§4.23 Inverse Trigonometric Functions

Keywords:: inverse trigonometric functions
Referenced by:: §22.15(i), §4.2(i)
Permalink:: http://dlmf.nist.gov/4.23

§4.23(i) General Definitions
§4.23(ii) Principal Values
§4.23(iii) Reflection Formulas
§4.23(iv) Logarithmic Forms
§4.23(v) Fundamental Property
§4.23(vi) Real and Imaginary Parts
§4.23(vii) Special Values and Interrelations
§4.23(viii) Gudermannian Function

§4.23(i) General Definitions

Keywords:: inverse trigonometric functions
Referenced by:: §4.23(iii), §4.23(vii)
Permalink:: http://dlmf.nist.gov/4.23.i

The general values of the inverse trigonometric functions are defined by

4.23.1 $\mathop{\mathrm{Arcsin}\/}\nolimits z=\int _{0}^{z}\frac{dt}{(1-t^{2})^{{1/2}}},$

Defines:: $\mathop{\mathrm{Arcsin}\/}\nolimits z$ : general arcsine function
Symbols:: $dx$ : 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

4.23.2 $\mathop{\mathrm{Arccos}\/}\nolimits z=\int _{z}^{1}\frac{dt}{(1-t^{2})^{{1/2}}},$

Defines:: $\mathop{\mathrm{Arccos}\/}\nolimits z$ : general arccosine function
Symbols:: $dx$ : 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

4.23.3 $\mathop{\mathrm{Arctan}\/}\nolimits z=\int _{0}^{z}\frac{dt}{1+t^{2}},$ $z\neq\pm i$ ,

Defines:: $\mathop{\mathrm{Arctan}\/}\nolimits z$ : general arctangent function
Symbols:: $dx$ : differential of $x$ , $\int$ : 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

4.23.4 $\mathop{\mathrm{Arccsc}\/}\nolimits z=\mathop{\mathrm{Arcsin}\/}\nolimits\!\left(1/z\right),$

Defines:: $\mathop{\mathrm{Arccsc}\/}\nolimits z$ : general arccosecant function
Symbols:: $\mathop{\mathrm{Arcsin}\/}\nolimits z$ : 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

4.23.5 $\mathop{\mathrm{Arcsec}\/}\nolimits z=\mathop{\mathrm{Arccos}\/}\nolimits\!\left(1/z\right),$

Defines:: $\mathop{\mathrm{Arcsec}\/}\nolimits z$ : general arcsecant function
Symbols:: $\mathop{\mathrm{Arccos}\/}\nolimits z$ : 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

4.23.6 $\mathop{\mathrm{Arccot}\/}\nolimits z=\mathop{\mathrm{Arctan}\/}\nolimits\!\left(1/z\right).$

Defines:: $\mathop{\mathrm{Arccot}\/}\nolimits z$ : general arccotangent function
Symbols:: $\mathop{\mathrm{Arctan}\/}\nolimits z$ : 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

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 $z$ . $\mathop{\mathrm{Arctan}\/}\nolimits z$ and $\mathop{\mathrm{Arccot}\/}\nolimits z$ have branch points at $z=\pm i$ ; the other four functions have branch points at $z=\pm 1$ .

§4.23(ii) Principal Values

Notes:: See Levinson and Redheffer (1970, pp. 68–69).
Defines:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function and $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function
Keywords:: inverse trigonometric functions, principal values
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

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 $\mathop{\mathrm{arcsin}\/}\nolimits z$ , $\mathop{\mathrm{arccos}\/}\nolimits z$ , $\mathop{\mathrm{arctan}\/}\nolimits z$ , 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 $\mathop{\mathrm{arccsc}\/}\nolimits z=\mathop{\mathrm{arcsin}\/}\nolimits\!\left(1/z\right),$

Defines:: $\mathop{\mathrm{arccsc}\/}\nolimits z$ : arccosecant function
Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : 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

4.23.8 $\mathop{\mathrm{arcsec}\/}\nolimits z=\mathop{\mathrm{arccos}\/}\nolimits\!\left(1/z\right).$

Defines:: $\mathop{\mathrm{arcsec}\/}\nolimits z$ : arcsecant function
Symbols:: $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.23.E8
Encodings:: TeX, pMML, png

4.23.9 $\mathop{\mathrm{arccot}\/}\nolimits z=\mathop{\mathrm{arctan}\/}\nolimits\!\left(1/z\right),$ $z\neq\pm i$ .

Defines:: $\mathop{\mathrm{arccot}\/}\nolimits z$ : arccotangent function
Symbols:: $\mathop{\mathrm{arctan}\/}\nolimits z$ : 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

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.


