§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 Ch.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 Ch.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 natural logarithm, $n$: 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 Ch.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 natural logarithm, $!$: 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 Ch.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 Ch.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).