# §4.6 Power Series

## §4.6(i) Logarithms

 4.6.1 $\mathop{\ln\/}\nolimits\!\left(1+z\right)=z-\tfrac{1}{2}z^{2}+\tfrac{1}{3}z^{3% }-\cdots,$ $|z|\leq 1$, $z\neq-1$, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 4.1.24 Referenced by: §4.45(i), §4.6(i) Permalink: http://dlmf.nist.gov/4.6.E1 Encodings: TeX, pMML, png See also: Annotations for 4.6(i)
 4.6.2 $\mathop{\ln\/}\nolimits z=\left(\frac{z-1}{z}\right)+\frac{1}{2}\left(\frac{z-% 1}{z}\right)^{2}+\frac{1}{3}\left(\frac{z-1}{z}\right)^{3}+\cdots,$ $\Re{z}\geq\frac{1}{2}$, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\Re{}$: real part and $z$: complex variable A&S Ref: 4.1.25 Permalink: http://dlmf.nist.gov/4.6.E2 Encodings: TeX, pMML, png See also: Annotations for 4.6(i)
 4.6.3 $\mathop{\ln\/}\nolimits z=(z-1)-\tfrac{1}{2}(z-1)^{2}+\tfrac{1}{3}(z-1)^{3}-\cdots,$ $|z-1|\leq 1$, $z\neq 0$, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 4.1.26 Permalink: http://dlmf.nist.gov/4.6.E3 Encodings: TeX, pMML, png See also: Annotations for 4.6(i)
 4.6.4 $\mathop{\ln\/}\nolimits z=2\left(\left(\frac{z-1}{z+1}\right)+\frac{1}{3}\left% (\frac{z-1}{z+1}\right)^{3}+\frac{1}{5}\left(\frac{z-1}{z+1}\right)^{5}+\cdots% \right),$ $\Re{z}\geq 0$, $z\neq 0$, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\Re{}$: real part and $z$: complex variable A&S Ref: 4.1.27 Permalink: http://dlmf.nist.gov/4.6.E4 Encodings: TeX, pMML, png See also: Annotations for 4.6(i)
 4.6.5 $\mathop{\ln\/}\nolimits\!\left(\frac{z+1}{z-1}\right)=2\left(\frac{1}{z}+\frac% {1}{3z^{3}}+\frac{1}{5z^{5}}+\cdots\right),$ $|z|\geq 1$, $z\neq\pm 1$, Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 4.1.28 Permalink: http://dlmf.nist.gov/4.6.E5 Encodings: TeX, pMML, png See also: Annotations for 4.6(i)
 4.6.6 $\mathop{\ln\/}\nolimits\!\left(z+a\right)=\mathop{\ln\/}\nolimits a+2\left(% \left(\frac{z}{2a+z}\right)+\frac{1}{3}\left(\frac{z}{2a+z}\right)^{3}+\frac{1% }{5}\left(\frac{z}{2a+z}\right)^{5}+\cdots\right),$ $a>0$, $\Re{z}\geq-a$, $z\neq-a$. Symbols: $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\Re{}$: real part, $a$: real or complex constant and $z$: complex variable A&S Ref: 4.1.29 Permalink: http://dlmf.nist.gov/4.6.E6 Encodings: TeX, pMML, png See also: Annotations for 4.6(i)

## §4.6(ii) Powers

### Binomial Expansion

 4.6.7 $(1+z)^{a}=1+\frac{a}{1!}z+\frac{a(a-1)}{2!}z^{2}+\frac{a(a-1)(a-2)}{3!}z^{3}+\cdots,$ Symbols: $!$: factorial (as in $n!$), $a$: real or complex constant and $z$: complex variable Referenced by: §4.6(ii), §4.6(ii) Permalink: http://dlmf.nist.gov/4.6.E7 Encodings: TeX, pMML, png See also: Annotations for 4.6(ii)

valid when $a$ is any real or complex constant and $|z|<1$. If $a=0,1,2,\dots$, then the series terminates and $z$ is unrestricted. Note that (4.6.7) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6).