4 Elementary Functions Trigonometric Functions 4.23 Inverse Trigonometric Functions 4.25 Continued Fractions

§4.24 Inverse Trigonometric Functions: Further Properties

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Referenced by:: §4.2(i), §4.38(i)
Permalink:: http://dlmf.nist.gov/4.24
See also:: Annotations for Ch.4

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

§4.24(i) Power Series

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Keywords:: inverse trigonometric functions, power series
Notes:: See Hobson (1928, pp. 279–280, 321).
Permalink:: http://dlmf.nist.gov/4.24.i
See also:: Annotations for §4.24 and Ch.4

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

4.24.2		$\operatorname{arccos}z=(2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3% \cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right),$
$\|1-z\|\leq 2$ .
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\operatorname{arccos}\NVar{z}$ : 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 and Ch.4

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

4.24.4		$\operatorname{arctan}z=\pm\frac{\pi}{2}-\frac{1}{z}+\frac{1}{3z^{3}}-\frac{1}{% 5z^{5}}+\cdots,$
$\Re z\gtrless 0$ , $\|z\|\geq 1$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arctan}\NVar{z}$ : 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 and Ch.4

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

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 and Ch.4

§4.24(ii) Derivatives

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Keywords:: derivatives, inverse trigonometric functions
Notes:: See Levinson and Redheffer (1970, pp. 68–70). For (4.24.10) and (4.24.11) note that the principal value of $(z^{2}-1)^{1/2}$ is discontinuous on the imaginary axis, hence we switch to the other branch when crossing this axis. This accounts for the two signs.
Permalink:: http://dlmf.nist.gov/4.24.ii
See also:: Annotations for §4.24 and Ch.4

4.24.7	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsin}z$	$\displaystyle=(1-z^{2})^{-1/2},$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arcsin}\NVar{z}$ : 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	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccos}z$	$\displaystyle=-(1-z^{2})^{-1/2},$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arccos}\NVar{z}$ : 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	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arctan}z$	$\displaystyle=\frac{1}{1+z^{2}}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arctan}\NVar{z}$ : 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	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsc}z$	$\displaystyle=\mp\frac{1}{z(z^{2}-1)^{1/2}},$
$\Re z\gtrless 0$ .
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arccsc}\NVar{z}$ : 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 and Ch.4
4.24.11	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsec}z$	$\displaystyle=\pm\frac{1}{z(z^{2}-1)^{1/2}},$
$\Re z\gtrless 0$ .
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arcsec}\NVar{z}$ : 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 and Ch.4
4.24.12	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccot}z$	$\displaystyle=-\frac{1}{1+z^{2}}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{arccot}\NVar{z}$ : 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(iii) Addition Formulas

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Keywords:: addition formulas, inverse trigonometric functions
Notes:: See Hobson (1928, pp. 54–55).
Referenced by:: §4.38(iii)
Permalink:: http://dlmf.nist.gov/4.24.iii
See also:: Annotations for §4.24 and Ch.4

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

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

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

4.24.16		$\operatorname{Arcsin}u\pm\operatorname{Arccos}v=\operatorname{Arcsin}\left(uv% \pm((1-u^{2})(1-v^{2}))^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/% 2}\mp u(1-v^{2})^{1/2}\right),$
ⓘ Symbols: $\operatorname{Arccos}\NVar{z}$ : general arccosine function and $\operatorname{Arcsin}\NVar{z}$ : 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 and Ch.4

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

4.23 Inverse Trigonometric Functions 4.25 Continued Fractions

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§4.24 Inverse Trigonometric Functions: Further Properties

Contents

§4.24(i) Power Series

§4.24(ii) Derivatives

§4.24(iii) Addition Formulas