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Natural logarithm

Table of contents: Definitions - Illustrations - Particular values - Functional equations and connection formulas - Analytic properties - Complex parts - Bounds and inequalities - Integral representations

Definitions

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Symbol: Log log ⁣(z)\log\!\left(z\right) Natural logarithm

Illustrations

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Image: X-ray of log ⁣(z)\log\!\left(z\right) on z[3,3]+[3,3]iz \in \left[-3, 3\right] + \left[-3, 3\right] i

Particular values

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log ⁣(1)=0\log\!\left(1\right) = 0
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log ⁣(e)=1\log\!\left(e\right) = 1
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Table of log ⁣(n)\log\!\left(n\right) to 50 digits for 1n501 \le n \le 50
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log ⁣(i)=πi2\log\!\left(i\right) = \frac{\pi i}{2}
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log ⁣(1)=πi\log\!\left(-1\right) = \pi i

Functional equations and connection formulas

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exp ⁣(log ⁣(z))=z\exp\!\left(\log\!\left(z\right)\right) = z
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log ⁣(ez)=z\log\!\left({e}^{z}\right) = z
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log ⁣(z)=log ⁣(z)+arg ⁣(z)i\log\!\left(z\right) = \log\!\left(\left|z\right|\right) + \arg\!\left(z\right) i
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log ⁣(cz)=log ⁣(c)+log ⁣(z)\log\!\left(c z\right) = \log\!\left(c\right) + \log\!\left(z\right)

Analytic properties

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HolomorphicDomain ⁣(log ⁣(z),z,C{~})=C(,0]\operatorname{HolomorphicDomain}\!\left(\log\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \left(-\infty, 0\right]
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Poles ⁣(log ⁣(z),z,C{~})={}\operatorname{Poles}\!\left(\log\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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EssentialSingularities ⁣(log ⁣(z),z,C{~})={}\operatorname{EssentialSingularities}\!\left(\log\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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BranchPoints ⁣(log ⁣(z),z,C{~})={~,0}\operatorname{BranchPoints}\!\left(\log\!\left(z\right), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}, 0\right\}
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BranchCuts ⁣(log ⁣(z),z,C)={(,0]}\operatorname{BranchCuts}\!\left(\log\!\left(z\right), z, \mathbb{C}\right) = \left\{\left(-\infty, 0\right]\right\}
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AnalyticContinuation ⁣(log ⁣(z),z,a,b)=log ⁣(z)+πi\operatorname{AnalyticContinuation}\!\left(\log\!\left(z\right), z, a, b\right) = \log\!\left(-z\right) + \pi i
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AnalyticContinuation ⁣(log ⁣(z),z,a,b)=log ⁣(z)πi\operatorname{AnalyticContinuation}\!\left(\log\!\left(z\right), z, a, b\right) = \log\!\left(-z\right) - \pi i
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zeroszClog ⁣(z)={1}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \log\!\left(z\right) = \left\{1\right\}

Complex parts

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log ⁣(z)=log ⁣(z)\log\!\left(\overline{z}\right) = \overline{\log\!\left(z\right)}
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Re ⁣(log ⁣(z))=log ⁣(z)\operatorname{Re}\!\left(\log\!\left(z\right)\right) = \log\!\left(\left|z\right|\right)
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Im ⁣(log ⁣(z))=arg ⁣(z)\operatorname{Im}\!\left(\log\!\left(z\right)\right) = \arg\!\left(z\right)
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log ⁣(z)=(log ⁣(z))2+(arg ⁣(z))2\left|\log\!\left(z\right)\right| = \sqrt{{\left(\log\!\left(\left|z\right|\right)\right)}^{2} + {\left(\arg\!\left(z\right)\right)}^{2}}

Bounds and inequalities

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log ⁣(x)x1\log\!\left(x\right) \le x - 1
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log ⁣(z)log ⁣(z)+π\left|\log\!\left(z\right)\right| \le \left|\log\!\left(\left|z\right|\right)\right| + \pi
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log ⁣(x+a)log ⁣(x)log ⁣(1+axa)\left|\log\!\left(x + a\right) - \log\!\left(x\right)\right| \le \log\!\left(1 + \frac{\left|a\right|}{x - \left|a\right|}\right)

Integral representations

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log ⁣(z)=1z1tdt\log\!\left(z\right) = \int_{1}^{z} \frac{1}{t} \, dt
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1zdz=log ⁣(z)+C\int \frac{1}{z} \, dz = \log\!\left(z\right) + \mathcal{C}
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1zdz=log ⁣(z)+C\int \frac{1}{z} \, dz = \log\!\left(-z\right) + \mathcal{C}
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1xdx=log ⁣(x)+C\int \frac{1}{x} \, dx = \log\!\left(\left|x\right|\right) + \mathcal{C}

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-19 20:12:49.583742 UTC