# §4.25 Continued Fractions

 4.25.1 $\tan z=\cfrac{z}{1-\cfrac{z^{2}}{3-\cfrac{z^{2}}{5-\cfrac{z^{2}}{7-}}}}\cdots,$ $z\neq\pm\tfrac{1}{2}\pi$, $\pm\tfrac{3}{2}\pi$, $\dots$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\tan\NVar{z}$: tangent function and $z$: complex variable A&S Ref: 4.3.94 Permalink: http://dlmf.nist.gov/4.25.E1 Encodings: TeX, pMML, png See also: Annotations for §4.25 and Ch.4
 4.25.2 $\tan\left(az\right)=\cfrac{a\tan z}{1+\cfrac{(1-a^{2}){\tan}^{2}z}{3+\cfrac{(4% -a^{2}){\tan}^{2}z}{5+\cfrac{(9-a^{2}){\tan}^{2}z}{7+}}}}\cdots,$ $|\Re z|<\tfrac{1}{2}\pi$, $az\neq\pm\tfrac{1}{2}\pi,\pm\tfrac{3}{2}\pi,\dots$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\Re$: real part, $\tan\NVar{z}$: tangent function, $a$: real or complex constant and $z$: complex variable A&S Ref: 4.3.95 Permalink: http://dlmf.nist.gov/4.25.E2 Encodings: TeX, pMML, png See also: Annotations for §4.25 and Ch.4
 4.25.3 $\frac{\operatorname{arcsin}z}{\sqrt{1-z^{2}}}=\cfrac{z}{1-\cfrac{1\cdot 2z^{2}% }{3-\cfrac{1\cdot 2z^{2}}{5-\cfrac{3\cdot 4z^{2}}{7-\cfrac{3\cdot 4z^{2}}{9-}}% }}}\cdots,$ ⓘ Symbols: $\operatorname{arcsin}\NVar{z}$: arcsine function and $z$: complex variable A&S Ref: 4.4.44 Permalink: http://dlmf.nist.gov/4.25.E3 Encodings: TeX, pMML, png See also: Annotations for §4.25 and Ch.4

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

 4.25.4 $\operatorname{arctan}z=\cfrac{z}{1+\cfrac{z^{2}}{3+\cfrac{4z^{2}}{5+\cfrac{9z^% {2}}{7+\cfrac{16z^{2}}{9+}}}}}\cdots,$ ⓘ Symbols: $\operatorname{arctan}\NVar{z}$: arctangent function and $z$: 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 See also: Annotations for §4.25 and Ch.4

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

 4.25.5 $e^{2a\operatorname{arctan}\left(1/z\right)}={1+\cfrac{2a}{z-a+\cfrac{a^{2}+1}{% 3z+\cfrac{a^{2}+4}{5z+\cfrac{a^{2}+9}{7z+}}}}\cdots,}$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\operatorname{arctan}\NVar{z}$: arctangent function, $a$: real or complex constant and $z$: complex variable A&S Ref: 4.2.42 Permalink: http://dlmf.nist.gov/4.25.E5 Encodings: TeX, pMML, png See also: Annotations for §4.25 and Ch.4

valid when $z$ 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).