Digital Library of Mathematical Functions
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4 Elementary FunctionsTrigonometric Functions

§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
Contents

§4.24(i) Power Series

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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.
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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:
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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.
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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:
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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.
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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:
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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.
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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:
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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},
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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

4.24.6x^{2}-y^{2}=-\tfrac{1}{2}.

§4.24(ii) Derivatives

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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
4.24.10\frac{d}{dz}\mathop{\mathrm{arccsc}\/}\nolimits z=\mp\frac{1}{z(z^{2}-1)^{{1/2% }}},\realpart{z}\gtrless 0.
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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:
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4.24.11\frac{d}{dz}\mathop{\mathrm{arcsec}\/}\nolimits z=\pm\frac{1}{z(z^{2}-1)^{{1/2% }}},\realpart{z}\gtrless 0.
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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:
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§4.24(iii) Addition Formulas

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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),
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Symbols:
\mathop{\mathrm{Arcsin}\/}\nolimits z: general arcsine function
A&S Ref:
4.4.32
Permalink:
http://dlmf.nist.gov/4.24.E13
Encodings:
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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),
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Symbols:
\mathop{\mathrm{Arccos}\/}\nolimits z: general arccosine function
A&S Ref:
4.4.33
Permalink:
http://dlmf.nist.gov/4.24.E14
Encodings:
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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),
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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:
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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),
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).

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.

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