(i) $\mathop{\mathrm{arcsin}\/}\nolimits z$ and $\mathop{\mathrm{arccos}\/}\nolimits z$	(ii) $\mathop{\mathrm{arctan}\/}\nolimits z$	(iii) $\mathop{\mathrm{arccsc}\/}\nolimits z$ and $\mathop{\mathrm{arcsec}\/}\nolimits z$	(iv) $\mathop{\mathrm{arccot}\/}\nolimits z$

Figure 4.23.1: $z$ -plane. Branch cuts for the inverse trigonometric functions.

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

Notes:: These results follow from the definitions in §§4.23(i), 4.23(ii).
Keywords:: inverse trigonometric functions
Permalink:: http://dlmf.nist.gov/4.23.iii

4.23.10 $\mathop{\mathrm{arcsin}\/}\nolimits\!\left(-z\right)=-\mathop{\mathrm{arcsin}\/}\nolimits z,$

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function and $z$ : complex variable
A&S Ref:: 4.4.14
Permalink:: http://dlmf.nist.gov/4.23.E10
Encodings:: TeX, pMML, png

4.23.11 $\mathop{\mathrm{arccos}\/}\nolimits\!\left(-z\right)=\pi-\mathop{\mathrm{arccos}\/}\nolimits z.$

Symbols:: $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function and $z$ : complex variable
A&S Ref:: 4.4.15
Permalink:: http://dlmf.nist.gov/4.23.E11
Encodings:: TeX, pMML, png

4.23.12 $\mathop{\mathrm{arctan}\/}\nolimits\!\left(-z\right)=-\mathop{\mathrm{arctan}\/}\nolimits z,$ $z\neq\pm i$ .

Symbols:: $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function and $z$ : complex variable
A&S Ref:: 4.4.16
Permalink:: http://dlmf.nist.gov/4.23.E12
Encodings:: TeX, pMML, png

4.23.13 $\mathop{\mathrm{arccsc}\/}\nolimits\!\left(-z\right)=-\mathop{\mathrm{arccsc}\/}\nolimits z,$

Symbols:: $\mathop{\mathrm{arccsc}\/}\nolimits z$ : arccosecant function and $z$ : complex variable
A&S Ref:: 4.4.17
Permalink:: http://dlmf.nist.gov/4.23.E13
Encodings:: TeX, pMML, png

4.23.14 $\mathop{\mathrm{arcsec}\/}\nolimits\!\left(-z\right)=\pi-\mathop{\mathrm{arcsec}\/}\nolimits z.$

Symbols:: $\mathop{\mathrm{arcsec}\/}\nolimits z$ : arcsecant function and $z$ : complex variable
A&S Ref:: 4.4.18
Permalink:: http://dlmf.nist.gov/4.23.E14
Encodings:: TeX, pMML, png

4.23.15 $\mathop{\mathrm{arccot}\/}\nolimits\!\left(-z\right)=-\mathop{\mathrm{arccot}\/}\nolimits z,$ $z\neq\pm i$ .

Symbols:: $\mathop{\mathrm{arccot}\/}\nolimits z$ : 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

4.23.16 $\mathop{\mathrm{arccos}\/}\nolimits z=\tfrac{1}{2}\pi-\mathop{\mathrm{arcsin}\/}\nolimits z,$

Symbols:: $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : 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

4.23.17 $\mathop{\mathrm{arcsec}\/}\nolimits z=\tfrac{1}{2}\pi-\mathop{\mathrm{arccsc}\/}\nolimits z.$

Symbols:: $\mathop{\mathrm{arccsc}\/}\nolimits z$ : arccosecant function, $\mathop{\mathrm{arcsec}\/}\nolimits z$ : arcsecant function and $z$ : complex variable
A&S Ref:: 4.4.9
Permalink:: http://dlmf.nist.gov/4.23.E17
Encodings:: TeX, pMML, png

4.23.18 $\mathop{\mathrm{arccot}\/}\nolimits z=\pm\tfrac{1}{2}\pi-\mathop{\mathrm{arctan}\/}\nolimits z,$ $\realpart{z}\gtrless 0$ .

