# Elementary functions Part of the [Fungrim Identities](/compute-engine/reference/fungrim/) reference — **212 identities** for elementary functions. :::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 - [Exponential function](#exponential-function) (16) - [Golden ratio](#golden-ratio) (5) - [Inverse tangent](#inverse-tangent) (44) - [Lambert W-function](#lambert-w-function) (15) - [Natural logarithm](#natural-logarithm) (11) - [Pi](#pi) (4) - [Powers](#powers) (8) - [Sinc function](#sinc-function) (24) - [Sine](#sine) (59) - [Square roots](#square-roots) (26) ## Exponential function $$\exponentialE^{z}=\cosh(z)+\sinh(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`1568e1` · Fungrim entry ↗](https://fungrim.org/entry/1568e1) --- $$\vert\exponentialE^{z}\vert=\exp(\Re(z))$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`1b3014` · Fungrim entry ↗](https://fungrim.org/entry/1b3014) --- $$\exp(z+2n\pi\imaginaryI)=\exponentialE^{z}$$ **Holds when** $z\in\C\land n\in\Z$. Used by the Compute Engine for simplification. [`1fa6b7` · Fungrim entry ↗](https://fungrim.org/entry/1fa6b7) --- $$\exp(\ln(z))=z$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification and equation solving. [`296627` · Fungrim entry ↗](https://fungrim.org/entry/296627) --- $$z\mapsto\exponentialE^{z}^{\prime}(z)=\exponentialE^{z}$$ **Holds when** $z\in\C\land n\in\N$. Used by the Compute Engine for simplification. [`4491b8` · Fungrim entry ↗](https://fungrim.org/entry/4491b8) --- $$\exp(z^\star)=\exponentialE^{z}^\star$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`52d827` · Fungrim entry ↗](https://fungrim.org/entry/52d827) --- $$\exp(\pi\imaginaryI)=-1$$ Used by the Compute Engine for simplification. [`54aaf1` · Fungrim entry ↗](https://fungrim.org/entry/54aaf1) --- $$\exp(a+b\imaginaryI)=\exponentialE^{a}(\cos(b)+\sin(b)\imaginaryI)$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`77d6bf` · Fungrim entry ↗](https://fungrim.org/entry/77d6bf) --- $$\exp(a+b)=\exponentialE^{a}\exponentialE^{b}$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`812707` · Fungrim entry ↗](https://fungrim.org/entry/812707) --- $$z\mapsto\exponentialE^{z}^{\prime}(z)=\exponentialE^{z}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`96af56` · Fungrim entry ↗](https://fungrim.org/entry/96af56) --- $$\exp(z+n\pi\imaginaryI)=(-1)^{n}\exponentialE^{z}$$ **Holds when** $z\in\C\land n\in\Z$. Used by the Compute Engine for simplification. [`97ba8d` · Fungrim entry ↗](https://fungrim.org/entry/97ba8d) --- $$\arg(\exponentialE^{z})=\Im(z)$$ **Holds when** $z\in\C\land\Im(z)\in\lparen-\pi, \pi\rbrack$. Used by the Compute Engine for expansion. [`a0d93c` · Fungrim entry ↗](https://fungrim.org/entry/a0d93c) --- $$\exp(\frac{\pi\imaginaryI}{2})=\imaginaryI$$ Used by the Compute Engine for simplification. [`a90f35` · Fungrim entry ↗](https://fungrim.org/entry/a90f35) --- $$\Re(\exponentialE^{z})=\exp(\Re(z))\cos(\Im(z))$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`b7d62b` · Fungrim entry ↗](https://fungrim.org/entry/b7d62b) --- $$\mathrm{sgn}(\exponentialE^{z})=\exp(\Im(z)\imaginaryI)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`caf706` · Fungrim entry ↗](https://fungrim.org/entry/caf706) --- $$\Im(\exponentialE^{z})=\exp(\Re(z))\sin(\Im(z))$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`e2fac7` · Fungrim entry ↗](https://fungrim.org/entry/e2fac7) --- ## Golden ratio $$\varphi^{n+1}=\varphi^{n}+\varphi^{n-1}$$ **Holds when** $n\in\C$. Used by the Compute Engine for simplification. [`0cd1a4` · Fungrim entry ↗](https://fungrim.org/entry/0cd1a4) --- $$\frac{1}{\varphi}=\varphi-1$$ Used by the Compute Engine for simplification. [`31f52c` · Fungrim entry ↗](https://fungrim.org/entry/31f52c) --- $$\varphi^{n}=\mathrm{Fibonacci}(n)\varphi+\mathrm{Fibonacci}(n-1)$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`6a11ce` · Fungrim entry ↗](https://fungrim.org/entry/6a11ce) --- $$\varphi^2-\varphi-1=0$$ Used by the Compute Engine for simplification. [`b464d3` · Fungrim entry ↗](https://fungrim.org/entry/b464d3) --- $$\mathrm{Spectrum}(\begin{pmatrix}1 & 1\\ 1 & 0\end{pmatrix})=\lbrace\varphi, 1-\varphi\rbrace$$ Used by the Compute Engine for simplification. [`ebfcd8` · Fungrim entry ↗](https://fungrim.org/entry/ebfcd8) --- ## Inverse tangent $$\arctan(x)-\arctan(y)=\mathrm{Arctan_2}(x-y, 1+xy)$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for expansion. [`00e608` · Fungrim entry ↗](https://fungrim.org/entry/00e608) --- $$\arctan(\imaginaryI z)=\imaginaryI\mathrm{artanh}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`072166` · Fungrim entry ↗](https://fungrim.org/entry/072166) --- $$\cos(\arctan(z))=\frac{1}{\sqrt{1+z^2}}$$ **Holds when** $z\in\C\setminus\lbrace-\imaginaryI, \imaginaryI\rbrace$. Used by the Compute Engine for simplification. [`0b829e` · Fungrim entry ↗](https://fungrim.org/entry/0b829e) --- $$\arctan(-z)=-\arctan(z)$$ **Holds when** $z\in\C\cup\lbrace-\infty, \infty\rbrace$. Used by the Compute Engine for expansion. [`0ee626` · Fungrim entry ↗](https://fungrim.org/entry/0ee626) --- $$\arctan(z)=-\frac{\imaginaryI}{2}\ln(\frac{1+\imaginaryI z}{1-\imaginaryI z})$$ **Holds