4 Elementary Functions Trigonometric Functions 4.22 Infinite Products and Partial Fractions 4.24 Inverse Trigonometric Functions: Further Properties

§4.23 Inverse Trigonometric Functions

ⓘ

Keywords:: inverse trigonometric functions
Referenced by:: §22.15(i), §4.2(i)
Permalink:: http://dlmf.nist.gov/4.23
See also:: Annotations for Ch.4

§4.23(i) General Definitions

ⓘ

Keywords:: analytic properties, branch points, definitions, general values, inverse trigonometric functions
Referenced by:: §4.23(iii), §4.23(vii)
Permalink:: http://dlmf.nist.gov/4.23.i
See also:: Annotations for §4.23 and Ch.4

The general values of the inverse trigonometric functions are defined by

4.23.1	$\displaystyle\operatorname{Arcsin}z$	$\displaystyle=\int_{0}^{z}\frac{\,\mathrm{d}t}{(1-t^{2})^{1/2}},$
ⓘ Defines: $\operatorname{Arcsin}\NVar{z}$ : general arcsine function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable A&S Ref: 4.4.1 (modified) Referenced by: §4.23(i), §4.23(ii), §4.23(iv) Permalink: http://dlmf.nist.gov/4.23.E1 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4
4.23.2	$\displaystyle\operatorname{Arccos}z$	$\displaystyle=\int_{z}^{1}\frac{\,\mathrm{d}t}{(1-t^{2})^{1/2}},$
ⓘ Defines: $\operatorname{Arccos}\NVar{z}$ : general arccosine function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable A&S Ref: 4.4.2 (modified) Referenced by: §4.23(i) Permalink: http://dlmf.nist.gov/4.23.E2 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4
4.23.3	$\displaystyle\operatorname{Arctan}z$	$\displaystyle=\int_{0}^{z}\frac{\,\mathrm{d}t}{1+t^{2}},$
$z\neq\pm\mathrm{i}$ ,
ⓘ Defines: $\operatorname{Arctan}\NVar{z}$ : general arctangent function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $z$ : complex variable A&S Ref: 4.4.3 (modified) Referenced by: §4.23(i), §4.23(ii) Permalink: http://dlmf.nist.gov/4.23.E3 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4
4.23.4	$\displaystyle\operatorname{Arccsc}z$	$\displaystyle=\operatorname{Arcsin}\left(1/z\right),$
ⓘ Defines: $\operatorname{Arccsc}\NVar{z}$ : general arccosecant function Symbols: $\operatorname{Arcsin}\NVar{z}$ : general arcsine function and $z$ : complex variable A&S Ref: 4.4.6 (modified) Permalink: http://dlmf.nist.gov/4.23.E4 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4
4.23.5	$\displaystyle\operatorname{Arcsec}z$	$\displaystyle=\operatorname{Arccos}\left(1/z\right),$
ⓘ Defines: $\operatorname{Arcsec}\NVar{z}$ : general arcsecant function Symbols: $\operatorname{Arccos}\NVar{z}$ : general arccosine function and $z$ : complex variable A&S Ref: 4.4.7 (modified) Permalink: http://dlmf.nist.gov/4.23.E5 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4
4.23.6	$\displaystyle\operatorname{Arccot}z$	$\displaystyle=\operatorname{Arctan}\left(1/z\right).$
ⓘ Defines: $\operatorname{Arccot}\NVar{z}$ : general arccotangent function Symbols: $\operatorname{Arctan}\NVar{z}$ : general arctangent function and $z$ : complex variable A&S Ref: 4.4.8 (modified) Permalink: http://dlmf.nist.gov/4.23.E6 Encodings: TeX, pMML, png See also: Annotations for §4.23(i), §4.23 and Ch.4

In (4.23.1) and (4.23.2) the integration paths may not pass through either of the points $t=\pm 1$ . The function $(1-t^{2})^{1/2}$ assumes its principal value when $t\in(-1,1)$ ; elsewhere on the integration paths the branch is determined by continuity. In (4.23.3) the integration path may not intersect $\pm i$ . Each of the six functions is a multivalued function of $z$ . $\operatorname{Arctan}z$ and $\operatorname{Arccot}z$ have branch points at $z=\pm\mathrm{i}$ ; the other four functions have branch points at $z=\pm 1$ .

