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Complex numbers

Part of the Fungrim Identities reference — 40 identities for complex numbers.

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Contents​

Complex parts​

\arg(-\imaginaryI)=-(\frac{\pi}{2})

Used by the Compute Engine for simplification. 089f85 · Fungrim entry ↗

\mathrm{sgn}(x)=\begin{cases}1&0\lt x\\-1&x\lt0\\0&x=0\end{cases}

Holds when x\in\R. Used by the Compute Engine for simplification. 18d335 · Fungrim entry ↗

\mathrm{sgn}(z)=\frac{z}{\vert z\vert}

Holds when z\in\C\setminus\lbrace0\rbrace. Used by the Compute Engine for simplification. 26565c · Fungrim entry ↗

\Re(z)=\frac{z+z^\star}{2}

Holds when z\in\C. Used by the Compute Engine for simplification. 3866dc · Fungrim entry ↗

\vert x+\imaginaryI y\vert=\sqrt{x^2+y^2}

Holds when x\in\R\land y\in\R. Used by the Compute Engine for simplification. 4f0049 · Fungrim entry ↗

\mathrm{sgn}(z)=\exp(\imaginaryI\arg(z))

Holds when z\in\C\setminus\lbrace0\rbrace. Used by the Compute Engine for simplification. 54340e · Fungrim entry ↗

\mathrm{Csgn}(z)=\begin{cases}\mathrm{sgn}(\Im(z))&\Re(z)=0\\\mathrm{sgn}(\Re(z))&\top\end{cases}

Holds when z\in\C. Symbols: Csgn — Real-valued sign function for complex numbers. Used by the Compute Engine for simplification. 59a5d6 · Fungrim entry ↗

\arg(z)=-(\imaginaryI\ln(\mathrm{sgn}(z)))

Holds when z\in\C\setminus\lbrace0\rbrace. Used by the Compute Engine for simplification. 60772e · Fungrim entry ↗

\vert z^\star\vert=\vert z\vert

Holds when z\in\C. Used by the Compute Engine for simplification. 6a894d · Fungrim entry ↗

\arg(\imaginaryI)=\frac{\pi}{2}

Used by the Compute Engine for simplification. 735409 · Fungrim entry ↗

\arg(cz)=\arg(z)

Holds when z\in\C\setminus\lbrace0\rbrace\land c\in\lparen0, \infty\rparen. Used by the Compute Engine for simplification. 8cac46 · Fungrim entry ↗

\Re(x+\imaginaryI y)=x

Holds when x\in\R\land y\in\R. Used by the Compute Engine for simplification. 8e6867 · Fungrim entry ↗

\vert ab\vert=\vert a\vert\vert b\vert

Holds when a\in\C\land b\in\C. Used by the Compute Engine for simplification. 98efc1 · Fungrim entry ↗

\arg(-1)=\pi

Used by the Compute Engine for simplification. a8b41c · Fungrim entry ↗

(x+\imaginaryI y)^\star=x-\imaginaryI y

Holds when x\in\R\land y\in\R. Used by the Compute Engine for simplification. acda23 · Fungrim entry ↗

\arg(x+\imaginaryI y)=\mathrm{Arctan_2}(y, x)

Holds when x\in\R\land y\in\R. Used by the Compute Engine for simplification. b2a880 · Fungrim entry ↗

\Im(x+\imaginaryI y)=y

Holds when x\in\R\land y\in\R. Used by the Compute Engine for simplification. ba6d81 · Fungrim entry ↗

zz^\star=\vert z\vert^2

Holds when z\in\C. Used by the Compute Engine for simplification. bcd22f · Fungrim entry ↗

\arg(1)=0

Used by the Compute Engine for simplification. c423d2 · Fungrim entry ↗

\mathrm{Csgn}(z)=\frac{\sqrt{z^2}}{z}

Holds when z\in\C\setminus\lbrace0\rbrace. Symbols: Csgn — Real-valued sign function for complex numbers. Used by the Compute Engine for simplification. e9465d · Fungrim entry ↗