Symbols:: $\mathop{\mathrm{arccot}\/}\nolimits z$ : arccotangent function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\realpart{}$ : 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

§4.23(iv) Logarithmic Forms

Notes:: To verify (4.23.19), denote the right-hand side by $\phi(z)$ , and the domain $\Complex\setminus(-\infty,-1]\cup[1,\infty)$ 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 $\mathop{\mathrm{Arcsin}\/}\nolimits$ replaced by $\mathop{\mathrm{arcsin}\/}\nolimits$ . 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).
Keywords:: inverse trigonometric functions
Referenced by:: ¶ ‣ §4.37(iv), §4.37(iv), §4.45(ii)
Permalink:: http://dlmf.nist.gov/4.23.iv

Throughout this subsection all quantities assume their principal values.

¶ Inverse Sine

4.23.19 $\mathop{\mathrm{arcsin}\/}\nolimits z=-i\mathop{\ln\/}\nolimits\!\left((1-z^{2})^{{1/2}}+iz\right),$ $z\in\Complex\setminus(-\infty,-1)\cup(1,\infty)$ ;

Symbols:: $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $(a,b)$ : 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

compare Figure 4.23.1(i). On the cuts

4.23.20 $\mathop{\mathrm{arcsin}\/}\nolimits x=\tfrac{1}{2}\pi\pm i\mathop{\ln\/}\nolimits\!\left((x^{2}-1)^{{1/2}}+x\right),$ $x\in[1,\infty)$ ,

Symbols:: $[a,b)$ : half-closed interval, $\in$ : element of, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.23.E20
Encodings:: TeX, pMML, png

4.23.21 $\mathop{\mathrm{arcsin}\/}\nolimits x=-\tfrac{1}{2}\pi\pm i\mathop{\ln\/}\nolimits\!\left((x^{2}-1)^{{1/2}}-x\right),$ $x\in(-\infty,-1]$ ,

Symbols:: $\in$ : element of, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $(a,b]$ : half-closed interval and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.23.E21
Encodings:: TeX, pMML, png

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

¶ Inverse Cosine

4.23.22 $\mathop{\mathrm{arccos}\/}\nolimits z=\tfrac{1}{2}\pi+i\mathop{\ln\/}\nolimits\!\left((1-z^{2})^{{1/2}}+iz\right),$ $z\in\Complex\setminus(-\infty,-1)\cup(1,\infty)$ ;

Symbols:: $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $(a,b)$ : 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

compare Figure 4.23.1(i). An equivalent definition is

4.23.23 $\mathop{\mathrm{arccos}\/}\nolimits z=-2i\mathop{\ln\/}\nolimits\!\left(\left(\frac{1+z}{2}\right)^{{1/2}}+i\left(\frac{1-z}{2}\right)^{{1/2}}\right),$ $z\in\Complex\setminus(-\infty,-1)\cup(1,\infty)$ ;

Symbols:: $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $(a,b)$ : 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 Kahan (1987).

On the cuts

4.23.24 $\mathop{\mathrm{arccos}\/}\nolimits x=\mp i\mathop{\ln\/}\nolimits\!\left((x^{2}-1)^{{1/2}}+x\right),$ $x\in[1,\infty)$ ,

Symbols:: $[a,b)$ : half-closed interval, $\in$ : element of, $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.23.E24
Encodings:: TeX, pMML, png

4.23.25 $\mathop{\mathrm{arccos}\/}\nolimits x=\pi\mp i\mathop{\ln\/}\nolimits\!\left((x^{2}-1)^{{1/2}}-x\right),$ $x\in(-\infty,-1]$ ,

Symbols:: $\in$ : element of, $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $(a,b]$ : half-closed interval and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.23.E25
Encodings:: TeX, pMML, png

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

¶ Inverse Tangent

4.23.26 $\mathop{\mathrm{arctan}\/}\nolimits z=\frac{i}{2}\mathop{\ln\/}\nolimits\!\left(\frac{i+z}{i-z}\right),$ $z/i\in\Complex\setminus(-\infty,-1]\cup[1,\infty)$ ;

Symbols:: $[a,b)$ : half-closed interval, $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $(a,b]$ : 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

compare Figure 4.23.1(ii). On the cuts

4.23.27 $\mathop{\mathrm{arctan}\/}\nolimits\!\left(iy\right)=\pm\frac{1}{2}\pi+\frac{i}{2}\mathop{\ln\/}\nolimits\!\left(\frac{y+1}{y-1}\right),$ $y\in(-\infty,-1)\cup(1,\infty)$ ,

Symbols:: $\in$ : element of, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $(a,b)$ : open interval, $\cup$ : union and $y$ : real variable
Permalink:: http://dlmf.nist.gov/4.23.E27
Encodings:: TeX, pMML, png

