4 Elementary Functions Trigonometric Functions 4.24 Inverse Trigonometric Functions: Further Properties 4.26 Integrals

§4.25 Continued Fractions

Notes:: See Jones and Thron (1980, pp. 202–203), Wall (1948, pp. 343–349).
Keywords:: continued fractions, inverse trigonometric functions, trigonometric functions
Permalink:: http://dlmf.nist.gov/4.25

4.25.1 $\mathop{\tan\/}\nolimits z=\cfrac{z}{1-\cfrac{z^{2}}{3-\cfrac{z^{2}}{5-\cfrac{% z^{2}}{7-}}}}\cdots,$ $z\neq\pm\tfrac{1}{2}\pi$ , $\pm\tfrac{3}{2}\pi$ , $\dots$ .

Symbols:: $\mathop{\tan\/}\nolimits z$ : tangent function and : complex variable
A&S Ref:: 4.3.94
Permalink:: http://dlmf.nist.gov/4.25.E1
Encodings:: TeX, pMML, png

4.25.2 $\mathop{\tan\/}\nolimits\!\left(az\right)=\cfrac{a\mathop{\tan\/}\nolimits z}{% 1+\cfrac{(1-a^{2}){\mathop{\tan\/}\nolimits^{{2}}}z}{3+\cfrac{(4-a^{2}){% \mathop{\tan\/}\nolimits^{{2}}}z}{5+\cfrac{(9-a^{2}){\mathop{\tan\/}\nolimits^% {{2}}}z}{7+}}}}\cdots,$ $|\realpart{z}|<\tfrac{1}{2}\pi$ , $az\neq\pm\tfrac{1}{2}\pi,\pm\tfrac{3}{2}\pi,\dots$ .

Symbols:: $\realpart{}$ : real part, $\mathop{\tan\/}\nolimits z$ : tangent function, : real or complex constant and : complex variable
A&S Ref:: 4.3.95
Permalink:: http://dlmf.nist.gov/4.25.E2
Encodings:: TeX, pMML, png

4.25.3 $\frac{\mathop{\mathrm{arcsin}\/}\nolimits z}{\sqrt{1-z^{2}}}=\cfrac{z}{1-% \cfrac{1\cdot 2z^{2}}{3-\cfrac{1\cdot 2z^{2}}{5-\cfrac{3\cdot 4z^{2}}{7-\cfrac% {3\cdot 4z^{2}}{9-}}}}}\cdots,$

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function and : complex variable
A&S Ref:: 4.4.44
Permalink:: http://dlmf.nist.gov/4.25.E3
Encodings:: TeX, pMML, png

valid when lies in the open cut plane shown in Figure 4.23.1(i).

4.25.4 $\mathop{\mathrm{arctan}\/}\nolimits z=\cfrac{z}{1+\cfrac{z^{2}}{3+\cfrac{4z^{2% }}{5+\cfrac{9z^{2}}{7+\cfrac{16z^{2}}{9+}}}}}\cdots,$

Symbols:: $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function and : complex variable
A&S Ref:: 4.4.43
Referenced by:: §3.10(ii)
Permalink:: http://dlmf.nist.gov/4.25.E4
Encodings:: TeX, pMML, png

valid when lies in the open cut plane shown in Figure 4.23.1(ii).

4.25.5 $e^{{2a\mathop{\mathrm{arctan}\/}\nolimits\!\left(1/z\right)}}={1+\cfrac{2a}{z-% a+\cfrac{a^{2}+1}{3z+\cfrac{a^{2}+4}{5z+\cfrac{a^{2}+9}{7z+}}}}\cdots,}$

Symbols:: : base of exponential function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, : real or complex constant and : complex variable
A&S Ref:: 4.2.42
Permalink:: http://dlmf.nist.gov/4.25.E5
Encodings:: TeX, pMML, png

valid when lies in the open cut plane shown in Figure 4.23.1(iv).

See Lorentzen and Waadeland (1992, pp. 560–571) for other continued fractions involving inverse trigonometric functions. See also Cuyt et al. (2008, pp. 201–203, 205–210).

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