4 Elementary Functions Logarithm, Exponential, Powers 4.5 Inequalities 4.7 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

§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		$\ln\left(1+z\right)=z-\tfrac{1}{2}z^{2}+\tfrac{1}{3}z^{3}-\cdots,$
$\|z\|\leq 1$ , $z\neq-1$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 4.1.24 Referenced by: §4.45(i), §4.6(i), (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		$\ln z=\left(\frac{z-1}{z}\right)+\frac{1}{2}\left(\frac{z-1}{z}\right)^{2}+% \frac{1}{3}\left(\frac{z-1}{z}\right)^{3}+\cdots,$
$\Re z\geq\frac{1}{2}$ ,
ⓘ Symbols: $\ln\NVar{z}$ : 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 and Ch.4

4.6.3		$\ln z=(z-1)-\tfrac{1}{2}(z-1)^{2}+\tfrac{1}{3}(z-1)^{3}-\cdots,$
$\|z-1\|\leq 1$ , $z\neq 0$ ,
ⓘ Symbols: $\ln\NVar{z}$ : 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 and Ch.4

4.6.4		$\ln z=2\left(\left(\frac{z-1}{z+1}\right)+\frac{1}{3}\left(\frac{z-1}{z+1}% \right)^{3}+\frac{1}{5}\left(\frac{z-1}{z+1}\right)^{5}+\cdots\right),$
$\Re z\geq 0$ , $z\neq 0$ ,
ⓘ Symbols: $\ln\NVar{z}$ : 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 and Ch.4

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

4.6.6		$\ln\left(z+a\right)=\ln a+2\left(\left(\frac{z}{2a+z}\right)+\frac{1}{3}\left(% \frac{z}{2a+z}\right)^{3}+\frac{1}{5}\left(\frac{z}{2a+z}\right)^{5}+\cdots% \right),$
$a>0$ , $\Re z\geq-a$ , $z\neq-a$ .
ⓘ Symbols: $\ln\NVar{z}$ : 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 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		$(1+z)^{a}=1+\frac{a}{1!}z+\frac{a(a-1)}{2!}z^{2}+\frac{a(a-1)(a-2)}{3!}z^{3}+\cdots,$
ⓘ 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 $|z|<1$ . 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).

4.5 Inequalities 4.7 Derivatives and Differential Equations

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§4.6 Power Series

Contents

§4.6(i) Logarithms

§4.6(ii) Powers

Binomial Expansion