when** $z\in\C\land z\imaginaryI\notin\lbrack1, \infty\rparen$. Used by the Compute Engine for simplification. [`12765e` · Fungrim entry ↗](https://fungrim.org/entry/12765e) --- $$\arctan(1)=\frac{\pi}{4}$$ Used by the Compute Engine for simplification. [`157c6c` · Fungrim entry ↗](https://fungrim.org/entry/157c6c) --- $$\tan(\arctan(z))=z$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification and equation solving. [`1f026d` · Fungrim entry ↗](https://fungrim.org/entry/1f026d) --- $$\mathrm{Arctan_2}(y, x)=\begin{cases}0&x=y=0\\\arctan(\frac{y}{x})&x\gt0\\\frac{\pi\mathrm{sgn}(y)}{2}-\arctan(x/y)&y\ne0\\\pi&y=0\land x\lt0\end{cases}$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`22fb4a` · Fungrim entry ↗](https://fungrim.org/entry/22fb4a) --- $$\arctan(x+y)=\arctan(x)+\arctan((y)(1+x(x+y))^{-1})$$ **Holds when** $x\in\C\land y\in\C\land\vert x+y\vert\lt1\land\vert x\vert\lt1$  _or_  $x\in\R\land y\in\R\land x(x+y)\gt-1$. Used by the Compute Engine for simplification. [`268c9e` · Fungrim entry ↗](https://fungrim.org/entry/268c9e) --- $$\arctan(z)=z\mathrm{Hypergeometric2F_1}(1, \frac{1}{2}, \frac{3}{2}, -z^2)$$ **Holds when** $z\in\C\setminus\lbrace-\imaginaryI, \imaginaryI\rbrace$. Used by the Compute Engine for simplification. [`34ff28` · Fungrim entry ↗](https://fungrim.org/entry/34ff28) --- $$z\mapsto\arctan(z)^{\prime}(z)=\frac{(-1)^{n}(n-1)!(\frac{1}{(z+\imaginaryI)^{n}}-\frac{1}{(z-\imaginaryI)^{n}})}{2\imaginaryI}$$ **Holds when** $n\in\N^*\land z\in\C\land\imaginaryI z\notin\lparen-\infty, -1\rbrack\cup\lbrack1, \infty\rparen$. Used by the Compute Engine for simplification. [`36171f` · Fungrim entry ↗](https://fungrim.org/entry/36171f) --- $$\arctan(\frac{1}{\sqrt{3}})=\frac{\pi}{6}$$ Used by the Compute Engine for simplification. [`3c1021` · Fungrim entry ↗](https://fungrim.org/entry/3c1021) --- $$\arctan(x)+\arctan(y)=\arctan(\frac{x+y}{1-xy})$$ **Holds when** $x\in\C\land y\in\C\land\vert x\vert\lt1\land\vert y\vert\lt1$  _or_  $x\in\R\land y\in\R\land xy\lt1$. Used by the Compute Engine for simplification. [`3ea11b` · Fungrim entry ↗](https://fungrim.org/entry/3ea11b) --- $$\vert\arctan(x+y)-\arctan(x)\vert=\mathrm{Arctan_2}(\vert y\vert, 1+x(x+y))$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`47331d` · Fungrim entry ↗](https://fungrim.org/entry/47331d) --- $$\arctan(z)=\frac{1}{2}(\imaginaryI\ln(\frac{1-\imaginaryI z}{1+\imaginaryI z}))$$ **Holds when** $z\in\C\land-z\imaginaryI\notin\lbrack1, \infty\rparen$. Used by the Compute Engine for simplification. [`500c0a` · Fungrim entry ↗](https://fungrim.org/entry/500c0a) --- $$\arctan(x)-\arctan(y)=\arctan(\frac{x-y}{1+xy})$$ **Holds when** $x\in\C\land y\in\C\land\vert x\vert\lt1\land\vert y\vert\lt1$  _or_  $x\in\R\land y\in\R\land xy\gt-1$. Used by the Compute Engine for simplification. [`503d4d` · Fungrim entry ↗](https://fungrim.org/entry/503d4d) --- $$\arctan(z^\star)=\arctan(z)^\star$$ **Holds when** $z\in\C\land\imaginaryI z\notin\lparen-\infty, -1\rparen\cup\lparen1, \infty\rparen$. Used by the Compute Engine for expansion. [`632063` · Fungrim entry ↗](https://fungrim.org/entry/632063) --- $$\arctan(0)=0$$ Used by the Compute Engine for simplification. [`645e30` · Fungrim entry ↗](https://fungrim.org/entry/645e30) --- $$\arctan(z)=2\arctan((z)(1+\sqrt{1+z^2})^{-1})$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`67c0be` · Fungrim entry ↗](https://fungrim.org/entry/67c0be) --- $$\arctan(\sqrt{3})=\frac{\pi}{3}$$ Used by the Compute Engine for simplification. [`706783` · Fungrim entry ↗](https://fungrim.org/entry/706783) --- $$\arctan(-\infty)=-(\frac{\pi}{2})$$ Used by the Compute Engine for simplification. [`7295b5` · Fungrim entry ↗](https://fungrim.org/entry/7295b5) --- $$\mathrm{Arctan_2}(y, 0)=\frac{\pi\mathrm{sgn}(y)}{2}$$ **Holds when** $y\in\R$. Used by the Compute Engine for simplification. [`77e519` · Fungrim entry ↗](https://fungrim.org/entry/77e519) --- $$\arctan(z)=\arcsin(\frac{z}{\sqrt{1+z^2}})$$ **Holds when** $z\in\C\land\imaginaryI z\notin\lparen-\infty, -1\rbrack\cup\lbrack1, \infty\rparen$. Used by the Compute Engine for simplification. [`7954ad` · Fungrim entry ↗](https://fungrim.org/entry/7954ad) --- $$\arctan(2-\sqrt{3})=\frac{\pi}{12}$$ Used by the Compute Engine for simplification. [`7dd050` · Fungrim entry ↗](https://fungrim.org/entry/7dd050) --- $$z\mapsto\arctan(z)^{\prime}(z)=\frac{1}{1+z^2}$$ **Holds when** $z\in\C\land\imaginaryI z\notin\lparen-\infty, -1\rbrack\cup\lbrack1, \infty\rparen$. Used by the Compute Engine for simplification. [`8fbf69` · Fungrim entry ↗](https://fungrim.org/entry/8fbf69) --- $$z\mapsto\arctan(z)^{\prime}(z)=\frac{(n-1)!