§4.24 Inverse Trigonometric Functions: Further Properties

Referenced by:: §4.2(i), §4.38(i)
Permalink:: http://dlmf.nist.gov/4.24

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

§4.24(i) Power Series

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

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:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : 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

4.24.2 $\mathop{\mathrm{arccos}\/}\nolimits z=(2(1-z))^{{1/2}}\times\left(1+\sum _{{n=1}}^{\infty}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{{2n}}(2n+1)n!}(1-z)^{n}\right),$ $|1-z|\leq 2$ .

Symbols:: $!$ : $n!$ : factorial, $\mathop{\mathrm{arccos}\/}\nolimits z$ : 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

4.24.3 $\mathop{\mathrm{arctan}\/}\nolimits z=z-\frac{z^{3}}{3}+\frac{z^{5}}{5}-\frac{z^{7}}{7}+\cdots,$ $\left|z\right|\leq 1$ , $z\neq\pm i$ .

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

4.24.4 $\mathop{\mathrm{arctan}\/}\nolimits z=\pm\frac{\pi}{2}-\frac{1}{z}+\frac{1}{3z^{3}}-\frac{1}{5z^{5}}+\cdots,$ $\realpart{z}\gtrless 0$ , $|z|\geq 1$ .

Symbols:: $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\realpart{}$ : real part and $z$ : complex variable
A&S Ref:: 4.4.42 (has an error. $\frac{\pi}{2}$ should have a negative sign when $\realpart{z}<0$ .)
Referenced by:: ¶ ‣ §4.45(i)
Permalink:: http://dlmf.nist.gov/4.24.E4
Encodings:: TeX, pMML, png

4.24.5 $\mathop{\mathrm{arctan}\/}\nolimits z=\frac{z}{z^{2}+1}\*\left(1+\frac{2}{3}\frac{z^{2}}{1+z^{2}}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}{1+z^{2}}\right)^{2}+\cdots\right),$ $\realpart{(z^{2})}>-\tfrac{1}{2}$ ,

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

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

§4.24(ii) Derivatives

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.
Keywords:: derivatives, inverse trigonometric functions
Permalink:: http://dlmf.nist.gov/4.24.ii

4.24.7 $\frac{d}{dz}\mathop{\mathrm{arcsin}\/}\nolimits z=(1-z^{2})^{{-1/2}},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function and $z$ : complex variable
A&S Ref:: 4.4.52
Permalink:: http://dlmf.nist.gov/4.24.E7
Encodings:: TeX, pMML, png

4.24.8 $\frac{d}{dz}\mathop{\mathrm{arccos}\/}\nolimits z=-(1-z^{2})^{{-1/2}},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function and $z$ : complex variable
A&S Ref:: 4.4.53
Permalink:: http://dlmf.nist.gov/4.24.E8
Encodings:: TeX, pMML, png

4.24.9 $\frac{d}{dz}\mathop{\mathrm{arctan}\/}\nolimits z=\frac{1}{1+z^{2}}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function and $z$ : complex variable
A&S Ref:: 4.4.54
Permalink:: http://dlmf.nist.gov/4.24.E9
Encodings:: TeX, pMML, png

4.24.10 $\frac{d}{dz}\mathop{\mathrm{arccsc}\/}\nolimits z=\mp\frac{1}{z(z^{2}-1)^{{1/2}}},$ $\realpart{z}\gtrless 0$ .

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arccsc}\/}\nolimits z$ : arccosecant function, $\realpart{}$ : 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

4.24.11 $\frac{d}{dz}\mathop{\mathrm{arcsec}\/}\nolimits z=\pm\frac{1}{z(z^{2}-1)^{{1/2}}},$ $\realpart{z}\gtrless 0$ .

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arcsec}\/}\nolimits z$ : arcsecant function, $\realpart{}$ : 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

4.24.12 $\frac{d}{dz}\mathop{\mathrm{arccot}\/}\nolimits z=-\frac{1}{1+z^{2}}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathrm{arccot}\/}\nolimits z$ : arccotangent function and $z$ : complex variable
A&S Ref:: 4.4.55
Permalink:: http://dlmf.nist.gov/4.24.E12
Encodings:: TeX, pMML, png

§4.24(iii) Addition Formulas

Notes:: See Hobson (1928, pp. 54–55).
Keywords:: inverse trigonometric functions
Referenced by:: §4.38(iii)
Permalink:: http://dlmf.nist.gov/4.24.iii

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:: $\mathop{\mathrm{Arcsin}\/}\nolimits z$ : general arcsine function
A&S Ref:: 4.4.32
Permalink:: http://dlmf.nist.gov/4.24.E13
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{Arccos}\/}\nolimits z$ : general arccosine function
A&S Ref:: 4.4.33
Permalink:: http://dlmf.nist.gov/4.24.E14
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{Arctan}\/}\nolimits z$ : 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

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:: $\mathop{\mathrm{Arccos}\/}\nolimits z$ : general arccosine function and $\mathop{\mathrm{Arcsin}\/}\nolimits z$ : general arcsine function
A&S Ref:: 4.4.35
Permalink:: http://dlmf.nist.gov/4.24.E16
Encodings:: TeX, pMML, png

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:: $\mathop{\mathrm{Arccot}\/}\nolimits z$ : general arccotangent function and $\mathop{\mathrm{Arctan}\/}\nolimits z$ : general arctangent function
A&S Ref:: 4.4.36
Permalink:: http://dlmf.nist.gov/4.24.E17
Encodings:: TeX, pMML, png

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.

§4.24 Inverse Trigonometric Functions: Further Properties

Contents

§4.24(i) Power Series

§4.24(ii) Derivatives

§4.24(iii) Addition Formulas