§4.24 Inverse Trigonometric Functions: Further Properties

4.24.1		$$\arcsin z=z+\frac{1}{2}\frac{z^3}{3}+\frac{1\cdot 3}{2\cdot 4}\frac{z^5}{5}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{z^7}{7}+\mathrm{\cdots },$$
$|z|\le 1$.
ⓘ Symbols: $\arcsin z$: arcsine function and $z$: complex variable A&S Ref: 4.4.40 (where the constraint is .) Permalink: http://dlmf.nist.gov/4.24.E1 Encodings: TeX, pMML, png See also: Annotations for §4.24(i), §4.24 and Ch.4

4.24.2		$$\arccos z={(2(1-z))}^{1/2}\left(1+\sum _{n=1}^{\mathrm{\infty }}\frac{1\cdot 3\cdot 5\mathrm{\cdots }(2n-1)}{2^{2n}(2n+1)n!}{(1-z)}^n\right),$$
$|1-z|\le 2$.
ⓘ Symbols: $!$: factorial (as in $n!$), $\arccos z$: 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 and Ch.4

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

4.24.4		$$\arctan z=\pm \frac{\pi }{2}-\frac{1}{z}+\frac{1}{3z^3}-\frac{1}{5z^5}+\mathrm{\cdots },$$
$\mathrm{\Re }z\gtrless 0$, $|z|\ge 1$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $z$: complex variable A&S Ref: 4.4.42 (has an error. $\frac{\pi }{2}$ should have a negative sign when .) 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 and Ch.4

4.24.5		$$\arctan z=\frac{z}{z^2+1}\left(1+\frac{2}{3}\frac{z^2}{1+z^2}+\frac{2\cdot 4}{3\cdot 5}{\left(\frac{z^2}{1+z^2}\right)}^2+\mathrm{\cdots }\right),$$
$\mathrm{\Re }\left(z^2\right)>-\frac{1}{2}$,
ⓘ Symbols: $\arctan z$: arctangent function, $\mathrm{\Re }$: 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), §4.24 and Ch.4

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

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

4.24.7		${\displaystyle \frac{d}{dz}}\arcsin z$	$={(1-z^2)}^{-1/2},$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\arcsin z$: arcsine function and $z$: complex variable A&S Ref: 4.4.52 Permalink: http://dlmf.nist.gov/4.24.E7 Encodings: TeX, pMML, png See also: Annotations for §4.24(ii), §4.24 and Ch.4
4.24.8		${\displaystyle \frac{d}{dz}}\arccos z$	$=-{(1-z^2)}^{-1/2},$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\arccos z$: arccosine function and $z$: complex variable A&S Ref: 4.4.53 Permalink: http://dlmf.nist.gov/4.24.E8 Encodings: TeX, pMML, png See also: Annotations for §4.24(ii), §4.24 and Ch.4
4.24.9		${\displaystyle \frac{d}{dz}}\arctan z$	$={\displaystyle \frac{1}{1+z^2}}.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\arctan z$: arctangent function and $z$: complex variable A&S Ref: 4.4.54 Permalink: http://dlmf.nist.gov/4.24.E9 Encodings: TeX, pMML, png See also: Annotations for §4.24(ii), §4.24 and Ch.4
4.24.10		${\displaystyle \frac{d}{dz}}\mathrm{arccsc}z$	$=\mp {\displaystyle \frac{1}{z{(z^2-1)}^{1/2}}},$
$\mathrm{\Re }z\gtrless 0$.
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{arccsc}z$: arccosecant function, $\mathrm{\Re }$: 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 and Ch.4
4.24.11		${\displaystyle \frac{d}{dz}}\mathrm{arcsec}z$	$=\pm {\displaystyle \frac{1}{z{(z^2-1)}^{1/2}}},$
$\mathrm{\Re }z\gtrless 0$.
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{arcsec}z$: arcsecant function, $\mathrm{\Re }$: 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 and Ch.4
4.24.12		${\displaystyle \frac{d}{dz}}\mathrm{arccot}z$	$=-{\displaystyle \frac{1}{1+z^2}}.$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{arccot}z$: arccotangent function and $z$: complex variable A&S Ref: 4.4.55 Permalink: http://dlmf.nist.gov/4.24.E12 Encodings: TeX, pMML, png See also: Annotations for §4.24(ii), §4.24 and Ch.4

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

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

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

4.24.16		$$\begin{array}{ll}\mathrm{Arcsin}u\pm \mathrm{Arccos}v& =\mathrm{Arcsin}\left(uv\pm {((1-u^2)(1-v^2))}^{1/2}\right)\\ & =\mathrm{Arccos}\left(v{(1-u^2)}^{1/2}\mp u{(1-v^2)}^{1/2}\right),\end{array}$$
ⓘ Symbols: $\mathrm{Arccos}z$: general arccosine function and $\mathrm{Arcsin}z$: 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 and Ch.4

4.24.17		$$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$
ⓘ Symbols: $\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: 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), §4.24 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.