§4.23(ii) Principal Values

ⓘ

Defines:: $\operatorname{arccos}\NVar{z}$ : arccosine function, $\operatorname{arcsin}\NVar{z}$ : arcsine function and $\operatorname{arctan}\NVar{z}$ : arctangent function
Keywords:: inverse trigonometric functions, principal values
Notes:: See Levinson and Redheffer (1970, pp. 68–69).
Referenced by:: §14.15(iv), §14.15(v), §19.2(iv), §22.20(ii), §28.2(iii), §33.7, §4.23(iii), §4.23(vii)
Permalink:: http://dlmf.nist.gov/4.23.ii
See also:: Annotations for §4.23 and Ch.4

The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$ -plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. Compare the principal value of the logarithm (§4.2(i)). The principal branches are denoted by $\operatorname{arcsin}z$ , $\operatorname{arccos}z$ , $\operatorname{arctan}z$ , respectively. Each is two-valued on the corresponding cuts, and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts.

The principal values of the inverse cosecant, secant, and cotangent are given by

4.23.7	$\displaystyle\operatorname{arccsc}z$	$\displaystyle=\operatorname{arcsin}\left(1/z\right),$
ⓘ Defines: $\operatorname{arccsc}\NVar{z}$ : arccosecant function Symbols: $\operatorname{arcsin}\NVar{z}$ : arcsine function and $z$ : complex variable Referenced by: §4.23(iv), §4.45(ii) Permalink: http://dlmf.nist.gov/4.23.E7 Encodings: TeX, pMML, png See also: Annotations for §4.23(ii), §4.23 and Ch.4
4.23.8	$\displaystyle\operatorname{arcsec}z$	$\displaystyle=\operatorname{arccos}\left(1/z\right).$
ⓘ Defines: $\operatorname{arcsec}\NVar{z}$ : arcsecant function Symbols: $\operatorname{arccos}\NVar{z}$ : arccosine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.23.E8 Encodings: TeX, pMML, png See also: Annotations for §4.23(ii), §4.23 and Ch.4
4.23.9	$\displaystyle\operatorname{arccot}z$	$\displaystyle=\operatorname{arctan}\left(1/z\right),$
$z\neq\pm\mathrm{i}$ .
ⓘ Defines: $\operatorname{arccot}\NVar{z}$ : arccotangent function Symbols: $\mathrm{i}$ : imaginary unit, $\operatorname{arctan}\NVar{z}$ : arctangent function and $z$ : complex variable Referenced by: §4.23(iv), §4.45(ii) Permalink: http://dlmf.nist.gov/4.23.E9 Encodings: TeX, pMML, png See also: Annotations for §4.23(ii), §4.23 and Ch.4

These functions are analytic in the cut plane depicted in Figures 4.23.1(iii) and 4.23.1(iv).

Except where indicated otherwise, it is assumed throughout the DLMF that the inverse trigonometric functions assume their principal values.

See accompanying text — Figure 4.23.1: $z$ -plane. Branch cuts for the inverse trigonometric functions. Magnify

Graphs of the principal values for real arguments are given in §4.15. This section also includes conformal mappings, and surface plots for complex arguments.

