§4.4 Special Values and Limits

Permalink:: http://dlmf.nist.gov/4.4

Contents

§4.4(i) Logarithms
§4.4(ii) Powers
§4.4(iii) Limits

§4.4(i) Logarithms

Notes:: See Levinson and Redheffer (1970, pp. 62–63).
Keywords:: logarithm function
Permalink:: http://dlmf.nist.gov/4.4.i

4.4.1 $\mathop{\ln\/}\nolimits 1=0,$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
A&S Ref:: 4.1.12
Permalink:: http://dlmf.nist.gov/4.4.E1
Encodings:: TeX, pMML, png

4.4.2 $\mathop{\ln\/}\nolimits\!\left(-1\pm i0\right)=\pm\pi i,$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
A&S Ref:: 4.1.14 (modified)
Permalink:: http://dlmf.nist.gov/4.4.E2
Encodings:: TeX, pMML, png

4.4.3 $\mathop{\ln\/}\nolimits\!\left(\pm i\right)=\pm\tfrac{1}{2}\pi i.$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
A&S Ref:: 4.1.15
Permalink:: http://dlmf.nist.gov/4.4.E3
Encodings:: TeX, pMML, png

§4.4(ii) Powers

Notes:: See Levinson and Redheffer (1970, p. 69).
Keywords:: exponential function, power function
Permalink:: http://dlmf.nist.gov/4.4.ii

4.4.4 $e^{0}=1,$

Symbols:: $e$ : base of exponential function
A&S Ref:: 4.2.23
Permalink:: http://dlmf.nist.gov/4.4.E4
Encodings:: TeX, pMML, png

4.4.5 $e^{{\pm\pi i}}=-1,$

Symbols:: $e$ : base of exponential function
A&S Ref:: 4.2.26
Permalink:: http://dlmf.nist.gov/4.4.E5
Encodings:: TeX, pMML, png

4.4.6 $e^{{\pm\pi i/2}}=\pm i,$

Symbols:: $e$ : base of exponential function
A&S Ref:: 4.2.27
Permalink:: http://dlmf.nist.gov/4.4.E6
Encodings:: TeX, pMML, png

4.4.7 $e^{{2\pi ki}}=1,$ $k\in\Integer$ ,

Symbols:: $\in$ : element of, $e$ : base of exponential function, $\Integer$ : 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

4.4.8 $e^{{\pm\pi i/3}}=\frac{1}{2}\pm i\frac{\sqrt{3}}{2},$

Symbols:: $e$ : base of exponential function
Permalink:: http://dlmf.nist.gov/4.4.E8
Encodings:: TeX, pMML, png

4.4.9 $e^{{\pm 2\pi i/3}}=-\frac{1}{2}\pm i\frac{\sqrt{3}}{2},$

Symbols:: $e$ : base of exponential function
Permalink:: http://dlmf.nist.gov/4.4.E9
Encodings:: TeX, pMML, png

4.4.10 $e^{{\pm\pi i/4}}=\frac{1}{\sqrt{2}}\pm i\frac{1}{\sqrt{2}},$

Symbols:: $e$ : base of exponential function
Permalink:: http://dlmf.nist.gov/4.4.E10
Encodings:: TeX, pMML, png

4.4.11 $e^{{\pm 3\pi i/4}}=-\frac{1}{\sqrt{2}}\pm i\frac{1}{\sqrt{2}},$

Symbols:: $e$ : base of exponential function
Permalink:: http://dlmf.nist.gov/4.4.E11
Encodings:: TeX, pMML, png

4.4.12 $i^{{\pm i}}=e^{{\mp\pi/2}}.$

Symbols:: $e$ : base of exponential function
Permalink:: http://dlmf.nist.gov/4.4.E12
Encodings:: TeX, pMML, png

§4.4(iii) Limits

Notes:: See Hardy (1952, pp. 403–420).
Keywords:: exponential function, logarithm function, power function
Permalink:: http://dlmf.nist.gov/4.4.iii

4.4.13 $\lim _{{x\to\infty}}x^{{-a}}\mathop{\ln\/}\nolimits x=0,$ $\realpart{a}>0$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : 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

4.4.14 $\lim _{{x\to 0}}x^{a}\mathop{\ln\/}\nolimits x=0,$ $\realpart{a}>0$ ,

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : 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

4.4.15 $\lim _{{x\to\infty}}x^{a}e^{{-x}}=0,$

Symbols:: $e$ : base of exponential function, $a$ : real or complex constant and $x$ : real variable
Permalink:: http://dlmf.nist.gov/4.4.E15
Encodings:: TeX, pMML, png

4.4.16 $\lim _{{z\to\infty}}z^{a}e^{{-z}}=0,$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta$ ( $<\tfrac{1}{2}\pi$ ),

Symbols:: $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

where $a$ ( $\in\Complex$ ) 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:: $e$ : base of exponential function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.2.21
Permalink:: http://dlmf.nist.gov/4.4.E17
Encodings:: TeX, pMML, png

4.4.18 $\lim _{{n\to\infty}}\left(1+\frac{1}{n}\right)^{n}=e.$

Symbols:: $e$ : base of exponential function and $n$ : integer
A&S Ref:: 4.1.17
Permalink:: http://dlmf.nist.gov/4.4.E18
Encodings:: TeX, pMML, png

4.4.19 $\lim _{{n\to\infty}}\left(\left(\sum^{n}_{{k=1}}\frac{1}{k}\right)-\mathop{\ln\/}\nolimits n\right)=\EulerConstant=0.57721\ 56649\ 0 1532\ 86060\dots,$

Notes:: For more digits see OEIS Sequence A001620; see also Sloane (2003).
Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $k$ : integer and $n$ : integer
A&S Ref:: 4.1.32 (with 10D value)
Permalink:: http://dlmf.nist.gov/4.4.E19
Encodings:: TeX, pMML, png

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