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

¶ Other Inverse Functions

Keywords:: inverse trigonometric functions

For the corresponding results for $\mathop{\mathrm{arccsc}\/}\nolimits z$ , $\mathop{\mathrm{arcsec}\/}\nolimits z$ , and $\mathop{\mathrm{arccot}\/}\nolimits z$ , 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

Notes:: See Hobson (1928, pp. 32–33), Levinson and Redheffer (1970, pp. 68–70).
Keywords:: inverse trigonometric functions
Permalink:: http://dlmf.nist.gov/4.23.v

With $k\in\Integer$ , the general solutions of the equations

4.23.28 $z=\mathop{\sin\/}\nolimits w,$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.23.E28
Encodings:: TeX, pMML, png

4.23.29 $z=\mathop{\cos\/}\nolimits w,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.23.E29
Encodings:: TeX, pMML, png

4.23.30 $z=\mathop{\tan\/}\nolimits w,$

Symbols:: $\mathop{\tan\/}\nolimits z$ : tangent function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/4.23.E30
Encodings:: TeX, pMML, png

are respectively

4.23.31 $w=\mathop{\mathrm{Arcsin}\/}\nolimits z=(-1)^{k}\mathop{\mathrm{arcsin}\/}\nolimits z+k\pi,$

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\mathrm{Arcsin}\/}\nolimits z$ : 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

4.23.32 $w=\mathop{\mathrm{Arccos}\/}\nolimits z=\pm\mathop{\mathrm{arccos}\/}\nolimits z+2k\pi,$

Symbols:: $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\mathrm{Arccos}\/}\nolimits z$ : 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

4.23.33 $w=\mathop{\mathrm{Arctan}\/}\nolimits z=\mathop{\mathrm{arctan}\/}\nolimits z+k\pi,$ $z\neq\pm i$ .

Symbols:: $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\mathrm{Arctan}\/}\nolimits z$ : 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

§4.23(vi) Real and Imaginary Parts

Notes:: See Hobson (1928, pp. 332–333).
Keywords:: inverse trigonometric functions
Permalink:: http://dlmf.nist.gov/4.23.vi

4.23.34 $\mathop{\mathrm{arcsin}\/}\nolimits z=\mathop{\mathrm{arcsin}\/}\nolimits\beta+i\mathop{\ln\/}\nolimits\!\left(\alpha+(\alpha^{2}-1)^{{1/2}}\right),$

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $z$ : complex variable, $\alpha$ and $\beta$
A&S Ref:: 4.4.37 (with the general value.)
Referenced by:: §4.23(vi)
Permalink:: http://dlmf.nist.gov/4.23.E34
Encodings:: TeX, pMML, png

4.23.35 $\mathop{\mathrm{arccos}\/}\nolimits z=\mathop{\mathrm{arccos}\/}\nolimits\beta-i\mathop{\ln\/}\nolimits\!\left(\alpha+(\alpha^{2}-1)^{{1/2}}\right),$

Symbols:: $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $z$ : complex variable, $\alpha$ and $\beta$
A&S Ref:: 4.4.38 (with the general value.)
Referenced by:: §4.23(vi)
Permalink:: http://dlmf.nist.gov/4.23.E35
Encodings:: TeX, pMML, png

4.23.36 $\mathop{\mathrm{arctan}\/}\nolimits z=\tfrac{1}{2}\mathop{\mathrm{arctan}\/}\nolimits\!\left(\frac{2x}{1-x^{2}-y^{2}}\right)+\tfrac{1}{4}i\mathop{\ln\/}\nolimits\!\left(\frac{x^{2}+(y+1)^{2}}{x^{2}+(y-1)^{2}}\right),$

Symbols:: $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\ln\/}\nolimits z$ : 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(vi)
Permalink:: http://dlmf.nist.gov/4.23.E36
Encodings:: TeX, pMML, png

where $z=x+iy$ and $x\in[-1,1]$ in (4.23.34) and (4.23.35), and $\left|z\right|<1$ in (4.23.36). Also,

4.23.37 $\alpha=\tfrac{1}{2}\left((x+1)^{2}+y^{2}\right)^{{1/2}}+\tfrac{1}{2}\left((x-1)^{2}+y^{2}\right)^{{1/2}},$

Symbols:: $x$ : real variable, $y$ : real variable and $\alpha$
Permalink:: http://dlmf.nist.gov/4.23.E37
Encodings:: TeX, pMML, png

4.23.38 $\beta=\tfrac{1}{2}\left((x+1)^{2}+y^{2}\right)^{{1/2}}-\tfrac{1}{2}\left((x-1)^{2}+y^{2}\right)^{{1/2}}.$

Symbols:: $x$ : real variable, $y$ : real variable and $\beta$
Permalink:: http://dlmf.nist.gov/4.23.E38
Encodings:: TeX, pMML, png

§4.23(vii) Special Values and Interrelations

Notes:: Table 4.23.1 is constructed from the definitions in §§4.23(i), 4.23(ii).
Keywords:: inverse trigonometric functions
Permalink:: http://dlmf.nist.gov/4.23.vii

Table 4.23.1: Inverse trigonometric functions: principal values at 0, $\pm 1$ , $\pm\infty$ .

