§4.24 Inverse Trigonometric Functions: Further Properties

§4.24(i) Power Series

 4.24.1 $\mathop{\mathrm{arcsin}\/}\nolimits z=z+\frac{1}{2}\frac{z^{3}}{3}+\frac{1% \cdot 3}{2\cdot 4}\frac{z^{5}}{5}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac% {z^{7}}{7}+\cdots,$ $|z|\leq 1$. Symbols: $\mathop{\mathrm{arcsin}\/}\nolimits\NVar{z}$: arcsine function and $z$: complex variable A&S Ref: 4.4.40 (where the constraint is $|z|<1$.) Permalink: http://dlmf.nist.gov/4.24.E1 Encodings: TeX, pMML, png See also: Annotations for 4.24(i)
 4.24.2 $\mathop{\mathrm{arccos}\/}\nolimits z=(2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{% \infty}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right),$ $|1-z|\leq 2$. Symbols: $!$: factorial (as in $n!$), $\mathop{\mathrm{arccos}\/}\nolimits\NVar{z}$: arccosine function, $n$: integer and $z$: complex variable A&S Ref: 4.4.41 (which is stated differently and the constraint on $z$ is more restrictive.) Permalink: http://dlmf.nist.gov/4.24.E2 Encodings: TeX, pMML, png See also: Annotations for 4.24(i)
 4.24.3 $\mathop{\mathrm{arctan}\/}\nolimits z=z-\frac{z^{3}}{3}+\frac{z^{5}}{5}-\frac{% z^{7}}{7}+\cdots,$ $\left|z\right|\leq 1$, $z\neq\pm\mathrm{i}$. Symbols: $\mathop{\mathrm{arctan}\/}\nolimits\NVar{z}$: arctangent function and $z$: complex variable A&S Ref: 4.4.42 Referenced by: §3.10(ii), §4.45(i), §4.45(i) Permalink: http://dlmf.nist.gov/4.24.E3 Encodings: TeX, pMML, png See also: Annotations for 4.24(i)
 4.24.4 $\mathop{\mathrm{arctan}\/}\nolimits z=\pm\frac{\pi}{2}-\frac{1}{z}+\frac{1}{3z% ^{3}}-\frac{1}{5z^{5}}+\cdots,$ $\Re{z}\gtrless 0$, $|z|\geq 1$. Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{arctan}\/}\nolimits\NVar{z}$: arctangent function, $\Re{}$: real part and $z$: complex variable A&S Ref: 4.4.42 (has an error. $\frac{\pi}{2}$ should have a negative sign when $\Re{z}<0$.) Referenced by: §4.45(i) Permalink: http://dlmf.nist.gov/4.24.E4 Encodings: TeX, pMML, png See also: Annotations for 4.24(i)
 4.24.5 $\mathop{\mathrm{arctan}\/}\nolimits z=\frac{z}{z^{2}+1}\*\left(1+\frac{2}{3}% \frac{z^{2}}{1+z^{2}}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}{1+z^{2}}% \right)^{2}+\cdots\right),$ $\Re{(z^{2})}>-\tfrac{1}{2}$, Symbols: $\mathop{\mathrm{arctan}\/}\nolimits\NVar{z}$: arctangent function, $\Re{}$: real part and $z$: complex variable A&S Ref: 4.4.42 (has an error in the conditions on $z$.) Permalink: http://dlmf.nist.gov/4.24.E5 Encodings: TeX, pMML, png See also: Annotations for 4.24(i)

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

 4.24.6 $x^{2}-y^{2}=-\tfrac{1}{2}.$ Symbols: $x$: real variable and $y$: real variable Permalink: http://dlmf.nist.gov/4.24.E6 Encodings: TeX, pMML, png See also: Annotations for 4.24(i)