\mathrm{ChebyshevU}(n-1, -(z/(z^2+1)^{1/2}))}{{(z^2+1)}^{\frac{n+1}{2}}}$$ **Holds when** $n\in\N^*\land z\in\C\land\imaginaryI z\notin\lparen-\infty, -1\rbrack\cup\lbrack1, \infty\rparen$. **Symbols:** **ChebyshevU** — Chebyshev polynomial of the second kind. Used by the Compute Engine for simplification. **Reference:** M. A. Boutiche and M. Rahmani (2017), On the higher derivatives of the inverse tangent function, [https://arxiv.org/abs/1712.03521,](https://arxiv.org/abs/1712.03521,) Theorem 9 [`90631b` · Fungrim entry ↗](https://fungrim.org/entry/90631b) --- $$\arctan(-\imaginaryI)=-\imaginaryI\infty$$ Used by the Compute Engine for simplification. [`9b0994` · Fungrim entry ↗](https://fungrim.org/entry/9b0994) --- $$\mathrm{Arctan_2}(y, x)=-\imaginaryI\ln(\mathrm{sgn}(x+y\imaginaryI))$$ **Holds when** $x\in\R\land y\in\R\land x+y\imaginaryI\ne0$. Used by the Compute Engine for simplification. [`9dec3e` · Fungrim entry ↗](https://fungrim.org/entry/9dec3e) --- $$\arctan(z)=\frac{1}{2}(\imaginaryI(\ln(1-\imaginaryI z)-\ln(1+\imaginaryI z)))$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`a18b77` · Fungrim entry ↗](https://fungrim.org/entry/a18b77) --- $$\arctan(\imaginaryI)=\imaginaryI\infty$$ Used by the Compute Engine for simplification. [`a2d208` · Fungrim entry ↗](https://fungrim.org/entry/a2d208) --- $$z\mapsto\arctan(z)^{\doubleprime}(z)=-(\frac{2z}{(1+z^2)^2})$$ **Holds when** $z\in\C\land\imaginaryI z\notin\lparen-\infty, -1\rbrack\cup\lbrack1, \infty\rparen$. Used by the Compute Engine for simplification. [`a4eb86` · Fungrim entry ↗](https://fungrim.org/entry/a4eb86) --- $$\mathrm{Arctan_2}(0, x)=\begin{cases}0&x\ge0\\\pi&x\lt0\end{cases}$$ **Holds when** $x\in\R$. Used by the Compute Engine for simplification. [`a6776b` · Fungrim entry ↗](https://fungrim.org/entry/a6776b) --- $$\arctan(\sqrt{2}-1)=\frac{\pi}{8}$$ Used by the Compute Engine for simplification. [`a9ecff` · Fungrim entry ↗](https://fungrim.org/entry/a9ecff) --- $$\arctan(2+\sqrt{3})=\frac{5\pi}{12}$$ Used by the Compute Engine for simplification. [`b0049f` · Fungrim entry ↗](https://fungrim.org/entry/b0049f) --- $$\Im(\arctan(x+y\imaginaryI))=\frac{1}{4}(\ln(\frac{x^2+(1+y)^2}{x^2+(1-y)^2}))$$ **Holds when** $x\in\R\land y\in\R\land x+y\imaginaryI\notin\lbrace-\imaginaryI, \imaginaryI\rbrace$. Used by the Compute Engine for simplification. [`b65d19` · Fungrim entry ↗](https://fungrim.org/entry/b65d19) --- $$\arctan(\frac{1}{z})=\frac{1}{2}(\pi\mathrm{Csgn}(1/z))-\arctan(z)$$ **Holds when** $z\in\C\land\imaginaryI z\notin\lbrace0\rbrace\cup\lparen-\infty, -1\rbrack\cup\lbrack1, \infty\rparen$. **Symbols:** **Csgn** — Real-valued sign function for complex numbers. Used by the Compute Engine for simplification. [`bfc13f` · Fungrim entry ↗](https://fungrim.org/entry/bfc13f) --- $$\arctan(z)=\mathrm{arcctg}(\frac{1}{z})$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`c580f4` · Fungrim entry ↗](https://fungrim.org/entry/c580f4) --- $$\arctan(\sqrt{2}+1)=\frac{3\pi}{8}$$ Used by the Compute Engine for simplification. [`c6c92a` · Fungrim entry ↗](https://fungrim.org/entry/c6c92a) --- $$\arctan(x)+\arctan(y)=\mathrm{Arctan_2}(x+y, 1-xy)$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`cf64b3` · Fungrim entry ↗](https://fungrim.org/entry/cf64b3) --- $$\arctan(\infty)=\frac{\pi}{2}$$ Used by the Compute Engine for simplification. [`d418d3` · Fungrim entry ↗](https://fungrim.org/entry/d418d3) --- $$\sin(\arctan(z))=\frac{z}{\sqrt{1+z^2}}$$ **Holds when** $z\in\C\setminus\lbrace-\imaginaryI, \imaginaryI\rbrace$. Used by the Compute Engine for simplification. [`d4b0b6` · Fungrim entry ↗](https://fungrim.org/entry/d4b0b6) --- $$\arctan(z)=\mathrm{Csgn}(z)\arccos(\frac{1}{\sqrt{1+z^2}})$$ **Holds when** $z\in\C\setminus\lbrace-\imaginaryI, \imaginaryI\rbrace$. **Symbols:** **Csgn** — Real-valued sign function for complex numbers. Used by the Compute Engine for simplification. [`ec7f2d` · Fungrim entry ↗](https://fungrim.org/entry/ec7f2d) --- $$\mathrm{Arctan_2}(y, x)=\Im(\ln(x+y\imaginaryI))$$ **Holds when** $x\in\R\land y\in\R\land x+y\imaginaryI\ne0$. Used by the Compute Engine for simplification. [`eca4ce` · Fungrim entry ↗](https://fungrim.org/entry/eca4ce) --- $$\arctan(\tan(\theta))=\theta$$ **Holds when** $\theta\in\C\land-(\frac{\pi}{2})\lt\Re(\theta)\lt\frac{\pi}{2}$. Used by the Compute Engine for simplification and equation solving. [`f516e3` · Fungrim entry ↗](https://fungrim.org/entry/f516e3) --- ## Lambert W-function $$\operatorname{W}(0)=0$$ Used by the Compute Engine for simplification. [`0be17d` · Fungrim entry ↗](https://fungrim.org/entry/0be17d) --- $$\operatorname{W}(x\ln(x))=\ln(x)$$ **Holds when** $x\in\lbrack\frac{1}{\exponentialE}, \infty\rparen$. Used by the Compute Engine for simplification. [`30bd5b` · Fungrim entry ↗](https://fungrim.org/entry/30bd5b) --- $$\mathrm{Map}(\lparen-\infty, -\exponentialE^{-1}\rparen, x\mapsto\operatorname{W}(x))=\mathrm{Map}(\lparen0, \pi\rparen, y\mapsto y\imaginaryI-y\cot(y))$$ Used by the Compute Engine for simplification. [`44ad09` · Fungrim entry ↗](https://fungrim.org/entry/44ad09) --- $$\mathrm{Map}(\lbrace-\exponentialE^{-1}\rbrace, x\mapsto\operatorname{W}(x))=\lbrace-1\rbrace$$ Used by the Compute Engine for simplification. [`55498b` · Fungrim entry ↗](https://fungrim.org/entry/55498b) --- $$\operatorname{W}(x\exponentialE^{x})=x$$ **Holds when** $x\in\lbrack-1, \infty\rparen$. Used by the Compute Engine for simplification and equation solving. [`8654a3` · Fungrim entry ↗](https://fungrim.org/entry/8654a3) --- $$z\mapsto\operatorname{W}(z)^{\prime}(0)=(-r)^{r-1}$$ **Holds when** $r\in\N^*$. Used by the Compute Engine for simplification. [`8e8a59` · Fungrim entry ↗](https://fungrim.org/entry/8e8a59) --- $$\operatorname{W}_{-1}(x\ln(x))=\ln(x)$$ **Holds when** $x\in\lparen0, \frac{1}{\exponentialE}\rbrack$. Used by the Compute Engine for simplification. [`a172c7` · Fungrim entry ↗](https://fungrim.org/entry/a172c7) --- $$\operatorname{W}(-(\frac{1}{\exponentialE}))=-1$$ Used by the Compute Engine for simplification. [`b93d09` · Fungrim entry ↗](https://fungrim.org/entry/b93d09) --- $$\operatorname{W}(\exponentialE)=1$$ Used by the Compute Engine for simplification. [`c95c4f` · Fungrim entry ↗](https://fungrim.org/entry/c95c4f) --- $$\operatorname{W}_{-1}(-(\frac{1}{\exponentialE}))=-1$$ Used by the Compute Engine for simplification. [`d09380` · Fungrim entry ↗](https://fungrim.org/entry/d09380) --- $$\mathrm{LambertWPuiseuxCoefficient}(k)=\frac{(k-1)(2\mathrm{LambertWPuiseuxCoefficient}(k-2)+\begin{cases}2&k-2=0\\-1&k-2=1\\\sum_{j=2}^{k-3}\mathrm{LambertWPuiseuxCoefficient}(j)\mathrm{LambertWPuiseuxCoefficient}(-j+k-1)&\top\end{cases})}{4(k+1)}-\frac{1}{2}(\begin{cases}2&k=0\\-1&k=1\\\sum_{j=2}^{k-1}\mathrm{LambertWPuiseuxCoefficient}(j)\mathrm{LambertWPuiseuxCoefficient}((k+1)-j)&\top\end{cases})-\frac{\mathrm{LambertWPuiseuxCoefficient}(k-1)}{k+1}$$ **Holds when** $k\in2..\infty$. **Symbols:** **LambertWPuiseuxCoefficient** — Coefficient in scaled Puiseux expansion of Lambert W-function. Used by the Compute Engine for simplification. [`d37d0f` · Fungrim entry ↗](https://fungrim.org/entry/d37d0f) --- $$\operatorname{W}(-(\frac{\pi}{2}))=\frac{\imaginaryI\pi}{2}$$ Used by the Compute Engine for simplification. [`e1dd64` · Fungrim entry ↗](https://fungrim.org/entry/e1dd64) --- $$\operatorname{W}_{-1}(x\exponentialE^{x})=x$$ **Holds when** $x\in\lparen-\infty, -1\rbrack$. Used by the Compute Engine for simplification and equation solving. [`ed7dac` · Fungrim entry ↗](https://fungrim.org/entry/ed7dac) --- $$\mathrm{Map}(\lparen-\exponentialE^{-1}, \infty\rparen, x\mapsto\operatorname{W}(x))=\lparen-1, \infty\rparen$$ Used by the Compute Engine for simplification. [`ee86fb` · Fungrim entry ↗](https://fungrim.org/entry/ee86fb) --- $$\operatorname{W}_{k}(0)=-\infty$$ **Holds when** $k\in\Z\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`f372e9` · Fungrim entry ↗](https://fungrim.org/entry/f372e9) --- ## Natural logarithm $$\Re(\ln(z))=\ln(\vert z\vert)$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`099b19` · Fungrim entry ↗](https://fungrim.org/entry/099b19) --- $$\ln(z^\star)=\ln(z)^\star$$ **Holds when** $z\in\C\setminus\lparen-\infty, 0\rbrack$. Used by the Compute Engine for expansion. [`13895b` · Fungrim entry ↗](https://fungrim.org/entry/13895b) --- $$\ln(-1)=\pi\imaginaryI$$ Used by the Compute Engine for simplification. [`2f1f7b` · Fungrim entry ↗](https://fungrim.org/entry/2f1f7b) --- $$\ln(\exponentialE^{z})=z$$ **Holds when** $z\in\C\land\Im(z)\in\lparen-\pi, \pi\rbrack$. Used by the Compute Engine for simplification and equation solving. [`4c1e1e` · Fungrim entry ↗](https://fungrim.org/entry/4c1e1e) --- $$\ln(\exponentialE)=1$$ Used by the Compute Engine for simplification. [`699c83` · Fungrim entry ↗](https://fungrim.org/entry/699c83) --- $$\ln(\exponentialE^{z})=z-2\pi\imaginaryI\lceil\Im(z)/(2\pi)-1/2\rceil$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`a3a253` · Fungrim entry ↗](https://fungrim.org/entry/a3a253) --- $$\ln(\imaginaryI)=\frac{\pi\imaginaryI}{2}$$ Used by the Compute Engine for simplification. [`c331da` · Fungrim entry ↗](https://fungrim.org/entry/c331da) --- $$\ln(z)=\ln(\vert z\vert)+\arg(z)\imaginaryI$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`c43533` · Fungrim entry ↗](https://fungrim.org/entry/c43533) --- $$\vert\ln(z)\vert=\sqrt{\ln(\vert z\vert)^2+\arg(z)^2}$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`dcc1e5` · Fungrim entry ↗](https://fungrim.org/entry/dcc1e5) --- $$\ln(cz)=\ln(c)+\ln(z)$$ **Holds when** $c\in\lparen0, \infty\rparen\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`f67fa2` · Fungrim entry ↗](https://fungrim.org/entry/f67fa2) --- $$\Im(\ln(z))=\arg(z)$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`fbfb81` · Fungrim entry ↗](https://fungrim.org/entry/fbfb81) --- ## Pi $$\exp(\pi\imaginaryI)+1=0$$ Used by the Compute Engine for simplification. [`271314` · Fungrim entry ↗](https://fungrim.org/entry/271314) --- $$\frac{1}{\pi}=\frac{1}{9}(2\sqrt{3}\mathrm{Hypergeometric2F_1}(-(1/3), \frac{1}{3}, 1, 1))$$ Used by the Compute Engine for simplification. [`68b73d` · Fungrim entry ↗](https://fungrim.org/entry/68b73d) --- $$\frac{1}{\pi}=\frac{1}{2}(\mathrm{Hypergeometric2F_1}(\frac{1}{2}, -(1/2), 1, 1))$$ Used by the Compute Engine for simplification. [`a7095f` · Fungrim entry ↗](https://fungrim.org/entry/a7095f) --- $$\frac{1}{\pi}=\frac{1}{4}(\mathrm{Hypergeometric2F_1}(-(1/2), -(1/2), 1, 1))$$ Used by the Compute Engine for simplification. [`c6c108` · Fungrim entry ↗](https://fungrim.org/entry/c6c108) --- ## Powers $$(a+b\imaginaryI)^{c+d\imaginaryI}=\vert a+b\imaginaryI\vert^{c}\exp(-(d\arg(a+b\imaginaryI)))(\cos(c\arg(a+b\imaginaryI)+d\ln(\vert a+b\imaginaryI\vert))+\imaginaryI\sin(c\arg(a+b\imaginaryI)+d\ln(\vert a+b\imaginaryI\vert)))$$ **Holds