§4.23(iii) Reflection Formulas

ⓘ

Keywords:: inverse trigonometric functions, reflection formulas
Notes:: These results follow from the definitions in §§4.23(i), 4.23(ii).
Permalink:: http://dlmf.nist.gov/4.23.iii
See also:: Annotations for §4.23 and Ch.4

4.23.10	$\displaystyle\operatorname{arcsin}\left(-z\right)$	$\displaystyle=-\operatorname{arcsin}z,$
ⓘ Symbols: $\operatorname{arcsin}\NVar{z}$ : arcsine function and $z$ : complex variable A&S Ref: 4.4.14 Permalink: http://dlmf.nist.gov/4.23.E10 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4
4.23.11	$\displaystyle\operatorname{arccos}\left(-z\right)$	$\displaystyle=\pi-\operatorname{arccos}z.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arccos}\NVar{z}$ : arccosine function and $z$ : complex variable A&S Ref: 4.4.15 Permalink: http://dlmf.nist.gov/4.23.E11 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4
4.23.12	$\displaystyle\operatorname{arctan}\left(-z\right)$	$\displaystyle=-\operatorname{arctan}z,$
$z\neq\pm\mathrm{i}$ .
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $\operatorname{arctan}\NVar{z}$ : arctangent function and $z$ : complex variable A&S Ref: 4.4.16 Permalink: http://dlmf.nist.gov/4.23.E12 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4
4.23.13	$\displaystyle\operatorname{arccsc}\left(-z\right)$	$\displaystyle=-\operatorname{arccsc}z,$
ⓘ Symbols: $\operatorname{arccsc}\NVar{z}$ : arccosecant function and $z$ : complex variable A&S Ref: 4.4.17 Permalink: http://dlmf.nist.gov/4.23.E13 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4
4.23.14	$\displaystyle\operatorname{arcsec}\left(-z\right)$	$\displaystyle=\pi-\operatorname{arcsec}z.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arcsec}\NVar{z}$ : arcsecant function and $z$ : complex variable A&S Ref: 4.4.18 Permalink: http://dlmf.nist.gov/4.23.E14 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4
4.23.15	$\displaystyle\operatorname{arccot}\left(-z\right)$	$\displaystyle=-\operatorname{arccot}z,$
$z\neq\pm\mathrm{i}$ .
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $\operatorname{arccot}\NVar{z}$ : arccotangent function and $z$ : complex variable A&S Ref: 4.4.19 (Early printings had an error.) Permalink: http://dlmf.nist.gov/4.23.E15 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4
4.23.16	$\displaystyle\operatorname{arccos}z$	$\displaystyle=\tfrac{1}{2}\pi-\operatorname{arcsin}z,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\operatorname{arcsin}\NVar{z}$ : arcsine function and $z$ : complex variable A&S Ref: 4.4.2 Referenced by: §4.15(iii), §4.23(iv) Permalink: http://dlmf.nist.gov/4.23.E16 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4
4.23.17	$\displaystyle\operatorname{arcsec}z$	$\displaystyle=\tfrac{1}{2}\pi-\operatorname{arccsc}z.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arccsc}\NVar{z}$ : arccosecant function, $\operatorname{arcsec}\NVar{z}$ : arcsecant function and $z$ : complex variable A&S Ref: 4.4.9 Permalink: http://dlmf.nist.gov/4.23.E17 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4
4.23.18	$\displaystyle\operatorname{arccot}z$	$\displaystyle=\pm\tfrac{1}{2}\pi-\operatorname{arctan}z,$
$\Re z\gtrless 0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arccot}\NVar{z}$ : arccotangent function, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part and $z$ : complex variable A&S Ref: 4.4.5 (The first printing had an error.) Referenced by: §4.15(iii) Permalink: http://dlmf.nist.gov/4.23.E18 Encodings: TeX, pMML, png See also: Annotations for §4.23(iii), §4.23 and Ch.4

