# §4.4 Special Values and Limits

## §4.4(i) Logarithms

 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

 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 $\displaystyle e^{2\pi k\mathrm{i}}$ $\displaystyle=1,$ $k\in\mathbb{Z}$, 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

 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 $\lim_{z\to\infty}z^{a}e^{-z}=0,$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$ ($<\tfrac{1}{2}\pi$),

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).