Digital Library of Mathematical Functions
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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.4 Special Values and Limits

Contents

§4.4(i) Logarithms

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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,
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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:
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4.4.2\mathop{\ln\/}\nolimits\!\left(-1\pm i0\right)=\pm\pi i,
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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:
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4.4.3\mathop{\ln\/}\nolimits\!\left(\pm i\right)=\pm\tfrac{1}{2}\pi i.
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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:
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§4.4(ii) Powers

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Notes:
See Levinson and Redheffer (1970, p. 69).
Keywords:
exponential function, power function
Permalink:
http://dlmf.nist.gov/4.4.ii
4.4.4e^{0}=1,
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Symbols:
e: base of exponential function
A&S Ref:
4.2.23
Permalink:
http://dlmf.nist.gov/4.4.E4
Encodings:
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4.4.5e^{{\pm\pi i}}=-1,
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Symbols:
e: base of exponential function
A&S Ref:
4.2.26
Permalink:
http://dlmf.nist.gov/4.4.E5
Encodings:
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4.4.6e^{{\pm\pi i/2}}=\pm i,
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Symbols:
e: base of exponential function
A&S Ref:
4.2.27
Permalink:
http://dlmf.nist.gov/4.4.E6
Encodings:
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4.4.8e^{{\pm\pi i/3}}=\frac{1}{2}\pm i\frac{\sqrt{3}}{2},
4.4.9e^{{\pm 2\pi i/3}}=-\frac{1}{2}\pm i\frac{\sqrt{3}}{2},
4.4.10e^{{\pm\pi i/4}}=\frac{1}{\sqrt{2}}\pm i\frac{1}{\sqrt{2}},
4.4.11e^{{\pm 3\pi i/4}}=-\frac{1}{\sqrt{2}}\pm i\frac{1}{\sqrt{2}},
4.4.12i^{{\pm i}}=e^{{\mp\pi/2}}.

§4.4(iii) Limits

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

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.
4.4.18\lim_{{n\to\infty}}\left(1+\frac{1}{n}\right)^{n}=e.
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Symbols:
e: base of exponential function and n: integer
A&S Ref:
4.1.17
Permalink:
http://dlmf.nist.gov/4.4.E18
Encodings:
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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\ 01532\ 86060\dots,
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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:
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where \EulerConstant is Euler’s constant; see (5.2.3).

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