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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(z) Natural logarithm

Illustrations

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Image: X-ray of log(z)\log(z) 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(1) = 0
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log(e)=1\log(e) = 1
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Table of log(n)\log(n) to 50 digits for 1n501 \le n \le 50
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log(i)=πi2\log(i) = \frac{\pi i}{2}
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log(1)=πi\log(-1) = \pi i

Functional equations and connection formulas

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exp ⁣(log(z))=z\exp\!\left(\log(z)\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(z) = \log\!\left(\left|z\right|\right) + \arg(z) i
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log ⁣(cz)=log(c)+log(z)\log\!\left(c z\right) = \log(c) + \log(z)

Analytic properties

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log(z) is holomorphic on zC(,0]\log(z) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(-\infty, 0\right]
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poleszC{~}log(z)={}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \log(z) = \left\{\right\}
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EssentialSingularities ⁣(log(z),z,C{~})={}\operatorname{EssentialSingularities}\!\left(\log(z), 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(z), 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(z), z, \mathbb{C}\right) = \left\{\left(-\infty, 0\right]\right\}
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Continuationz:ablog(z)=log ⁣(b)+πi\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) + \pi i
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Continuationz:ablog(z)=log ⁣(b)πi\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) - \pi i
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Continuationz:(Reit,t:0θ)log(z)=log(R)+θi\mathop{\text{Continuation}}\limits_{\displaystyle{z: \left(R {e}^{i t},\, t : 0 \rightsquigarrow \theta\right)}} \, \log(z) = \log(R) + \theta i
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Continuationt:0θlog ⁣(Reit)=log(R)+θi\mathop{\text{Continuation}}\limits_{\displaystyle{t: 0 \rightsquigarrow \theta}} \, \log\!\left(R {e}^{i t}\right) = \log(R) + \theta i
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zeroszClog(z)={1}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \log(z) = \left\{1\right\}

Complex parts

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log ⁣(z)=log(z)\log\!\left(\overline{z}\right) = \overline{\log(z)}
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Re ⁣(log(z))=log ⁣(z)\operatorname{Re}\!\left(\log(z)\right) = \log\!\left(\left|z\right|\right)
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Im ⁣(log(z))=arg(z)\operatorname{Im}\!\left(\log(z)\right) = \arg(z)
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log(z)=log2 ⁣(z)+(arg(z))2\left|\log(z)\right| = \sqrt{\log^{2}\!\left(\left|z\right|\right) + {\left(\arg(z)\right)}^{2}}

Bounds and inequalities

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

Integral representations

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log(z)=1z1tdt\log(z) = \int_{1}^{z} \frac{1}{t} \, dt
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1zdz=log(z)+C\int \frac{1}{z} \, dz = \log(z) + \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.

2019-11-11 15:50:15.016492 UTC