{}$ : 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

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:: $\mathop{\mathrm{arctanh}\/}\nolimits z$ : 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

4.38.6 $\mathop{\mathrm{arctanh}\/}\nolimits z=\pm i\frac{\pi}{2}+\frac{1}{z}+\frac{1}{3z^{3}}+\frac{1}{5z^{5}}+\cdots,$ $\imagpart{z}\gtrless 0$ , $\left|z\right|\geq 1$ .

Symbols:: $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function, $\imagpart{}$ : imaginary part and $z$ : complex variable
A&S Ref:: 4.6.33
Permalink:: http://dlmf.nist.gov/4.38.E6
Encodings:: TeX, pMML, png

4.38.7 $\mathop{\mathrm{arctanh}\/}\nolimits z=\frac{z}{1-z^{2}}\*{\left(1+\frac{2}{3}\frac{z^{2}}{z^{2}-1}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}{z^{2}-1}\right)^{2}+\cdots\right)},$ $\realpart{(z^{2})}<\tfrac{1}{2}$ ,

Symbols:: $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §4.38(i)
Permalink:: http://dlmf.nist.gov/4.38.E7
Encodings:: TeX, pMML, png

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

§4.38(ii) Derivatives

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

In the following equations square roots have their principal values.

4.38.9 $\frac{d}{dz}\mathop{\mathrm{arcsinh}\/}\nolimits z=(1+z^{2})^{{-1/2}}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : 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

4.38.10 $\frac{d}{dz}\mathop{\mathrm{arccosh}\/}\nolimits z=\pm(z^{2}-1)^{{-1/2}},$ $\realpart{z}\gtrless 0$ .

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\realpart{}$ : 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

4.38.11 $\frac{d}{dz}\mathop{\mathrm{arctanh}\/}\nolimits z=\frac{1}{1-z^{2}}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arctanh}\/}\nolimits z$ : 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

4.38.12 $\frac{d}{dz}\mathop{\mathrm{arccsch}\/}\nolimits z=\mp\frac{1}{z(1+z^{2})^{{1/2}}},$ $\realpart{z}\gtrless 0$ .

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arccsch}\/}\nolimits z$ : inverse hyperbolic cosecant function, $\realpart{}$ : real part and $z$ : complex variable
A&S Ref:: 4.6.40
Permalink:: http://dlmf.nist.gov/4.38.E12
Encodings:: TeX, pMML, png

4.38.13 $\frac{d}{dz}\mathop{\mathrm{arcsech}\/}\nolimits z=-\frac{1}{z(1-z^{2})^{{1/2}}}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arcsech}\/}\nolimits z$ : 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

4.38.14 $\frac{d}{dz}\mathop{\mathrm{arccoth}\/}\nolimits z=\frac{1}{1-z^{2}}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arccoth}\/}\nolimits z$ : 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

§4.38(iii) Addition Formulas

Notes:: Similar analysis to that used in §4.24(iii) for inverse trigonometric functions applies here.
Keywords:: inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.38.iii

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:: $\mathop{\mathrm{Arcsinh}\/}\nolimits z$ : general inverse hyperbolic sine function
A&S Ref:: 4.6.26
Permalink:: http://dlmf.nist.gov/4.38.E15
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{Arccosh}\/}\nolimits z$ : general inverse hyperbolic cosine function
A&S Ref:: 4.6.27
Permalink:: http://dlmf.nist.gov/4.38.E16
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{Arctanh}\/}\nolimits z$ : general inverse hyperbolic tangent function
A&S Ref:: 4.6.28
Permalink:: http://dlmf.nist.gov/4.38.E17
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{Arccosh}\/}\nolimits z$ : general inverse hyperbolic cosine function and $\mathop{\mathrm{Arcsinh}\/}\nolimits z$ : general inverse hyperbolic sine function
A&S Ref:: 4.6.29
Permalink:: http://dlmf.nist.gov/4.38.E18
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{Arccoth}\/}\nolimits z$ : general inverse hyperbolic cotangent function and $\mathop{\mathrm{Arctanh}\/}\nolimits z$ : general inverse hyperbolic tangent function
A&S Ref:: 4.6.30
Permalink:: http://dlmf.nist.gov/4.38.E19
Encodings:: TeX, pMML, png

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.

§4.38 Inverse Hyperbolic Functions: Further Properties

Contents

§4.38(i) Power Series

§4.38(ii) Derivatives

§4.38(iii) Addition Formulas