# §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).