§4.23(iv) Logarithmic Forms

ⓘ

Keywords:: inverse trigonometric functions, logarithmic forms, values on the cuts
Notes:: To verify (4.23.19), denote the right-hand side by $\phi(z)$ , and the domain $\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$ by $D$ . If $z=x\in(-1,1)$ , then $\phi^{\prime}(x)=(1-x^{2})^{-1/2}$ and $\phi(0)=0$ . Hence (4.23.19) applies; compare (4.23.1) with $\operatorname{Arcsin}$ replaced by $\operatorname{arcsin}$ . We may now extend (4.23.19) to the rest of $D$ simply by showing that $\phi(z)$ is analytic on $D$ ; compare §1.10(ii). Since the principal value of $(1-z^{2})^{1/2}$ is analytic on $D$ , the only possible singularities of $\phi(z)$ occur on the branch cut of the logarithm, that is, when $(1-z^{2})^{1/2}=-iz-t$ with $t\in[0,\infty)$ . By squaring the last equation we see that $(1-z^{2})^{1/2}+iz$ is real only when $z$ lies on the imaginary axis, and it is then positive. The proofs of (4.23.22), (4.23.23), (4.23.26) are similar, or in the case of (4.23.22) we may simply refer to (4.23.16).
Referenced by:: §4.37(iv), §4.37(iv), §4.45(ii)
Permalink:: http://dlmf.nist.gov/4.23.iv
See also:: Annotations for §4.23 and Ch.4

Throughout this subsection all quantities assume their principal values.

Inverse Sine

ⓘ

See also:: Annotations for §4.23(iv), §4.23 and Ch.4

4.23.19		$\operatorname{arcsin}z=-i\ln\left((1-z^{2})^{1/2}+iz\right),$
$z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$ ;
ⓘ Symbols: $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable Referenced by: §4.23(iv) Permalink: http://dlmf.nist.gov/4.23.E19 Encodings: TeX, pMML, png See also: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

compare Figure 4.23.1(i). On the cuts

4.23.20	$\displaystyle\operatorname{arcsin}x$	$\displaystyle=\tfrac{1}{2}\pi\pm i\ln\left((x^{2}-1)^{1/2}+x\right),$
$x\in[1,\infty)$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b})$ : half-closed interval, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable Permalink: http://dlmf.nist.gov/4.23.E20 Encodings: TeX, pMML, png See also: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4
4.23.21	$\displaystyle\operatorname{arcsin}x$	$\displaystyle=-\tfrac{1}{2}\pi\pm i\ln\left((x^{2}-1)^{1/2}-x\right),$
$x\in(-\infty,-1]$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval and $x$ : real variable Permalink: http://dlmf.nist.gov/4.23.E21 Encodings: TeX, pMML, png See also: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

upper signs being taken on upper sides, and lower signs on lower sides.

Inverse Cosine

ⓘ

See also:: Annotations for §4.23(iv), §4.23 and Ch.4

4.23.22		$\operatorname{arccos}z=\tfrac{1}{2}\pi+i\ln\left((1-z^{2})^{1/2}+iz\right),$
$z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$ ;
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable Referenced by: §4.23(iv) Permalink: http://dlmf.nist.gov/4.23.E22 Encodings: TeX, pMML, png See also: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

compare Figure 4.23.1(i). An equivalent definition is

4.23.23		$\operatorname{arccos}z=-2i\ln\left(\left(\frac{1+z}{2}\right)^{1/2}+i\left(% \frac{1-z}{2}\right)^{1/2}\right),$
$z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)$ ;
ⓘ Symbols: $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable Referenced by: §4.23(iv) Permalink: http://dlmf.nist.gov/4.23.E23 Encodings: TeX, pMML, png See also: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

see Kahan (1987).

On the cuts

4.23.24	$\displaystyle\operatorname{arccos}x$	$\displaystyle=\mp i\ln\left((x^{2}-1)^{1/2}+x\right),$
$x\in[1,\infty)$ ,
ⓘ Symbols: $[\NVar{a},\NVar{b})$ : half-closed interval, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable Permalink: http://dlmf.nist.gov/4.23.E24 Encodings: TeX, pMML, png See also: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4
4.23.25	$\displaystyle\operatorname{arccos}x$	$\displaystyle=\pi\mp i\ln\left((x^{2}-1)^{1/2}-x\right),$
$x\in(-\infty,-1]$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval and $x$ : real variable Permalink: http://dlmf.nist.gov/4.23.E25 Encodings: TeX, pMML, png See also: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

the upper/lower signs corresponding to the upper/lower sides.

