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Exponential function

Table of contents: Definitions - Illustrations - Domain and range - 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: Plot of ex{e}^{x} on x[4,4]x \in \left[-4, 4\right]
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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

Domain and range

Numbers

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z{0}        ez{1}z \in \left\{0\right\} \;\implies\; {e}^{z} \in \left\{1\right\}
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zR        ez(0,)z \in \mathbb{R} \;\implies\; {e}^{z} \in \left(0, \infty\right)
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zC        ezC{0}z \in \mathbb{C} \;\implies\; {e}^{z} \in \mathbb{C} \setminus \left\{0\right\}

Infinities

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z{}        ez{}z \in \left\{\infty\right\} \;\implies\; {e}^{z} \in \left\{\infty\right\}
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z{}        ez{0}z \in \left\{-\infty\right\} \;\implies\; {e}^{z} \in \left\{0\right\}

Formal power series

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(zQ[[x]]  and  [x0]z=0)    (ezQ[[x]]  and  [x0]ez=1)\left(z \in \mathbb{Q}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z = 0\right) \implies \left({e}^{z} \in \mathbb{Q}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] {e}^{z} = 1\right)
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(zR[[x]])    (ezR[[x]]  and  [x0]ez0)\left(z \in \mathbb{R}[[x]]\right) \implies \left({e}^{z} \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] {e}^{z} \ne 0\right)
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(zC[[x]])    (ezC[[x]]  and  [x0]ez0)\left(z \in \mathbb{C}[[x]]\right) \implies \left({e}^{z} \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] {e}^{z} \ne 0\right)

Matrices

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AMn×n ⁣(R)        eAGLn ⁣(R)A \in \operatorname{M}_{n \times n}\!\left(\mathbb{R}\right) \;\implies\; {e}^{A} \in \operatorname{GL}_{n}\!\left(\mathbb{R}\right)
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AMn×n ⁣(C)        eAGLn ⁣(C)A \in \operatorname{M}_{n \times n}\!\left(\mathbb{C}\right) \;\implies\; {e}^{A} \in \operatorname{GL}_{n}\!\left(\mathbb{C}\right)

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
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eQe \notin \mathbb{Q}
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eQe \notin \overline{\mathbb{Q}}
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eαQ   for all αQ{0}{e}^{\alpha} \notin \mathbb{Q} \;\text{ for all } \alpha \in \mathbb{Q} \setminus \left\{0\right\}
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eαQ   for all αQ{0}{e}^{\alpha} \notin \overline{\mathbb{Q}} \;\text{ for all } \alpha \in \overline{\mathbb{Q}} \setminus \left\{0\right\}

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(b) + \sin(b) 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(z) + \sinh(z)
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eiz=cos(z)+isin(z){e}^{i z} = \cos(z) + i \sin(z)
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elog(z)=z{e}^{\log(z)} = z
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(Im(z)(π,π])    (log ⁣(ez)=z)\left(\operatorname{Im}(z) \in \left(-\pi, \pi\right]\right) \implies \left(\log\!\left({e}^{z}\right) = z\right)

Analytic properties

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ez is holomorphic on zC{e}^{z} \text{ is holomorphic on } z \in \mathbb{C}
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poleszC{~}ez={}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} {e}^{z} = \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=eRe(z)\left|{e}^{z}\right| = {e}^{\operatorname{Re}(z)}
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sgn ⁣(ez)=eIm(z)i\operatorname{sgn}\!\left({e}^{z}\right) = {e}^{\operatorname{Im}(z) i}
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Re ⁣(ez)=eRe(z)cos ⁣(Im(z))\operatorname{Re}\!\left({e}^{z}\right) = {e}^{\operatorname{Re}(z)} \cos\!\left(\operatorname{Im}(z)\right)
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Im ⁣(ez)=eRe(z)sin ⁣(Im(z))\operatorname{Im}\!\left({e}^{z}\right) = {e}^{\operatorname{Re}(z)} \sin\!\left(\operatorname{Im}(z)\right)
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arg ⁣(ez)=Im(z)\arg\!\left({e}^{z}\right) = \operatorname{Im}(z)
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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.

2020-04-08 16:14:44.404316 UTC