\Im(z)=\frac{z-z^\star}{2\imaginaryI}

Holds when z\in\C. Used by the Compute Engine for simplification. f1a29b · Fungrim entry ↗

Complex plane​

\mathrm{BernsteinEllipse}(\rho)=\lbrace\frac{1}{2}(\rho\exp(\imaginaryI\theta)+\frac{\exp(-(\imaginaryI\theta))}{\rho}), \theta\in\lbrack0, 2\pi\rparen\rbrace

Holds when 1\lt\rho\land\rho\in\R. Used by the Compute Engine for simplification. 40baa9 · Fungrim entry ↗

\mathrm{OpenDisk}(z, r)=\lbrace t, t\in\C\in\vert z-t\vert\lt r\rbrace

Holds when 0\lt r\land z\in\C\land r\in\R. Used by the Compute Engine for simplification. c98bad · Fungrim entry ↗

\mathrm{ClosedDisk}(z, r)=\lbrace t, t\in\C\in\vert z-t\vert\le r\rbrace

Holds when 0\le r\land z\in\C\land r\in\R. Used by the Compute Engine for simplification. d1cf0c · Fungrim entry ↗

Imaginary unit​

\mathrm{sgn}(\imaginaryI)=\imaginaryI

Used by the Compute Engine for simplification. 09c107 · Fungrim entry ↗

\imaginaryI^{z}=\imaginaryI\sin(\frac{\pi z}{2})+\cos(\frac{\pi z}{2})

Holds when z\in\C. Used by the Compute Engine for simplification. 15f92d · Fungrim entry ↗

\mathrm{PolyLog}(2, \imaginaryI)=\imaginaryI G-\frac{\pi^2}{48}

Used by the Compute Engine for simplification. 208da7 · Fungrim entry ↗

\Re(\imaginaryI)=0

Used by the Compute Engine for simplification. 249fd6 · Fungrim entry ↗

\imaginaryI^2=-1

Used by the Compute Engine for simplification. 31b0df · Fungrim entry ↗

\Im(\mathrm{Digamma}(\imaginaryI))=\frac{1}{2}(1+\pi\coth(\pi))

Used by the Compute Engine for simplification. 3ac0ce · Fungrim entry ↗

\imaginaryI^\star=-\imaginaryI

Used by the Compute Engine for simplification. 44ae4a · Fungrim entry ↗

\Im(\imaginaryI)=1

Used by the Compute Engine for simplification. 61784f · Fungrim entry ↗

\vert\imaginaryI\vert=1

Used by the Compute Engine for simplification. 65bbd6 · Fungrim entry ↗

\frac{1}{\imaginaryI}=-\imaginaryI

Used by the Compute Engine for simplification. 67c262 · Fungrim entry ↗

\imaginaryI^3=-\imaginaryI

Used by the Compute Engine for simplification. 8be138 · Fungrim entry ↗

\vert\Gamma(\imaginaryI)\vert=\sqrt{\frac{\pi}{\sinh(\pi)}}

Used by the Compute Engine for simplification. 9c93bb · Fungrim entry ↗

\imaginaryI^{\imaginaryI}=\exp(-(\frac{\pi}{2}))

Used by the Compute Engine for simplification. a39534 · Fungrim entry ↗

\imaginaryI^{n}=\begin{cases}1&\mathrm{CongruentMod}(n, 0, 4)\\\imaginaryI&\mathrm{CongruentMod}(n, 1, 4)\\-1&\mathrm{CongruentMod}(n, 2, 4)\\-\imaginaryI&\mathrm{CongruentMod}(n, 3, 4)\end{cases}

Holds when n\in\Z. Used by the Compute Engine for simplification. c12a41 · Fungrim entry ↗

\imaginaryI^4=1

Used by the Compute Engine for simplification. e0425a · Fungrim entry ↗

\imaginaryI^{z}=\exp(\frac{\imaginaryI\pi z}{2})

Holds when z\in\C. Used by the Compute Engine for simplification. f8a56f · Fungrim entry ↗