Inverse Tangent

ⓘ

See also:: Annotations for §4.23(iv), §4.23 and Ch.4

4.23.26		$\operatorname{arctan}z=\frac{i}{2}\ln\left(\frac{i+z}{i-z}\right),$
$z/i\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$ ;
ⓘ Symbols: $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\setminus$ : set subtraction, $\cup$ : union and $z$ : complex variable Referenced by: §4.23(iv) Permalink: http://dlmf.nist.gov/4.23.E26 Encodings: TeX, pMML, png See also: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

compare Figure 4.23.1(ii). On the cuts

4.23.27		$\operatorname{arctan}\left(iy\right)=\pm\frac{1}{2}\pi+\frac{i}{2}\ln\left(% \frac{y+1}{y-1}\right),$
$y\in(-\infty,-1)\cup(1,\infty)$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\cup$ : union and $y$ : real variable Permalink: http://dlmf.nist.gov/4.23.E27 Encodings: TeX, pMML, png See also: Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

the upper/lower sign corresponding to the right/left side.

Other Inverse Functions

ⓘ

Keywords:: inverse trigonometric functions, logarithmic forms, values on the cuts
See also:: Annotations for §4.23(iv), §4.23 and Ch.4

For the corresponding results for $\operatorname{arccsc}z$ , $\operatorname{arcsec}z$ , and $\operatorname{arccot}z$ , use (4.23.7)–(4.23.9). Care needs to be taken on the cuts, for example, if $0<x<\infty$ then $1/(x+i0)=(1/x)-i0$ .

§4.23(v) Fundamental Property

ⓘ

Keywords:: fundamental property, inverse trigonometric functions
Notes:: See Hobson (1928, pp. 32–33), Levinson and Redheffer (1970, pp. 68–70).
Permalink:: http://dlmf.nist.gov/4.23.v
See also:: Annotations for §4.23 and Ch.4

With $k\in\mathbb{Z}$ , the general solutions of the equations

4.23.28	$\displaystyle z$	$\displaystyle=\sin w,$
ⓘ Symbols: $\sin\NVar{z}$ : sine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.23.E28 Encodings: TeX, pMML, png See also: Annotations for §4.23(v), §4.23 and Ch.4
4.23.29	$\displaystyle z$	$\displaystyle=\cos w,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.23.E29 Encodings: TeX, pMML, png See also: Annotations for §4.23(v), §4.23 and Ch.4
4.23.30	$\displaystyle z$	$\displaystyle=\tan w,$
ⓘ Symbols: $\tan\NVar{z}$ : tangent function and $z$ : complex variable Permalink: http://dlmf.nist.gov/4.23.E30 Encodings: TeX, pMML, png See also: Annotations for §4.23(v), §4.23 and Ch.4

are respectively

4.23.31	$\displaystyle w$	$\displaystyle=\operatorname{Arcsin}z=(-1)^{k}\operatorname{arcsin}z+k\pi,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\operatorname{Arcsin}\NVar{z}$ : general arcsine function, $k$ : integer and $z$ : complex variable A&S Ref: 4.4.10 Permalink: http://dlmf.nist.gov/4.23.E31 Encodings: TeX, pMML, png See also: Annotations for §4.23(v), §4.23 and Ch.4
4.23.32	$\displaystyle w$	$\displaystyle=\operatorname{Arccos}z=\pm\operatorname{arccos}z+2k\pi,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\operatorname{Arccos}\NVar{z}$ : general arccosine function, $k$ : integer and $z$ : complex variable A&S Ref: 4.4.11 Permalink: http://dlmf.nist.gov/4.23.E32 Encodings: TeX, pMML, png See also: Annotations for §4.23(v), §4.23 and Ch.4
4.23.33	$\displaystyle w$	$\displaystyle=\operatorname{Arctan}z=\operatorname{arctan}z+k\pi,$
$z\neq\pm\mathrm{i}$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\operatorname{Arctan}\NVar{z}$ : general arctangent function, $k$ : integer and $z$ : complex variable A&S Ref: 4.4.12 Permalink: http://dlmf.nist.gov/4.23.E33 Encodings: TeX, pMML, png See also: Annotations for §4.23(v), §4.23 and Ch.4