$x$	$\mathop{\mathrm{arcsin}\/}\nolimits x$	$\mathop{\mathrm{arccos}\/}\nolimits x$	$\mathop{\mathrm{arctan}\/}\nolimits x$	$\mathop{\mathrm{arccsc}\/}\nolimits x$	$\mathop{\mathrm{arcsec}\/}\nolimits x$	$\mathop{\mathrm{arccot}\/}\nolimits x$
$-\infty$	–	–	$-\tfrac{1}{2}\pi$	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:: $\mathop{\mathrm{arccsc}\/}\nolimits z$ : arccosecant function, $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\mathop{\mathrm{arccot}\/}\nolimits z$ : arccotangent function, $\mathop{\mathrm{arcsec}\/}\nolimits z$ : arcsecant function, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function and $x$ : real variable
Referenced by:: §4.23(vii)
Permalink:: http://dlmf.nist.gov/4.23.T1

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

§4.23(viii) 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 $x\to 0$ .
Keywords:: Gudermannian function, inverse Gudermannian function
Referenced by:: §19.10(ii), §19.6(ii), ¶ ‣ §22.16(i)
Permalink:: http://dlmf.nist.gov/4.23.viii

The Gudermannian $\mathop{\mathrm{gd}\/}\nolimits\!\left(x\right)$ is defined by

4.23.39 $\mathop{\mathrm{gd}\/}\nolimits\!\left(x\right)=\int _{0}^{x}\mathop{\mathrm{sech}\/}\nolimits tdt,$ $-\infty<x<\infty$ .

Defines:: $\mathop{\mathrm{gd}\/}\nolimits x$ : Gudermannian function
Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{sech}\/}\nolimits z$ : 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

Equivalently,

4.23.40 $\mathop{\mathrm{gd}\/}\nolimits\!\left(x\right)=2\mathop{\mathrm{arctan}\/}\nolimits\!\left(e^{x}\right)-\tfrac{1}{2}\pi\\ =\mathop{\mathrm{arcsin}\/}\nolimits\!\left(\mathop{\tanh\/}\nolimits x\right)=\mathop{\mathrm{arccsc}\/}\nolimits\!\left(\mathop{\coth\/}\nolimits x\right)\\ =\mathop{\mathrm{arccos}\/}\nolimits\!\left(\mathop{\mathrm{sech}\/}\nolimits x\right)=\mathop{\mathrm{arcsec}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits x\right)\\ =\mathop{\mathrm{arctan}\/}\nolimits\!\left(\mathop{\sinh\/}\nolimits x\right)=\mathop{\mathrm{arccot}\/}\nolimits\!\left(\mathop{\mathrm{csch}\/}\nolimits x\right).$

The inverse Gudermannian function is given by

4.23.41 $\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(x\right)=\int _{0}^{x}\mathop{\sec\/}\nolimits tdt,$ $-\frac{1}{2}\pi<x<\frac{1}{2}\pi$ .

Defines:: $\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(x\right)$ : inverse Gudermannian function
Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sec\/}\nolimits z$ : 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

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

4.23.42 $\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(x\right)=\mathop{\ln\/}\nolimits\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}x+\tfrac{1}{4}\pi\right)=\mathop{\ln\/}\nolimits\!\left(\mathop{\sec\/}\nolimits x+\mathop{\tan\/}\nolimits x\right)=\mathop{\mathrm{arcsinh}\/}\nolimits\!\left(\mathop{\tan\/}\nolimits x\right)=\mathop{\mathrm{arccsch}\/}\nolimits\!\left(\mathop{\cot\/}\nolimits x\right)=\mathop{\mathrm{arccosh}\/}\nolimits\!\left(\mathop{\sec\/}\nolimits x\right)=\mathop{\mathrm{arcsech}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits x\right)=\mathop{\mathrm{arctanh}\/}\nolimits\!\left(\mathop{\sin\/}\nolimits x\right)=\mathop{\mathrm{arccoth}\/}\nolimits\!\left(\mathop{\csc\/}\nolimits x\right).$