4 Elementary FunctionsLogarithm, Exponential, Powers4.5 Inequalities4.7 Derivatives and Differential Equations

§4.6 Power Series

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Permalink:

http://dlmf.nist.gov/4.6

See also:

Annotations for Ch.4

Contents

1. §4.6(i) Logarithms
2. §4.6(ii) Powers

§4.6(i) Logarithms

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Keywords:

logarithm function, power series

Notes:

See Hardy (1952, pp. 471–473) for (4.6.1). The other equations are variations of this.

Permalink:

http://dlmf.nist.gov/4.6.i

See also:

Annotations for §4.6 and Ch.4

4.6.1		$$\ln \left(1+z\right)=z-\frac{1}{2}{z}^2+\frac{1}{3}{z}^3-\mathrm{\cdots },$$
$|z|\le 1$, $z\ne -1$,
ⓘ Symbols: $\ln z$: principal branch of logarithm function and $z$: complex variable A&S Ref: 4.1.24 Referenced by: §4.45(i), §4.6(i), (8.15.2) Permalink: http://dlmf.nist.gov/4.6.E1 Encodings: TeX, pMML, png See also: Annotations for §4.6(i), §4.6 and Ch.4

4.6.2		$$\ln z=\left(\frac{z-1}{z}\right)+\frac{1}{2}{\left(\frac{z-1}{z}\right)}^2+\frac{1}{3}{\left(\frac{z-1}{z}\right)}^3+\mathrm{\cdots },$$
$\mathrm{\Re }z\ge \frac{1}{2}$,
ⓘ Symbols: $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: 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 and Ch.4

4.6.3		$$\ln z=(z-1)-\frac{1}{2}{(z-1)}^2+\frac{1}{3}{(z-1)}^3-\mathrm{\cdots },$$
$|z-1|\le 1$, $z\ne 0$,
ⓘ Symbols: $\ln z$: 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 and Ch.4

4.6.4		$$\ln z=2\left(\left(\frac{z-1}{z+1}\right)+\frac{1}{3}{\left(\frac{z-1}{z+1}\right)}^3+\frac{1}{5}{\left(\frac{z-1}{z+1}\right)}^5+\mathrm{\cdots }\right),$$
$\mathrm{\Re }z\ge 0$, $z\ne 0$,
ⓘ Symbols: $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: 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 and Ch.4

4.6.5		$$\ln \left(\frac{z+1}{z-1}\right)=2\left(\frac{1}{z}+\frac{1}{3z^3}+\frac{1}{5z^5}+\mathrm{\cdots }\right),$$
$|z|\ge 1$, $z\ne \pm 1$,
ⓘ Symbols: $\ln z$: 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 and Ch.4

4.6.6		$$\ln \left(z+a\right)=\ln a+2\left(\left(\frac{z}{2a+z}\right)+\frac{1}{3}{\left(\frac{z}{2a+z}\right)}^3+\frac{1}{5}{\left(\frac{z}{2a+z}\right)}^5+\mathrm{\cdots }\right),$$
$a>0$, $\mathrm{\Re }z\ge -a$, $z\ne -a$.
ⓘ Symbols: $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: 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 and Ch.4

§4.6(ii) Powers

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Notes:

See Hardy (1952, pp. 476–477).

Referenced by:

Erratum (V1.0.11) for Clarifications

Permalink:

http://dlmf.nist.gov/4.6.ii

Addition (effective with 1.0.11):

A sentence was added after (4.6.7) to explain that it is a generalization of (1.2.2) using (1.2.6)

See also:

Annotations for §4.6 and Ch.4

Binomial Expansion

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Keywords:

binomial expansion

See also:

Annotations for §4.6(ii), §4.6 and Ch.4

4.6.7		$${(1+z)}^a=1+\frac{a}{1!}z+\frac{a(a-1)}{2!}{z}^2+\frac{a(a-1)(a-2)}{3!}{z}^3+\mathrm{\cdots },$$
ⓘ 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), §4.6(ii), §4.6 and Ch.4

valid when $a$ is any real or complex constant and . If $a=0,1,2,\mathrm{\dots }$, 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).

4.5 Inequalities4.7 Derivatives and Differential Equations

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