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Exponential function

Table of contents: Definitions - Illustrations - Particular values - Functional equations and connection formulas - Analytic properties - Complex parts - Taylor series - Derivatives and integrals - Approximations - Bounds and inequalities

Definitions

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Symbol: Exp ez{e}^{z} Exponential function
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Symbol: ConstE ee The constant e (2.718...)

Illustrations

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Image: X-ray of ez{e}^{z} on z[5,5]+[5,5]iz \in \left[-5, 5\right] + \left[-5, 5\right] i

Particular values

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e0=1{e}^{0} = 1
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e1=e{e}^{1} = e
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eπi=1{e}^{\pi i} = -1
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eπi/2=i{e}^{\pi i / 2} = i

Functional equations and connection formulas

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ea+b=eaeb{e}^{a + b} = {e}^{a} {e}^{b}
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(ez)n=enz{\left({e}^{z}\right)}^{n} = {e}^{n z}
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ez=1ez{e}^{-z} = \frac{1}{{e}^{z}}
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ea+bi=ea(cos ⁣(b)+sin ⁣(b)i){e}^{a + b i} = {e}^{a} \left(\cos\!\left(b\right) + \sin\!\left(b\right) i\right)
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ez+nπi=(1)nez{e}^{z + n \pi i} = {\left(-1\right)}^{n} {e}^{z}
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ez+2nπi=ez{e}^{z + 2 n \pi i} = {e}^{z}
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ez=cosh ⁣(z)+sinh ⁣(z){e}^{z} = \cosh\!\left(z\right) + \sinh\!\left(z\right)
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eiz=cos ⁣(z)+isin ⁣(z){e}^{i z} = \cos\!\left(z\right) + i \sin\!\left(z\right)

Analytic properties

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HolomorphicDomain ⁣(ez,z,C{~})=C\operatorname{HolomorphicDomain}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C}
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Poles ⁣(ez,z,C{~})={}\operatorname{Poles}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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EssentialSingularities ⁣(ez,z,C{~})={~}\operatorname{EssentialSingularities}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}
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BranchPoints ⁣(ez,z,C{~})={}\operatorname{BranchPoints}\!\left({e}^{z}, z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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BranchCuts ⁣(ez,z,C)={}\operatorname{BranchCuts}\!\left({e}^{z}, z, \mathbb{C}\right) = \left\{\right\}
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zeroszCez={}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} {e}^{z} = \left\{\right\}

Complex parts

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ez=exp ⁣(Re ⁣(z))\left|{e}^{z}\right| = \exp\!\left(\operatorname{Re}\!\left(z\right)\right)
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sgn ⁣(ez)=exp ⁣(Im ⁣(z)i)\operatorname{sgn}\!\left({e}^{z}\right) = \exp\!\left(\operatorname{Im}\!\left(z\right) i\right)
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Re ⁣(ez)=exp ⁣(Re ⁣(z))cos ⁣(Im ⁣(z))\operatorname{Re}\!\left({e}^{z}\right) = \exp\!\left(\operatorname{Re}\!\left(z\right)\right) \cos\!\left(\operatorname{Im}\!\left(z\right)\right)
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Im ⁣(ez)=exp ⁣(Re ⁣(z))sin ⁣(Im ⁣(z))\operatorname{Im}\!\left({e}^{z}\right) = \exp\!\left(\operatorname{Re}\!\left(z\right)\right) \sin\!\left(\operatorname{Im}\!\left(z\right)\right)
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arg ⁣(ez)=Im ⁣(z)\arg\!\left({e}^{z}\right) = \operatorname{Im}\!\left(z\right)
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exp ⁣(z)=ez\exp\!\left(\overline{z}\right) = \overline{{e}^{z}}

Taylor series

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ez=k=0zkk!{e}^{z} = \sum_{k=0}^{\infty} \frac{{z}^{k}}{k !}
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ec+z=eck=0zkk!{e}^{c + z} = {e}^{c} \sum_{k=0}^{\infty} \frac{{z}^{k}}{k !}

Derivatives and integrals

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abezdz=ebea\int_{a}^{b} {e}^{z} \, dz = {e}^{b} - {e}^{a}
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ddzez=ez\frac{d}{d z}\, {e}^{z} = {e}^{z}
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dndznez=ez\frac{d^{n}}{{d z}^{n}} {e}^{z} = {e}^{z}

Approximations

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ezk=0N1zkk!zNN!(1zN)\left|{e}^{z} - \sum_{k=0}^{N - 1} \frac{{z}^{k}}{k !}\right| \le \frac{{\left|z\right|}^{N}}{N ! \left(1 - \frac{\left|z\right|}{N}\right)}

Bounds and inequalities

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ezez\left|{e}^{z}\right| \le {e}^{\left|z\right|}
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ex+aexex(ea1)\left|{e}^{x + a} - {e}^{x}\right| \le {e}^{x} \left({e}^{\left|a\right|} - 1\right)

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

2019-08-21 11:44:15.926409 UTC