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Imaginary unit

Table of contents: Definitions - Domain - Quadratic equations - Numerical value - Complex parts - Transformations - Special functions at this value

Definitions

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Symbol: ConstI ii Imaginary unit

Domain

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iCi \in \mathbb{C}
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iQi \in \overline{\mathbb{Q}}
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iRi \notin \mathbb{R}

Quadratic equations

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solutionsxC[x2+1=0]={i,i}\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[{x}^{2} + 1 = 0\right] = \left\{i, -i\right\}

Numerical value

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i=1i = \sqrt{-1}
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i=(1)1/2i = {\left(-1\right)}^{1 / 2}

Complex parts

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i=1\left|i\right| = 1
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Re(i)=0\operatorname{Re}(i) = 0
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Im(i)=1\operatorname{Im}(i) = 1
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arg(i)=π2\arg(i) = \frac{\pi}{2}
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arg ⁣(i)=π2\arg\!\left(-i\right) = -\frac{\pi}{2}
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sgn(i)=i\operatorname{sgn}(i) = i

Transformations

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i2=1{i}^{2} = -1
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i3=i{i}^{3} = -i
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i4=1{i}^{4} = 1
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in={1,n0(mod4)i,n1(mod4)1,n2(mod4)i,n3(mod4){i}^{n} = \begin{cases} 1, & n \equiv 0 \pmod {4}\\i, & n \equiv 1 \pmod {4}\\-1, & n \equiv 2 \pmod {4}\\-i, & n \equiv 3 \pmod {4}\\ \end{cases}
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i=i\overline{i} = -i
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1i=i\frac{1}{i} = -i
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iz=eπiz/2{i}^{z} = {e}^{\pi i z / 2}
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iz=cos ⁣(π2z)+sin ⁣(π2z)i{i}^{z} = \cos\!\left(\frac{\pi}{2} z\right) + \sin\!\left(\frac{\pi}{2} z\right) i
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i=12(1+i)\sqrt{i} = \frac{1}{\sqrt{2}} \left(1 + i\right)
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ii=eπ/2{i}^{i} = {e}^{-\pi / 2}

Special functions at this value

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log(i)=πi2\log(i) = \frac{\pi i}{2}
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Γ(i)=πsinh(π)\left|\Gamma(i)\right| = \sqrt{\frac{\pi}{\sinh(\pi)}}
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Im ⁣(ψ ⁣(i))=12(πcoth(π)+1)\operatorname{Im}\!\left(\psi\!\left(i\right)\right) = \frac{1}{2} \left(\pi \coth(\pi) + 1\right)
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Li2 ⁣(i)=π248+Gi\operatorname{Li}_{2}\!\left(i\right) = -\frac{{\pi}^{2}}{48} + G i

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