§4.24(ii) Derivatives

 4.24.7 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arcsin}\/}\nolimits z$ $\displaystyle=(1-z^{2})^{-1/2},$ 4.24.8 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arccos}\/}\nolimits z$ $\displaystyle=-(1-z^{2})^{-1/2},$ 4.24.9 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arctan}\/}\nolimits z$ $\displaystyle=\frac{1}{1+z^{2}}.$ 4.24.10 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arccsc}\/}\nolimits z$ $\displaystyle=\mp\frac{1}{z(z^{2}-1)^{1/2}},$ $\Re{z}\gtrless 0$. Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $\mathop{\mathrm{arccsc}\/}\nolimits\NVar{z}$: arccosecant function, $\Re{}$: real part and $z$: complex variable A&S Ref: 4.4.57 (has an error.) Referenced by: §4.24(ii) Permalink: http://dlmf.nist.gov/4.24.E10 Encodings: TeX, pMML, png See also: Annotations for 4.24(ii) 4.24.11 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arcsec}\/}\nolimits z$ $\displaystyle=\pm\frac{1}{z(z^{2}-1)^{1/2}},$ $\Re{z}\gtrless 0$. Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $\mathop{\mathrm{arcsec}\/}\nolimits\NVar{z}$: arcsecant function, $\Re{}$: real part and $z$: complex variable A&S Ref: 4.4.56 (has an error.) Referenced by: §4.24(ii) Permalink: http://dlmf.nist.gov/4.24.E11 Encodings: TeX, pMML, png See also: Annotations for 4.24(ii) 4.24.12 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{arccot}\/}\nolimits z$ $\displaystyle=-\frac{1}{1+z^{2}}.$

 4.24.13 $\mathop{\mathrm{Arcsin}\/}\nolimits u\pm\mathop{\mathrm{Arcsin}\/}\nolimits v=% \mathop{\mathrm{Arcsin}\/}\nolimits\!\left(u(1-v^{2})^{1/2}\pm v(1-u^{2})^{1/2% }\right),$ Symbols: $\mathop{\mathrm{Arcsin}\/}\nolimits\NVar{z}$: general arcsine function A&S Ref: 4.4.32 Permalink: http://dlmf.nist.gov/4.24.E13 Encodings: TeX, pMML, png See also: Annotations for 4.24(iii)
 4.24.14 $\mathop{\mathrm{Arccos}\/}\nolimits u\pm\mathop{\mathrm{Arccos}\/}\nolimits v=% \mathop{\mathrm{Arccos}\/}\nolimits\!\left(uv\mp((1-u^{2})(1-v^{2}))^{1/2}% \right),$ Symbols: $\mathop{\mathrm{Arccos}\/}\nolimits\NVar{z}$: general arccosine function A&S Ref: 4.4.33 Permalink: http://dlmf.nist.gov/4.24.E14 Encodings: TeX, pMML, png See also: Annotations for 4.24(iii)
 4.24.15 $\mathop{\mathrm{Arctan}\/}\nolimits u\pm\mathop{\mathrm{Arctan}\/}\nolimits v=% \mathop{\mathrm{Arctan}\/}\nolimits\!\left(\frac{u\pm v}{1\mp uv}\right),$ Symbols: $\mathop{\mathrm{Arctan}\/}\nolimits\NVar{z}$: general arctangent function A&S Ref: 4.4.34 Referenced by: §4.45(i) Permalink: http://dlmf.nist.gov/4.24.E15 Encodings: TeX, pMML, png See also: Annotations for 4.24(iii)
 4.24.16 $\mathop{\mathrm{Arcsin}\/}\nolimits u\pm\mathop{\mathrm{Arccos}\/}\nolimits v=% \mathop{\mathrm{Arcsin}\/}\nolimits\!\left(uv\pm((1-u^{2})(1-v^{2}))^{1/2}% \right)=\mathop{\mathrm{Arccos}\/}\nolimits\!\left(v(1-u^{2})^{1/2}\mp u(1-v^{% 2})^{1/2}\right),$ Symbols: $\mathop{\mathrm{Arccos}\/}\nolimits\NVar{z}$: general arccosine function and $\mathop{\mathrm{Arcsin}\/}\nolimits\NVar{z}$: general arcsine function A&S Ref: 4.4.35 Permalink: http://dlmf.nist.gov/4.24.E16 Encodings: TeX, pMML, png See also: Annotations for 4.24(iii)
 4.24.17 $\mathop{\mathrm{Arctan}\/}\nolimits u\pm\mathop{\mathrm{Arccot}\/}\nolimits v=% \mathop{\mathrm{Arctan}\/}\nolimits\!\left(\frac{uv\pm 1}{v\mp u}\right)=% \mathop{\mathrm{Arccot}\/}\nolimits\!\left(\frac{v\mp u}{uv\pm 1}\right).$ Symbols: $\mathop{\mathrm{Arccot}\/}\nolimits\NVar{z}$: general arccotangent function and $\mathop{\mathrm{Arctan}\/}\nolimits\NVar{z}$: general arctangent function A&S Ref: 4.4.36 Permalink: http://dlmf.nist.gov/4.24.E17 Encodings: TeX, pMML, png See also: Annotations for 4.24(iii)