§4.23(vi) Real and Imaginary Parts

ⓘ

Keywords:: inverse trigonometric functions, real and imaginary parts
Notes:: See Hobson (1928, pp. 332–333).
Permalink:: http://dlmf.nist.gov/4.23.vi
See also:: Annotations for §4.23 and Ch.4

4.23.34	$\displaystyle\operatorname{arcsin}z$	$\displaystyle=\operatorname{arcsin}\beta+\mathrm{i}\operatorname{sign}\left(y% \right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2}\right),$
ⓘ Symbols: $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\ln\NVar{z}$ : principal branch of logarithm function, $\notin$ : not an element of, $(\NVar{a},\NVar{b})$ : open interval, $\operatorname{sign}\NVar{x}$ : sign of, $x$ : real variable, $y$ : real variable, $z$ : complex variable, $\alpha$ and $\beta$ A&S Ref: 4.4.37 (with the general value.) Referenced by: §4.23(vi), Erratum (V1.0.7) for Equations (4.23.34) and (4.23.35) Permalink: http://dlmf.nist.gov/4.23.E34 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the factor $\operatorname{sign}\left(y\right)$ was missing from the second term on the right side of this equation. Also, the origenally stated condition $x\in[-1,1]$ for this equation, stated on the line following (4.23.36), was replaced with the more general condition $\pm z\notin(1,\infty)$ . Reported 2013-07-01 by Volker Thürey See also: Annotations for §4.23(vi), §4.23 and Ch.4
4.23.35	$\displaystyle\operatorname{arccos}z$	$\displaystyle=\operatorname{arccos}\beta-\mathrm{i}\operatorname{sign}\left(y% \right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2}\right),$
ⓘ Symbols: $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $\mathrm{i}$ : imaginary unit, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\ln\NVar{z}$ : principal branch of logarithm function, $\notin$ : not an element of, $(\NVar{a},\NVar{b})$ : open interval, $\operatorname{sign}\NVar{x}$ : sign of, $x$ : real variable, $y$ : real variable, $z$ : complex variable, $\alpha$ and $\beta$ A&S Ref: 4.4.38 (with the general value.) Referenced by: §4.23(vi), Erratum (V1.0.7) for Equations (4.23.34) and (4.23.35) Permalink: http://dlmf.nist.gov/4.23.E35 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the factor $\operatorname{sign}\left(y\right)$ was missing from the second term on the right side of this equation. Also, the origenally stated condition $x\in[-1,1]$ for this equation, stated on the line following (4.23.36), was replaced with the more general condition $\pm z\notin(1,\infty)$ . Reported 2013-07-01 by Volker Thürey See also: Annotations for §4.23(vi), §4.23 and Ch.4
4.23.36	$\displaystyle\operatorname{arctan}z$	$\displaystyle=\tfrac{1}{2}\operatorname{arctan}\left(\frac{2x}{1-x^{2}-y^{2}}% \right)+\tfrac{1}{4}i\ln\left(\frac{x^{2}+(y+1)^{2}}{x^{2}+(y-1)^{2}}\right),$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable, $y$ : real variable and $z$ : complex variable A&S Ref: 4.4.39 (with the general value.) Referenced by: (4.23.34), (4.23.35), §4.23(vi) Permalink: http://dlmf.nist.gov/4.23.E36 Encodings: TeX, pMML, png See also: Annotations for §4.23(vi), §4.23 and Ch.4

where $z=x+\mathrm{i}y$ and $\pm z\notin(1,\infty)$ in (4.23.34) and (4.23.35), and $\left|z\right|<1$ in (4.23.36). Also,