when** $a\in\R\land b\in\R\land c\in\R\land d\in\R\land a+b\imaginaryI\ne0$. Used by the Compute Engine for simplification. [`0aac97` · Fungrim entry ↗](https://fungrim.org/entry/0aac97) --- $$\Im((a+b\imaginaryI)^{c+d\imaginaryI})=\vert a+b\imaginaryI\vert^{c}\exp(-(d\arg(a+b\imaginaryI)))\sin(c\arg(a+b\imaginaryI)+d\ln(\vert a+b\imaginaryI\vert))$$ **Holds when** $a\in\R\land b\in\R\land c\in\R\land d\in\R\land a+b\imaginaryI\ne0$. Used by the Compute Engine for simplification. [`18873d` · Fungrim entry ↗](https://fungrim.org/entry/18873d) --- $$(xy)^{a}=x^{a}y^{a}\exp(2\pi\imaginaryI a\lfloor\frac{\pi-\arg(x)-\arg(y)}{2\pi}\rfloor)$$ **Holds when** $x\in\C\setminus\lbrace0\rbrace\land y\in\C\setminus\lbrace0\rbrace\land a\in\C$. Used by the Compute Engine for simplification. [`2090c3` · Fungrim entry ↗](https://fungrim.org/entry/2090c3) --- $$z^0=1$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`310f36` · Fungrim entry ↗](https://fungrim.org/entry/310f36) --- $$a^{b}=\exp(b\ln(a))$$ **Holds when** $a\in\C\setminus\lbrace0\rbrace\land b\in\C$. Used by the Compute Engine for simplification. [`4d6416` · Fungrim entry ↗](https://fungrim.org/entry/4d6416) --- $$z^{n+1}=z^{n}z$$ **Holds when** $z\in\C\land n\in\N$  _or_  $z\in R\land R\in\mathrm{Rings}\land n\in\N$. Used by the Compute Engine for simplification. [`6c2b31` · Fungrim entry ↗](https://fungrim.org/entry/6c2b31) --- $$\vert(a+b\imaginaryI)^{c+d\imaginaryI}\vert=\vert a+b\imaginaryI\vert^{c}\exp(-(d\arg(a+b\imaginaryI)))$$ **Holds when** $a\in\R\land b\in\R\land c\in\R\land d\in\R\land a+b\imaginaryI\ne0$. Used by the Compute Engine for simplification. [`bc4d0a` · Fungrim entry ↗](https://fungrim.org/entry/bc4d0a) --- $$\Re((a+b\imaginaryI)^{c+d\imaginaryI})=\vert a+b\imaginaryI\vert^{c}\exp(-(d\arg(a+b\imaginaryI)))\cos(c\arg(a+b\imaginaryI)+d\ln(\vert a+b\imaginaryI\vert))$$ **Holds when** $a\in\R\land b\in\R\land c\in\R\land d\in\R\land a+b\imaginaryI\ne0$. Used by the Compute Engine for simplification. [`caf8cf` · Fungrim entry ↗](https://fungrim.org/entry/caf8cf) --- ## Sinc function $$\mathrm{sinc}(z)=\begin{cases}\frac{\sin(z)}{z}&z\ne0\\1&z=0\end{cases}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`01422b` · Fungrim entry ↗](https://fungrim.org/entry/01422b) --- $$\mathrm{sinc}(z)=\frac{\operatorname{J}_{\frac{1}{2}}(z)}{\sqrt{\frac{2z}{\pi}}}$$ **Holds when** $z\in\C\land z\ne0$. Used by the Compute Engine for simplification. [`19d7d9` · Fungrim entry ↗](https://fungrim.org/entry/19d7d9) --- $$z\mapsto\mathrm{sinc}(z)^{\prime}(0)=\begin{cases}\frac{(-1)^{\lfloor n/2\rfloor}}{n+1}&\mathrm{IsEven}(n)\\0&\mathrm{IsOdd}(n)\end{cases}$$ **Holds when** $n\in\N$. Used by the Compute Engine for simplification. [`1c3766` · Fungrim entry ↗](https://fungrim.org/entry/1c3766) --- $$\mathrm{ArgMin}(x\mapsto\mathrm{sinc}(x), \R)=\lbrace-\mathrm{BesselJZero}(3/2, 1), \mathrm{BesselJZero}(\frac{3}{2}, 1)\rbrace$$ **Symbols:** **ArgMin** — Locations of minimum value. Used by the Compute Engine for simplification. [`1e6344` · Fungrim entry ↗](https://fungrim.org/entry/1e6344) --- $$\mathrm{sinc}(\frac{\pi}{3})=\frac{3\sqrt{3}}{2\pi}$$ Used by the Compute Engine for simplification. [`340936` · Fungrim entry ↗](https://fungrim.org/entry/340936) --- $$\mathrm{sinc}(z^\star)=\mathrm{sinc}(z)^\star$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`3a428f` · Fungrim entry ↗](https://fungrim.org/entry/3a428f) --- $$\mathrm{sinc}(\frac{\pi}{6})=\frac{3}{\pi}$$ Used by the Compute Engine for simplification. [`45740a` · Fungrim entry ↗](https://fungrim.org/entry/45740a) --- $$z\mapsto\mathrm{sinc}(z)^{\prime}(z)=-(\frac{1}{3}(z\mathrm{Hypergeometric0F_1}(5/2, -(z^2/4))))$$ **Holds when** $z\in\C$. **Symbols:** **Hypergeometric0F1** — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. [`50f72f` · Fungrim entry ↗](https://fungrim.org/entry/50f72f) --- $$\mathrm{sinc}(\pi n)=\begin{cases}1&n=0\\0&n\ne0\end{cases}$$ **Holds when** $n\in\Z$. Used by the Compute Engine for simplification. [`593e63` · Fungrim entry ↗](https://fungrim.org/entry/593e63) --- $$\max(\mathrm{Map}(\R, x\mapsto\mathrm{sinc}(x)))=1$$ Used by the Compute Engine for simplification. [`632d1c` · Fungrim entry ↗](https://fungrim.org/entry/632d1c) --- $$z\mapsto\mathrm{sinc}(z)^{\prime}(z)=\begin{cases}\frac{\cos(z)}{z}-\frac{\sin(z)}{z^2}&z\ne0\\0&z=0\end{cases}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`768c77` · Fungrim entry ↗](https://fungrim.org/entry/768c77) --- $$z\mapsto\mathrm{sinc}(z)^{\doubleprime}(z)=\begin{cases}(2/z^3-1/z)\sin(z)-\frac{2\cos(z)}{z^2}&z\ne0\\-(\frac{1}{3})&z=0\end{cases}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`90c66a` · Fungrim entry ↗](https://fungrim.org/entry/90c66a) --- $$\mathrm{sinc}(0)=1$$ Used by the Compute Engine for simplification. [`b18020` · Fungrim entry ↗](https://fungrim.org/entry/b18020) --- $$\mathrm{ArgMaxUnique}(x\mapsto\mathrm{sinc}(x), \R)=0$$ **Symbols:** **ArgMaxUnique** — Unique location of maximum value. Used by the Compute Engine for simplification. [`b1a260` · Fungrim entry ↗](https://fungrim.org/entry/b1a260) --- $$\mathrm{sinc}(\imaginaryI z)=\frac{\sinh(z)}{z}$$ **Holds when** $z\in\C\land