4 Elementary FunctionsLogarithm, Exponential, Powers4.3 Graphics4.5 Inequalities

§4.4 Special Values and Limits

ⓘ

Permalink:

http://dlmf.nist.gov/4.4

See also:

Annotations for Ch.4

Contents

1. §4.4(i) Logarithms
2. §4.4(ii) Powers
3. §4.4(iii) Limits

§4.4(i) Logarithms

ⓘ

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		$\ln 1$	$=0,$
ⓘ Symbols: $\ln z$: 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		$\ln \left(-1\pm \mathrm{i}0\right)$	$=\pm \pi \mathrm{i},$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\ln z$: 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		$\ln \left(\pm \mathrm{i}\right)$	$=\pm \frac{1}{2}\pi \mathrm{i}.$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\ln z$: 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

ⓘ

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		${\mathrm{e}}^0$	$=1,$
ⓘ 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		${\mathrm{e}}^{\pm \pi \mathrm{i}}$	$=-1,$
ⓘ 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		${\mathrm{e}}^{\pm \pi \mathrm{i}/2}$	$=\pm \mathrm{i},$
ⓘ 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		${\mathrm{e}}^{2\pi k\mathrm{i}}$	$=1,$
$k\in \mathbb{Z}$,
ⓘ 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		${\mathrm{e}}^{\pm \pi \mathrm{i}/3}$	$={\displaystyle \frac{1}{2}}\pm \mathrm{i}{\displaystyle \frac{\sqrt{3}}{2}},$
ⓘ 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		${\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}$	$=-{\displaystyle \frac{1}{2}}\pm \mathrm{i}{\displaystyle \frac{\sqrt{3}}{2}},$
ⓘ 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		${\mathrm{e}}^{\pm \pi \mathrm{i}/4}$	$={\displaystyle \frac{1}{\sqrt{2}}}\pm \mathrm{i}{\displaystyle \frac{1}{\sqrt{2}}},$
ⓘ 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		${\mathrm{e}}^{\pm 3\pi \mathrm{i}/4}$	$=-{\displaystyle \frac{1}{\sqrt{2}}}\pm \mathrm{i}{\displaystyle \frac{1}{\sqrt{2}}},$
ⓘ 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		${\mathrm{i}}^{\pm \mathrm{i}}$	$={\mathrm{e}}^{\mp \pi /2}.$
ⓘ 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

ⓘ

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		$$\underset{x\to \mathrm{\infty }}{lim}{x}^{-a}\ln x=0,$$
$\mathrm{\Re }a>0$,
ⓘ Symbols: $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: 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		$$\underset{x\to 0}{lim}{x}^a\ln x=0,$$
$\mathrm{\Re }a>0$,
ⓘ Symbols: $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: 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		$$\underset{x\to \mathrm{\infty }}{lim}{x}^a{\mathrm{e}}^{-x}=0,$$
ⓘ 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		$$\underset{z\to \mathrm{\infty }}{lim}{z}^a{\mathrm{e}}^{-z}=0,$$
$|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ (),
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: 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 $a$ ($\in ℂ$) and $\delta$ ($\in (0,\frac{1}{2}\pi ]$) are constants.

4.4.17		$$\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{z}{n}\right)}^n={\mathrm{e}}^z,$$
$z=$ 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		$$\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{1}{n}\right)}^n=\mathrm{e}.$$
ⓘ 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		$$\underset{n\to \mathrm{\infty }}{lim}\left(\left(\sum _{k=1}^n\frac{1}{k}\right)-\ln n\right)=\gamma =\mathrm{0.57721\; 56649\; 01532\; 86060}\mathrm{\dots },$$
ⓘ Symbols: $\gamma$: Euler’s constant, $\ln z$: 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 Graphics4.5 Inequalities

© 2010–2024 NIST / Disclaimer / Feedback; Version 1.2.3; Release date 2024-12-15.

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy