§4.9 Continued Fractions

§4.9(i) Logarithms

 4.9.1 $\ln\left(1+z\right)=\cfrac{z}{1+\cfrac{z}{2+\cfrac{z}{3+\cfrac{4z}{4+\cfrac{4z% }{5+\cfrac{9z}{6+\cfrac{9z}{7+}}}}}}}\cdots,$ $|\operatorname{ph}\left(1+z\right)|<\pi$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function, $\operatorname{ph}$: phase and $z$: complex variable A&S Ref: 4.1.39 Permalink: http://dlmf.nist.gov/4.9.E1 Encodings: TeX, pMML, png See also: Annotations for 4.9(i), 4.9 and 4
 4.9.2 $\ln\left(\frac{1+z}{1-z}\right)=\cfrac{2z}{1-\cfrac{z^{2}}{3-\cfrac{4z^{2}}{5-% \cfrac{9z^{2}}{7-\cfrac{16z^{2}}{9-}}}}}\cdots,$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 4.1.40 Permalink: http://dlmf.nist.gov/4.9.E2 Encodings: TeX, pMML, png See also: Annotations for 4.9(i), 4.9 and 4

valid when $z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$; see Figure 4.23.1(i).

For other continued fractions involving logarithms see Lorentzen and Waadeland (1992, pp. 566–568). See also Cuyt et al. (2008, pp. 196–200).

§4.9(ii) Exponentials

For $z\in\mathbb{C}$,

 4.9.3 $\displaystyle e^{z}$ $\displaystyle=\cfrac{1}{1-\cfrac{z}{1+\cfrac{z}{2-\cfrac{z}{3+\cfrac{z}{2-% \cfrac{z}{5+\cfrac{z}{2-}}}}}}}\cdots$ $\displaystyle=1+\cfrac{z}{1-\cfrac{z}{2+\cfrac{z}{3-\cfrac{z}{2+\cfrac{z}{5-% \cfrac{z}{2+\cfrac{z}{7-}}}}}}}\cdots$ $\displaystyle=1+\cfrac{z}{1-(z/2)+\cfrac{z^{2}/(4\cdot 3)}{1+\cfrac{z^{2}/(4% \cdot 15)}{1+\cfrac{z^{2}/(4\cdot 35)}{1+}}}}\cdots\cfrac{z^{2}/(4(4n^{2}-1))}% {1+}\cdots$ ⓘ Symbols: $\mathrm{e}$: base of exponential function, $n$: integer and $z$: complex variable A&S Ref: 4.2.40 Permalink: http://dlmf.nist.gov/4.9.E3 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 4.9(ii), 4.9 and 4
 4.9.4 $e^{z}-e_{n-1}(z)={\cfrac{z^{n}}{n!-\cfrac{n!z}{(n+1)+\cfrac{z}{(n+2)-\cfrac{(n% +1)z}{(n+3)+\cfrac{2z}{(n+4)-}}}}}\cfrac{(n+2)z}{(n+5)+\cfrac{3z}{(n+6)-}}% \cdots},$ ⓘ Symbols: $\mathrm{e}$: base of exponential function, $!$: factorial (as in $n!$), $n$: integer, $z$: complex variable and $e_{n}(z)$: expansion of exponential A&S Ref: 4.2.41 Permalink: http://dlmf.nist.gov/4.9.E4 Encodings: TeX, pMML, png See also: Annotations for 4.9(ii), 4.9 and 4

where

 4.9.5 $e_{n}(z)=\sum_{k=0}^{n}\frac{z^{k}}{k!}.$ ⓘ Defines: $e_{n}(z)$: expansion of exponential (locally) Symbols: $!$: factorial (as in $n!$), $k$: integer, $n$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/4.9.E5 Encodings: TeX, pMML, png See also: Annotations for 4.9(ii), 4.9 and 4

For other continued fractions involving the exponential function see Lorentzen and Waadeland (1992, pp. 563–564). See also Cuyt et al. (2008, pp. 193–195).

§4.9(iii) Powers

See Cuyt et al. (2008, pp. 217–220).