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DLMF: §4.4 Special Values and Limits ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions

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4 Elementary Functions Logarithm, Exponential, Powers 4.3 Graphics 4.5 Inequalities

§4.4 Special Values and Limits

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Permalink:: http://dlmf.nist.gov/4.4
See also:: Annotations for Ch.4

Contents

§4.4(i) Logarithms

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Keywords:: logarithm function, special values
Notes:: See Levinson and Redheffer (1970, pp. 62–63).
Permalink:: http://dlmf.nist.gov/4.4.i
See also:: Annotations for §4.4 and Ch.4

4.4.1		$\displaystyle\ln 1$	$\displaystyle=0,$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function A&S Ref: 4.1.12 Permalink: http://dlmf.nist.gov/4.4.E1 Encodings: TeX, pMML, png See also: Annotations for §4.4(i), §4.4 and Ch.4
4.4.2		$\displaystyle\ln\left(-1\pm\mathrm{i}0\right)$	$\displaystyle=\pm\pi\mathrm{i},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit and $\ln\NVar{z}$ : principal branch of logarithm function A&S Ref: 4.1.14 (modified) Permalink: http://dlmf.nist.gov/4.4.E2 Encodings: TeX, pMML, png See also: Annotations for §4.4(i), §4.4 and Ch.4
4.4.3		$\displaystyle\ln\left(\pm\mathrm{i}\right)$	$\displaystyle=\pm\tfrac{1}{2}\pi\mathrm{i}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit and $\ln\NVar{z}$ : principal branch of logarithm function A&S Ref: 4.1.15 Permalink: http://dlmf.nist.gov/4.4.E3 Encodings: TeX, pMML, png See also: Annotations for §4.4(i), §4.4 and Ch.4

§4.4(ii) Powers

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Keywords:: exponential function, power function, special values
Notes:: See Levinson and Redheffer (1970, p. 69).
Permalink:: http://dlmf.nist.gov/4.4.ii
See also:: Annotations for §4.4 and Ch.4

4.4.4		$\displaystyle e^{0}$	$\displaystyle=1,$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm A&S Ref: 4.2.23 Permalink: http://dlmf.nist.gov/4.4.E4 Encodings: TeX, pMML, png See also: Annotations for §4.4(ii), §4.4 and Ch.4
4.4.5		$\displaystyle e^{\pm\pi\mathrm{i}}$	$\displaystyle=-1,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $\mathrm{i}$ : imaginary unit A&S Ref: 4.2.26 Permalink: http://dlmf.nist.gov/4.4.E5 Encodings: TeX, pMML, png See also: Annotations for §4.4(ii), §4.4 and Ch.4
4.4.6		$\displaystyle e^{\pm\pi\mathrm{i}/2}$	$\displaystyle=\pm\mathrm{i},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $\mathrm{i}$ : imaginary unit A&S Ref: 4.2.27 Permalink: http://dlmf.nist.gov/4.4.E6 Encodings: TeX, pMML, png See also: Annotations for §4.4(ii), §4.4 and Ch.4
4.4.7		$\displaystyle e^{2\pi k\mathrm{i}}$	$\displaystyle=1,$
$k\in\mathbb{Z}$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers and $k$ : integer A&S Ref: 4.2.28 Permalink: http://dlmf.nist.gov/4.4.E7 Encodings: TeX, pMML, png See also: Annotations for §4.4(ii), §4.4 and Ch.4
4.4.8		$\displaystyle e^{\pm\pi\mathrm{i}/3}$	$\displaystyle=\frac{1}{2}\pm\mathrm{i}\frac{\sqrt{3}}{2},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $\mathrm{i}$ : imaginary unit Permalink: http://dlmf.nist.gov/4.4.E8 Encodings: TeX, pMML, png See also: Annotations for §4.4(ii), §4.4 and Ch.4
4.4.9		$\displaystyle e^{\pm 2\pi\mathrm{i}/3}$	$\displaystyle=-\frac{1}{2}\pm\mathrm{i}\frac{\sqrt{3}}{2},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $\mathrm{i}$ : imaginary unit Permalink: http://dlmf.nist.gov/4.4.E9 Encodings: TeX, pMML, png See also: Annotations for §4.4(ii), §4.4 and Ch.4
4.4.10		$\displaystyle e^{\pm\pi\mathrm{i}/4}$	$\displaystyle=\frac{1}{\sqrt{2}}\pm\mathrm{i}\frac{1}{\sqrt{2}},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $\mathrm{i}$ : imaginary unit Permalink: http://dlmf.nist.gov/4.4.E10 Encodings: TeX, pMML, png See also: Annotations for §4.4(ii), §4.4 and Ch.4
4.4.11		$\displaystyle e^{\pm 3\pi\mathrm{i}/4}$	$\displaystyle=-\frac{1}{\sqrt{2}}\pm\mathrm{i}\frac{1}{\sqrt{2}},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $\mathrm{i}$ : imaginary unit Permalink: http://dlmf.nist.gov/4.4.E11 Encodings: TeX, pMML, png See also: Annotations for §4.4(ii), §4.4 and Ch.4
4.4.12		$\displaystyle{\mathrm{i}}^{\pm\mathrm{i}}$	$\displaystyle=e^{\mp\pi/2}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $\mathrm{i}$ : imaginary unit Permalink: http://dlmf.nist.gov/4.4.E12 Encodings: TeX, pMML, png See also: Annotations for §4.4(ii), §4.4 and Ch.4

