# Complex numbers Part of the [Fungrim Identities](/compute-engine/reference/fungrim/) reference — **40 identities** for complex numbers. :::info[Generated reference] This page is generated from the compiled Fungrim artifact by `scripts/fungrim/gen-reference-doc.ts` (upstream snapshot `3a299164c683`, translator `grim2mathjson 0.1.0`). Do not edit it by hand. The corpus is MIT-licensed; see `data/fungrim/LICENSE`. ::: ## Contents - [Complex parts](#complex-parts) (21) - [Complex plane](#complex-plane) (3) - [Imaginary unit](#imaginary-unit) (16) ## Complex parts $$\arg(-\imaginaryI)=-(\frac{\pi}{2})$$ Used by the Compute Engine for simplification. [`089f85` · Fungrim entry ↗](https://fungrim.org/entry/089f85) --- $$\mathrm{sgn}(x)=\begin{cases}1&x\gt0\\-1&x\lt0\\0&x=0\end{cases}$$ **Holds when** $x\in\R$. Used by the Compute Engine for simplification. [`18d335` · Fungrim entry ↗](https://fungrim.org/entry/18d335) --- $$\mathrm{sgn}(z)=\frac{z}{\vert z\vert}$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`26565c` · Fungrim entry ↗](https://fungrim.org/entry/26565c) --- $$\Re(z)=\frac{z+z^\star}{2}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`3866dc` · Fungrim entry ↗](https://fungrim.org/entry/3866dc) --- $$\vert x+y\imaginaryI\vert=\sqrt{x^2+y^2}$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`4f0049` · Fungrim entry ↗](https://fungrim.org/entry/4f0049) --- $$\mathrm{sgn}(z)=\exp(\imaginaryI\arg(z))$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`54340e` · Fungrim entry ↗](https://fungrim.org/entry/54340e) --- $$\mathrm{Csgn}(z)=\begin{cases}\mathrm{sgn}(\Im(z))&\Re(z)=0\\\mathrm{sgn}(\Re(z))&\top\end{cases}$$ **Holds when** $z\in\C$. **Symbols:** **Csgn** — Real-valued sign function for complex numbers. Used by the Compute Engine for simplification. [`59a5d6` · Fungrim entry ↗](https://fungrim.org/entry/59a5d6) --- $$\arg(z)=-(\imaginaryI\ln(\mathrm{sgn}(z)))$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`60772e` · Fungrim entry ↗](https://fungrim.org/entry/60772e) --- $$\vert z^\star\vert=\vert z\vert$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`6a894d` · Fungrim entry ↗](https://fungrim.org/entry/6a894d) --- $$\arg(\imaginaryI)=\frac{\pi}{2}$$ Used by the Compute Engine for simplification. [`735409` · Fungrim entry ↗](https://fungrim.org/entry/735409) --- $$\arg(cz)=\arg(z)$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace\land c\in\lparen0, \infty\rparen$. Used by the Compute Engine for simplification. [`8cac46` · Fungrim entry ↗](https://fungrim.org/entry/8cac46) --- $$\Re(x+y\imaginaryI)=x$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`8e6867` · Fungrim entry ↗](https://fungrim.org/entry/8e6867) --- $$\vert ab\vert=\vert a\vert\vert b\vert$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`98efc1` · Fungrim entry ↗](https://fungrim.org/entry/98efc1) --- $$\arg(-1)=\pi$$ Used by the Compute Engine for simplification. [`a8b41c` · Fungrim entry ↗](https://fungrim.org/entry/a8b41c) --- $$(x+y\imaginaryI)^\star=x-y\imaginaryI$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`acda23` · Fungrim entry ↗](https://fungrim.org/entry/acda23) --- $$\arg(x+y\imaginaryI)=\mathrm{Arctan_2}(y, x)$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`b2a880` · Fungrim entry ↗](https://fungrim.org/entry/b2a880) --- $$\Im(x+y\imaginaryI)=y$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`ba6d81` · Fungrim entry ↗](https://fungrim.org/entry/ba6d81) --- $$zz^\star=\vert z\vert^2$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`bcd22f` · Fungrim entry ↗](https://fungrim.org/entry/bcd22f) --- $$\arg(1)=0$$ Used by the Compute Engine for simplification. [`c423d2` · Fungrim entry ↗](https://fungrim.org/entry/c423d2) --- $$\mathrm{Csgn}(z)=\frac{\sqrt{z^2}}{z}$$ **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 ↗](https://fungrim.org/entry/e9465d) --- $$\Im(z)=\frac{z-z^\star}{2\imaginaryI}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`f1a29b` · Fungrim entry ↗](https://fungrim.org/entry/f1a29b) --- ## Complex plane $$\mathrm{BernsteinEllipse}(\rho)=\mathrm{Map}(\lbrack0, 2\pi\rparen, \theta\mapsto\frac{1}{2}(\rho\exp(\imaginaryI\theta)+\exp(-(\imaginaryI\theta))/\rho))$$ **Holds when** $\rho\in\R\land\rho\gt1$. Used by the Compute Engine for simplification. [`40baa9` · Fungrim entry ↗](https://fungrim.org/entry/40baa9) --- $$\mathrm{OpenDisk}(z, r)=\mathrm{Filter}(\C, t\mapsto\vert z-t\vert\lt r)$$ **Holds when** $z\in\C\land r\in\R\land r\gt0$. Used by the Compute Engine for simplification. [`c98bad` · Fungrim entry ↗](https://fungrim.org/entry/c98bad) --- $$\mathrm{ClosedDisk}(z, r)=\mathrm{Filter}(\C, t\mapsto\vert z-t\vert\le r)$$ **Holds when** $z\in\C\land r\in\R\land r\ge0$. Used by the Compute Engine for simplification. [`d1cf0c` · Fungrim entry ↗](https://fungrim.org/entry/d1cf0c) --- ## Imaginary unit $$\mathrm{sgn}(\imaginaryI)=\imaginaryI$$ Used by the Compute Engine for simplification. [`09c107` · Fungrim entry ↗](https://fungrim.org/entry/09c107) --- $$\imaginaryI^{z}=\cos(\frac{\pi z}{2})+\sin(\frac{\pi z}{2})\imaginaryI$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`15f92d` · Fungrim entry ↗](https://fungrim.org/entry/15f92d) --- $$\operatorname{Li}_{2}(\imaginaryI)=G\imaginaryI-\frac{\pi^2}{48}$$ Used by the Compute Engine for simplification. [`208da7` · Fungrim entry ↗](https://fungrim.org/entry/208da7) --- $$\Re(\imaginaryI)=0$$ Used by the Compute Engine for simplification. [`249fd6` · Fungrim entry ↗](https://fungrim.org/entry/249fd6) --- $$\imaginaryI^2=-1$$ Used by the Compute Engine for simplification. [`31b0df` · Fungrim entry ↗](https://fungrim.org/entry/31b0df) --- $$\Im(\mathrm{Digamma}(\imaginaryI))=\frac{1}{2}(\pi\coth(\pi)+1)$$ Used by the Compute Engine for simplification. [`3ac0ce` · Fungrim entry ↗](https://fungrim.org/entry/3ac0ce) --- $$\imaginaryI^\star=-\imaginaryI$$ Used by the Compute Engine for simplification. [`44ae4a` · Fungrim entry ↗](https://fungrim.org/entry/44ae4a) --- $$\Im(\imaginaryI)=1$$ Used by the Compute Engine for simplification. [`61784f` · Fungrim entry ↗](https://fungrim.org/entry/61784f) --- $$\vert\imaginaryI\vert=1$$ Used by the Compute Engine for simplification. [`65bbd6` · Fungrim entry ↗](https://fungrim.org/entry/65bbd6) --- $$\frac{1}{\imaginaryI}=-\imaginaryI$$ Used by the Compute Engine for simplification. [`67c262` · Fungrim entry ↗](https://fungrim.org/entry/67c262) --- $$\imaginaryI^3=-\imaginaryI$$ Used by the Compute Engine for simplification. [`8be138` · Fungrim entry ↗](https://fungrim.org/entry/8be138) --- $$\vert\Gamma(\imaginaryI)\vert=\sqrt{\frac{\pi}{\sinh(\pi)}}$$ Used by the Compute Engine for simplification. [`9c93bb` · Fungrim entry ↗](https://fungrim.org/entry/9c93bb) --- $$\imaginaryI^{\imaginaryI}=\exp(-(\frac{\pi}{2}))$$ Used by the Compute Engine for simplification. [`a39534` · Fungrim entry ↗](https://fungrim.org/entry/a39534) --- $$\imaginaryI^{n}=\begin{cases}1&\mathrm{CongruentMod}(n, 0, 4)\\\imaginaryI&\mathrm{CongruentMod}(n, 1, 4)\\-1&\mathrm{CongruentMod}(n, 2, 4)\\-\imaginaryI&\mathrm{CongruentMod}(n, 3, 4)\end{cases}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`c12a41` · Fungrim entry ↗](https://fungrim.org/entry/c12a41) --- $$\imaginaryI^4=1$$ Used by the Compute Engine for simplification. [`e0425a` · Fungrim entry ↗](https://fungrim.org/entry/e0425a) --- $$\imaginaryI^{z}=\exp(\frac{\pi\imaginaryI z}{2})$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`f8a56f` · Fungrim entry ↗](https://fungrim.org/entry/f8a56f) ---