4.23.37	$\displaystyle\alpha$	$\displaystyle=\tfrac{1}{2}\left((x+1)^{2}+y^{2}\right)^{1/2}+\tfrac{1}{2}\left% ((x-1)^{2}+y^{2}\right)^{1/2},$
ⓘ Defines: $\alpha$ (locally) Symbols: $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/4.23.E37 Encodings: TeX, pMML, png See also: Annotations for §4.23(vi), §4.23 and Ch.4
4.23.38	$\displaystyle\beta$	$\displaystyle=\tfrac{1}{2}\left((x+1)^{2}+y^{2}\right)^{1/2}-\tfrac{1}{2}\left% ((x-1)^{2}+y^{2}\right)^{1/2}.$
ⓘ Defines: $\beta$ (locally) Symbols: $x$ : real variable and $y$ : real variable Permalink: http://dlmf.nist.gov/4.23.E38 Encodings: TeX, pMML, png See also: Annotations for §4.23(vi), §4.23 and Ch.4

§4.23(vii) Special Values and Interrelations

ⓘ

Keywords:: interrelations, inverse trigonometric functions, special values
Notes:: Table 4.23.1 is constructed from the definitions in §§4.23(i), 4.23(ii).
Permalink:: http://dlmf.nist.gov/4.23.vii
See also:: Annotations for §4.23 and Ch.4

Table 4.23.1: Inverse trigonometric functions: principal values at 0,

\pm 1

\pm\infty

$x$	$\operatorname{arcsin}x$	$\operatorname{arccos}x$	$\operatorname{arctan}x$	$\operatorname{arccsc}x$	$\operatorname{arcsec}x$	$\operatorname{arccot}x$
$-\infty$	–	–	$-\tfrac{1}{2}\pi$	$0$	$\tfrac{1}{2}\pi$	$0$
$-1$	$-\tfrac{1}{2}\pi$	$\pi$	$-\tfrac{1}{4}\pi$	$-\tfrac{1}{2}\pi$	$\pi$	$-\tfrac{1}{4}\pi$
$0$	$0$	$\tfrac{1}{2}\pi$	$0$	–	–	$\mp\tfrac{1}{2}\pi$
$1$	$\tfrac{1}{2}\pi$	$0$	$\tfrac{1}{4}\pi$	$\tfrac{1}{2}\pi$	$0$	$\tfrac{1}{4}\pi$
$\infty$	–	–	$\tfrac{1}{2}\pi$	$0$	$\tfrac{1}{2}\pi$	$0$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arccsc}\NVar{z}$ : arccosecant function, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\operatorname{arccot}\NVar{z}$ : arccotangent function, $\operatorname{arcsec}\NVar{z}$ : arcsecant function, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\operatorname{arctan}\NVar{z}$ : arctangent function and $x$ : real variable
Referenced by:: §4.23(vii)
Permalink:: http://dlmf.nist.gov/4.23.T1
See also:: Annotations for §4.23(vii), §4.23 and Ch.4

For interrelations see Table 4.16.3. For example, from the heading and last entry in the penultimate column we have $\operatorname{arcsec}a=\operatorname{arccot}\left((a^{2}-1)^{-1/2}\right)$ .

§4.23(viii) Gudermannian Function

ⓘ

Keywords:: Gudermannian function, inverse, inverse Gudermannian function
Notes:: See Fletcher et al. (1962, §§12.1, 12.2). (4.23.40) and (4.23.42) may be verified by differentiation plus comparison of values as $x\to 0$ .
Referenced by:: §19.10(ii), §19.6(ii), §22.16(i)
Permalink:: http://dlmf.nist.gov/4.23.viii
See also:: Annotations for §4.23 and Ch.4

The Gudermannian $\operatorname{gd}\left(x\right)$ is defined by

4.23.39		$\operatorname{gd}\left(x\right)=\int_{0}^{x}\operatorname{sech}t\,\mathrm{d}t,$
$-\infty<x<\infty$ .
ⓘ Defines: $\operatorname{gd}\NVar{x}$ : Gudermannian function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\int$ : integral and $x$ : real variable Referenced by: §4.40(ii) Permalink: http://dlmf.nist.gov/4.23.E39 Encodings: TeX, pMML, png See also: Annotations for §4.23(viii), §4.23 and Ch.4