§4.4(iii) Limits

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Keywords:: exponential function, limits, logarithm function, power function
Notes:: See Hardy (1952, pp. 403–420).
Permalink:: http://dlmf.nist.gov/4.4.iii
See also:: Annotations for §4.4 and Ch.4

4.4.13		$\lim_{x\to\infty}x^{-a}\ln x=0,$
$\Re a>0$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part, $a$ : real or complex constant and $x$ : real variable A&S Ref: 4.1.30 Permalink: http://dlmf.nist.gov/4.4.E13 Encodings: TeX, pMML, png See also: Annotations for §4.4(iii), §4.4 and Ch.4

4.4.14		$\lim_{x\to 0}x^{a}\ln x=0,$
$\Re a>0$ ,
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part, $a$ : real or complex constant and $x$ : real variable A&S Ref: 4.1.31 Permalink: http://dlmf.nist.gov/4.4.E14 Encodings: TeX, pMML, png See also: Annotations for §4.4(iii), §4.4 and Ch.4

4.4.15		$\lim_{x\to\infty}x^{a}e^{-x}=0,$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $a$ : real or complex constant and $x$ : real variable Permalink: http://dlmf.nist.gov/4.4.E15 Encodings: TeX, pMML, png See also: Annotations for §4.4(iii), §4.4 and Ch.4

4.4.16		$\lim_{z\to\infty}z^{a}e^{-z}=0,$
$\|\operatorname{ph}z\|\leq\tfrac{1}{2}\pi-\delta$ ( $<\tfrac{1}{2}\pi$ ),
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $a$ : real or complex constant, $z$ : complex variable and $\delta$ : constant A&S Ref: 4.2.20 Permalink: http://dlmf.nist.gov/4.4.E16 Encodings: TeX, pMML, png See also: Annotations for §4.4(iii), §4.4 and Ch.4

where $a$ ( $\in\mathbb{C}$ ) and $\delta$ ( $\in(0,\tfrac{1}{2}\pi]$ ) are constants.

4.4.17		$\lim_{n\to\infty}\left(1+\frac{z}{n}\right)^{n}=e^{z},$
$z=$ constant.
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $n$ : integer and $z$ : complex variable A&S Ref: 4.2.21 Permalink: http://dlmf.nist.gov/4.4.E17 Encodings: TeX, pMML, png See also: Annotations for §4.4(iii), §4.4 and Ch.4

4.4.18		$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}=e.$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm and $n$ : integer A&S Ref: 4.1.17 Permalink: http://dlmf.nist.gov/4.4.E18 Encodings: TeX, pMML, png See also: Annotations for §4.4(iii), §4.4 and Ch.4

4.4.19		$\lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1}{k}\right)-\ln n\right)=% \gamma=0.57721\ 56649\ 01532\ 86060\dots,$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : integer and $n$ : integer A&S Ref: 4.1.32 (with 10D value) Notes: For more digits see OEIS Sequence A001620; see also Sloane (2003). Permalink: http://dlmf.nist.gov/4.4.E19 Encodings: TeX, pMML, png See also: Annotations for §4.4(iii), §4.4 and Ch.4

where $\gamma$ is Euler’s constant; see (5.2.3).

4.3 Graphics 4.5 Inequalities

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