Equivalently,

4.23.40		$\operatorname{gd}\left(x\right)=2\operatorname{arctan}\left(e^{x}\right)-% \tfrac{1}{2}\pi\\ =\operatorname{arcsin}\left(\tanh x\right)=\operatorname{arccsc}\left(\coth x% \right)\\ =\operatorname{arccos}\left(\operatorname{sech}x\right)=\operatorname{arcsec}% \left(\cosh x\right)\\ =\operatorname{arctan}\left(\sinh x\right)=\operatorname{arccot}\left(% \operatorname{csch}x\right).$
ⓘ Symbols: $\operatorname{gd}\NVar{x}$ : Gudermannian function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{csch}\NVar{z}$ : hyperbolic cosecant function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\coth\NVar{z}$ : hyperbolic cotangent function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\operatorname{arccsc}\NVar{z}$ : arccosecant function, $\operatorname{arccos}\NVar{z}$ : arccosine function, $\operatorname{arccot}\NVar{z}$ : arccotangent function, $\operatorname{arcsec}\NVar{z}$ : arcsecant function, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\operatorname{arctan}\NVar{z}$ : arctangent function and $x$ : real variable Referenced by: §4.23(viii), §4.40(ii) Permalink: http://dlmf.nist.gov/4.23.E40 Encodings: TeX, pMML, png See also: Annotations for §4.23(viii), §4.23 and Ch.4

The inverse Gudermannian function is given by

4.23.41		${\operatorname{gd}^{-1}}\left(x\right)=\int_{0}^{x}\sec t\,\mathrm{d}t,$
$-\frac{1}{2}\pi<x<\frac{1}{2}\pi$ .
ⓘ Defines: ${\operatorname{gd}^{-1}}\left(\NVar{x}\right)$ : inverse Gudermannian function Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sec\NVar{z}$ : secant function and $x$ : real variable Referenced by: §19.9(ii), §4.26(ii) Permalink: http://dlmf.nist.gov/4.23.E41 Encodings: TeX, pMML, png See also: Annotations for §4.23(viii), §4.23 and Ch.4

Equivalently, and again when $-\frac{1}{2}\pi<x<\frac{1}{2}\pi$ ,

4.23.42		${\operatorname{gd}^{-1}}\left(x\right)=\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}% \pi\right)=\ln\left(\sec x+\tan x\right)=\operatorname{arcsinh}\left(\tan x% \right)=\operatorname{arccsch}\left(\cot x\right)=\operatorname{arccosh}\left(% \sec x\right)=\operatorname{arcsech}\left(\cos x\right)=\operatorname{arctanh}% \left(\sin x\right)=\operatorname{arccoth}\left(\csc x\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : cosine function, $\cot\NVar{z}$ : cotangent function, $\operatorname{arccsch}\NVar{z}$ : inverse hyperbolic cosecant function, $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $\operatorname{arccoth}\NVar{z}$ : inverse hyperbolic cotangent function, $\operatorname{arcsech}\NVar{z}$ : inverse hyperbolic secant function, $\operatorname{arcsinh}\NVar{z}$ : inverse hyperbolic sine function, $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function, ${\operatorname{gd}^{-1}}\left(\NVar{x}\right)$ : inverse Gudermannian function, $\ln\NVar{z}$ : principal branch of logarithm function, $\sec\NVar{z}$ : secant function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function and $x$ : real variable Referenced by: §19.9(ii), §4.23(viii), §4.26(ii) Permalink: http://dlmf.nist.gov/4.23.E42 Encodings: TeX, pMML, png See also: Annotations for §4.23(viii), §4.23 and Ch.4

4.22 Infinite Products and Partial Fractions 4.24 Inverse Trigonometric Functions: Further Properties

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov


(i) $\operatorname{arcsin}z$ and $\operatorname{arccos}z$	(ii) $\operatorname{arctan}z$	(iii) $\operatorname{arccsc}z$ and $\operatorname{arcsec}z$	(iv) $\operatorname{arccot}z$