z\ne0$. Used by the Compute Engine for expansion. [`b41d08` · Fungrim entry ↗](https://fungrim.org/entry/b41d08) --- $$zz\mapsto\mathrm{sinc}(z)^{\doubleprime}(z)+2z\mapsto\mathrm{sinc}(z)^{\prime}(z)+z\mathrm{sinc}(z)=0$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`c6e6b2` · Fungrim entry ↗](https://fungrim.org/entry/c6e6b2) --- $$\mathrm{sinc}(\frac{\pi}{4})=\frac{2\sqrt{2}}{\pi}$$ Used by the Compute Engine for simplification. [`c9ead2` · Fungrim entry ↗](https://fungrim.org/entry/c9ead2) --- $$\mathrm{sinc}(2z)=\mathrm{sinc}(z)\cos(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`d5000a` · Fungrim entry ↗](https://fungrim.org/entry/d5000a) --- $$\min(\mathrm{Map}(\R, x\mapsto\mathrm{sinc}(x)))=\mathrm{sinc}(\mathrm{BesselJZero}(\frac{3}{2}, 1))$$ Used by the Compute Engine for simplification. [`da7fb1` · Fungrim entry ↗](https://fungrim.org/entry/da7fb1) --- $$\mathrm{sinc}(z)=\mathrm{Hypergeometric0F_1}(\frac{3}{2}, -(\frac{z^2}{4}))$$ **Holds when** $z\in\C$. **Symbols:** **Hypergeometric0F1** — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. [`e2878f` · Fungrim entry ↗](https://fungrim.org/entry/e2878f) --- $$\mathrm{sinc}(-z)=\mathrm{sinc}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`f19e0a` · Fungrim entry ↗](https://fungrim.org/entry/f19e0a) --- $$\mathrm{sinc}(z)=\frac{\sin(z)}{z}$$ **Holds when** $z\in\C\land z\ne0$. Used by the Compute Engine for simplification. [`fa9283` · Fungrim entry ↗](https://fungrim.org/entry/fa9283) --- $$\mathrm{sinc}(\frac{\pi}{2})=\frac{2}{\pi}$$ Used by the Compute Engine for simplification. [`fdc94c` · Fungrim entry ↗](https://fungrim.org/entry/fdc94c) --- $$\frac{z(n^2+5n+6)z\mapsto\mathrm{sinc}(z)^{\prime}(z)}{(n+3)!}+\frac{(n^2+5n+6)z\mapsto\mathrm{sinc}(z)^{\prime}(z)}{(n+2)!}+\frac{zz\mapsto\mathrm{sinc}(z)^{\prime}(z)}{(n+1)!}+\frac{1}{n!}(z\mapsto\mathrm{sinc}(z)^{\prime}(z))=0$$ **Holds when** $z\in\C\land n\in\N$. Used by the Compute Engine for simplification. [`ff5e82` · Fungrim entry ↗](https://fungrim.org/entry/ff5e82) --- ## Sine $$\sin(a)\cos(b)=\frac{1}{2}(\sin(a+b)+\sin(a-b))$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`012eba` · Fungrim entry ↗](https://fungrim.org/entry/012eba) --- $$\Im(\sin(x+\imaginaryI y))=\cos(x)\sinh(y)$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`037a6e` · Fungrim entry ↗](https://fungrim.org/entry/037a6e) --- $$\sin(z)=\sqrt{\frac{\pi z}{2}}\operatorname{J}_{\frac{1}{2}}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`0fbd15` · Fungrim entry ↗](https://fungrim.org/entry/0fbd15) --- $$\sin(z)=\frac{\exp(\imaginaryI z)-\exp(-\imaginaryI z)}{2\imaginaryI}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`18f40c` · Fungrim entry ↗](https://fungrim.org/entry/18f40c) --- $$\sin(2z)=2\sin(z)\cos(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`1b11be` · Fungrim entry ↗](https://fungrim.org/entry/1b11be) --- $$\sin(\pi+z)=-\sin(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`1c22f1` · Fungrim entry ↗](https://fungrim.org/entry/1c22f1) --- $$z\mapsto\sin(z)^{\doubleprime}(z)+\sin(z)=0$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`21f156` · Fungrim entry ↗](https://fungrim.org/entry/21f156) --- $$\sin(a)^2-\sin(b)^2=\sin(a+b)\sin(a-b)$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`2392f5` · Fungrim entry ↗](https://fungrim.org/entry/2392f5) --- $$\sin(z)^2=1-\cos(z)^2$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`244127` · Fungrim entry ↗](https://fungrim.org/entry/244127) --- $$\min(\mathrm{Map}(\R, x\mapsto\sin(x)))=-1$$ Used by the Compute Engine for simplification. [`27766c` · Fungrim entry ↗](https://fungrim.org/entry/27766c) --- $$z\mapsto\sin(z)^{\doubleprime}(z)=-\sin(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`297b3c` · Fungrim entry ↗](https://fungrim.org/entry/297b3c) --- $$\sin(x)=\Im(\exp(\imaginaryI x))$$ **Holds when** $x\in\R$. Used by the Compute Engine for simplification. [`299209` · Fungrim entry ↗](https://fungrim.org/entry/299209) --- $$\sin(z)^3=\frac{1}{4}(3\sin(z)-\sin(3z))$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`2a6702` · Fungrim entry ↗](https://fungrim.org/entry/2a6702) --- $$\sin(z+\pi k)=(-1)^{k}\sin(z)$$ **Holds when** $z\in\C\land k\in\Z$. Used by the Compute Engine for simplification. [`393b62` · Fungrim entry ↗](https://fungrim.org/entry/393b62) --- $$\sin(a+b\imaginaryI)=\sin(a)\cosh(b)+\imaginaryI\cos(a)\sinh(b)$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`3b839c` · Fungrim entry ↗](https://fungrim.org/entry/3b839c) --- $$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$$ Used by the Compute Engine for simplification. [`3c833f` · Fungrim entry ↗](https://fungrim.org/entry/3c833f) --- $$\sin(z)^2+\cos(z)^2=1$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`4948ea` · Fungrim entry ↗](https://fungrim.org/entry/4948ea) --- $$\sin(\frac{\pi}{2}+\pi k)=(-1)^{k}$$ **Holds when** $k\in\Z$. Used by the Compute Engine for simplification. [`506d0c` · Fungrim entry ↗](https://fungrim.org/entry/506d0c) --- $$\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`508e2c` · Fungrim entry ↗](https://fungrim.org/entry/508e2c) --- $$\sin(z)=z\mathrm{Hypergeometric0F_1}(\frac{3}{2}, \frac{-z^2}{4})$$ **Holds when** $z\in\C$. **Symbols:** **Hypergeometric0F1** — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. [`54daa9` · Fungrim entry ↗](https://fungrim.org/entry/54daa9) --- $$\sin(z)^{2n}=\frac{\binom{2n}{n}}{4^{n}}+\frac{2(\sum_{k=0}^{n-1}\cos(2z(n-k))\binom{2n}{k}\times(-1)^{k+n})}{4^{n}}$$ **Holds when** $z\in\C\land n\in\N$. Used by the Compute Engine for simplification. [`54f420` · Fungrim entry ↗](https://fungrim.org/entry/54f420) --- $$\sin(\frac{3\pi}{2})=-1$$ Used by the Compute Engine for simplification. [`56667c` · Fungrim entry ↗](https://fungrim.org/entry/56667c) --- $$\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}$$ Used by the Compute Engine for simplification. [`5fc688` · Fungrim entry ↗](https://fungrim.org/entry/5fc688) --- $$z\mapsto\sin(z)^{\prime}(z)=\sin(z+\frac{\pi r}{2})$$ **Holds when** $z\in\C\land r\in\N$. Used by the Compute Engine for simplification. [`612b21` · Fungrim entry ↗](https://fungrim.org/entry/612b21) --- $$\sin(\frac{\pi}{2})=1$$ Used by the Compute Engine for simplification. [`69c5ef` · Fungrim entry ↗](https://fungrim.org/entry/69c5ef) --- $$\sin(z+2\pi k)=\sin(z)$$ **Holds when** $z\in\C\land k\in\Z$. Used by the Compute Engine for simplification. [`6a8889` · Fungrim entry ↗](https://fungrim.org/entry/6a8889) --- $$\sin(z)-\cos(z)=\sqrt{2}\sin(z-\frac{\pi}{4})$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`6c3ba9` · Fungrim entry ↗](https://fungrim.org/entry/6c3ba9) --- $$\sin(3z)=3\sin(z)-4\sin(z)^3$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`729215` · Fungrim entry ↗](https://fungrim.org/entry/729215) --- $$\Re(\sin(x+\imaginaryI y))=\sin(x)\cosh(y)$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`729b70` · Fungrim entry ↗](https://fungrim.org/entry/729b70) --- $$\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`742943` · Fungrim entry ↗](https://fungrim.org/entry/742943) --- $$\sin(\imaginaryI z)=\imaginaryI\sinh(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`755655` · Fungrim entry ↗](https://fungrim.org/entry/755655) --- $$\sin(z^\star)=\sin(z)^\star$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`82c83f` · Fungrim entry ↗](https://fungrim.org/entry/82c83f) --- $$\sin(z)=\cos(\frac{\pi}{2}-z)=\cos(z-\frac{\pi}{2})=-\cos(z+\frac{\pi}{2})$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`925e5b` · Fungrim entry ↗](https://fungrim.org/entry/925e5b) --- $$\sin(z)^2-\cos(z)^2=-\cos(2z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`954066` · Fungrim entry ↗](https://fungrim.org/entry/954066) --- $$\sin(\pi-z)=\sin(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`9cc0f2` · Fungrim entry ↗](https://fungrim.org/entry/9cc0f2) --- $$\sin(-z)=-\sin(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`a2a30d` · Fungrim entry ↗](https://fungrim.org/entry/a2a30d) --- $$z\mapsto\sin(z)^{\prime}(z)=-z\mapsto\sin(z)^{\prime}(z)$$ **Holds when** $z\in\C\land r\in\N$. Used by the Compute Engine for expansion. [`a6667d` · Fungrim entry ↗](https://fungrim.org/entry/a6667d) --- $$\vert\sin(x+\imaginaryI y)\vert=\sqrt{\sin(x)^2+\sinh(y)^2}$$ **Holds when** $x\in\R\land y\in\R$. Used by the Compute Engine for simplification. [`abaf91` · Fungrim entry ↗](https://fungrim.org/entry/abaf91) --- $$\mathrm{ArgMin}(x\mapsto\sin(x), \R)=\mathrm{Map}(\Z, n\mapsto\pi(2n-1/2))$$ **Symbols:** **ArgMin** — Locations of minimum value. Used by the Compute Engine for simplification. [`ad04bd` · Fungrim entry ↗](https://fungrim.org/entry/ad04bd) --- $$\sin(\frac{\pi}{6})=\frac{1}{2}$$ Used by the Compute Engine for simplification. [`ad6b74` · Fungrim entry ↗](https://fungrim.org/entry/ad6b74) --- $$\sin(a)\sin(b)=\frac{1}{2}(\cos(a-b)-\cos(a+b))$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`ad6c1c` · Fungrim entry ↗](https://fungrim.org/entry/ad6c1c) --- $$\cos(z)+\imaginaryI\sin(z)=\exp(\imaginaryI z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`adbc1a` · Fungrim entry ↗](https://fungrim.org/entry/adbc1a) --- $$\sin(\frac{\pi}{2}+z)=\cos(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`bae475` · Fungrim entry ↗](https://fungrim.org/entry/bae475) --- $$\max(\mathrm{Map}(\R, x\mapsto\sin(x)))=1$$ Used by the Compute Engine for simplification. [`bfe28b` · Fungrim entry ↗](https://fungrim.org/entry/bfe28b) --- $$\sin(0)=0$$ Used by the Compute Engine for simplification. [`c52772` · Fungrim entry ↗](https://fungrim.org/entry/c52772) --- $$\mathrm{ArgMax}(x\mapsto\sin(x), \R)=\mathrm{Map}(\Z, n\mapsto\pi(2n+\frac{1}{2}))$$ **Symbols:** **ArgMax** — Locations of maximum value. Used by the Compute Engine for simplification. [`c5bdcc` · Fungrim entry ↗](https://fungrim.org/entry/c5bdcc) --- $$\sin(\pi k)=0$$ **Holds when** $k\in\Z$. Used by the Compute Engine for simplification. [`c62afa` · Fungrim entry ↗](https://fungrim.org/entry/c62afa) --- $$\sin(z)^2=\frac{1}{2}(1-\cos(2z))$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`cf6e35` · Fungrim entry ↗](https://fungrim.org/entry/cf6e35) --- $$\sin(z)=-\imaginaryI\sinh(\imaginaryI z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`cfc5c3` · Fungrim entry ↗](https://fungrim.org/entry/cfc5c3) --- $$(\cos(z)+\imaginaryI\sin(z))^{n}=\cos(nz)+\imaginaryI\sin(nz)$$ **Holds when** $z\in\C\land n\in\Z$. Used by the Compute Engine for simplification. [`d0505f` · Fungrim entry ↗](https://fungrim.org/entry/d0505f) --- $$\sin(\pi z)=(\pi)(\Gamma(z)\Gamma(1-z))^{-1}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`d38a03` · Fungrim entry ↗](https://fungrim.org/entry/d38a03) --- $$\sin(a)+\sin(b)=2\sin(\frac{a+b}{2})\cos(\frac{a-b}{2})$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`d59bd9` · Fungrim entry ↗](https://fungrim.org/entry/d59bd9) --- $$z\mapsto\sin(z)^{\prime}(z)=z\mapsto\sin(z)^{\prime}(z)$$ **Holds when** $z\in\C\land r\in\N$. Used by the Compute Engine for expansion. [`d81355` · Fungrim entry ↗](https://fungrim.org/entry/d81355) --- $$\sin(\frac{\pi}{2}-z)=\cos(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`da58f7` · Fungrim entry ↗](https://fungrim.org/entry/da58f7) --- $$\sin(\pi)=0$$ Used by the Compute Engine for simplification. [`e2161b` · Fungrim entry ↗](https://fungrim.org/entry/e2161b) --- $$\sin(a)-\sin(b)=2\cos(\frac{a+b}{2})\sin(\frac{a-b}{2})$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`e69cf6` · Fungrim entry ↗](https://fungrim.org/entry/e69cf6) --- $$\sin(z)+\cos(z)=\sqrt{2}\sin(z+\frac{\pi}{4})$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`f183d0` · Fungrim entry ↗](https://fungrim.org/entry/f183d0) --- $$\sin(a)^2-\cos(b)^2=-\cos(a+b)\cos(a-b)$$ **Holds when** $a\in\C\land b\in\C$. Used by the Compute Engine for simplification. [`f6d0c6` · Fungrim entry ↗](https://fungrim.org/entry/f6d0c6) --- $$z\mapsto\sin(z)^{\prime}(z)=\cos(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`f7ab32` · Fungrim entry ↗](https://fungrim.org/entry/f7ab32) --- ## Square roots $$\sqrt{z}^2=z$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`0984ef` · Fungrim entry ↗](https://fungrim.org/entry/0984ef) --- $$\sqrt{\imaginaryI}=\frac{1+\imaginaryI}{\sqrt{2}}$$ Used by the Compute Engine for simplification. [`0ad836` · Fungrim entry ↗](https://fungrim.org/entry/0ad836) --- $$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$$ **Holds when** $a\in\C\land b\in\lparen0, \infty\rparen$  _or_  $a\in\lbrack0, \infty\rparen\land b\in\C\setminus\lparen-\infty, 0\rbrack$  _or_  $a\in\C\land b\in\C\setminus\lbrace0\rbrace\land\arg(a)-\arg(b)\in\lparen-\pi, \pi\rbrack$. Used by the Compute Engine for simplification. [`0d8e03` · Fungrim entry ↗](https://fungrim.org/entry/0d8e03) --- $$\sqrt{r\exp(\imaginaryI\theta)}=\sqrt{r}\exp(\frac{\imaginaryI\theta}{2})$$ **Holds when** $r\in\lbrack0, \infty\rparen\land\theta\in\lparen-\pi, \pi\rbrack$. Used by the Compute Engine for simplification. [`1232f7` · Fungrim entry ↗](https://fungrim.org/entry/1232f7) --- $$\sqrt{\frac{z}{c-z}}=\sqrt{z}\sqrt{\frac{1}{c-z}}$$ **Holds when** $z\in\R\land c\in\lbrack0, \infty\rparen\land c-z\ne0$. Used by the Compute Engine for simplification. [`185efc` · Fungrim entry ↗](https://fungrim.org/entry/185efc) --- $$\arg(\sqrt{z})=\frac{\arg(z)}{2}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`22e0be` · Fungrim entry ↗](https://fungrim.org/entry/22e0be) --- $$z\mapsto\sqrt{z}^{\prime}(z)=\frac{1}{2\sqrt{z}}$$ **Holds when** $z\in\C\setminus\lparen-\infty, 0\rbrack$. Used by the Compute Engine for simplification. [`2a11ab` · Fungrim entry ↗](https://fungrim.org/entry/2a11ab) --- $$\sqrt{-1}=\imaginaryI$$ Used by the Compute Engine for simplification. [`2eb54a` · Fungrim entry ↗](https://fungrim.org/entry/2eb54a) --- $$\sqrt{\tilde\infty}=\tilde\infty$$ Used by the Compute Engine for simplification. [`31a8ca` · Fungrim entry ↗](https://fungrim.org/entry/31a8ca) --- $$\sqrt{x^2}=\vert x\vert$$ **Holds when** $x\in\R$. Used by the Compute Engine for simplification. [`3cc884` · Fungrim entry ↗](https://fungrim.org/entry/3cc884) --- $$z\mapsto\sqrt{z}^{\doubleprime}(z)=-((4z^{1/2}^{3})^{-1})$$ **Holds when** $z\in\C\setminus\lparen-\infty, 0\rbrack$. Used by the Compute Engine for simplification. [`3e71f4` · Fungrim entry ↗](https://fungrim.org/entry/3e71f4) --- $$\Re(\sqrt{z})=\sqrt{\frac{\vert z\vert+\Re(z)}{2}}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`4ed6a8` · Fungrim entry ↗](https://fungrim.org/entry/4ed6a8) --- $$\sqrt{\frac{z}{2}}=\frac{\sqrt{z}}{\sqrt{2}}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`616bcb` · Fungrim entry ↗](https://fungrim.org/entry/616bcb) --- $$\sqrt{z}=\exp(\frac{\ln(z)}{2})$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`627c9c` · Fungrim entry ↗](https://fungrim.org/entry/627c9c) --- $$\sqrt{\frac{z}{z-c}}=\frac{\sqrt{-z}}{\sqrt{c-z}}$$ **Holds when** $z\in\C\land c\in\lbrack0, \infty\rparen\land z-c\ne0$. Used by the Compute Engine for simplification. [`6f63dd` · Fungrim entry ↗](https://fungrim.org/entry/6f63dd) --- $$z\mapsto\sqrt{z}^{\prime}(z)=(-1)^{r}\mathrm{RisingFactorial}(-(\frac{1}{2}), r)z^{r-\frac{1}{2}}$$ **Holds when** $z\in\C\setminus\lparen-\infty, 0\rbrack\land r\in\N$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`83abff` · Fungrim entry ↗](https://fungrim.org/entry/83abff) --- $$\mathrm{sgn}(\sqrt{z})=\sqrt{\mathrm{sgn}(z)}$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`8c1ee5` · Fungrim entry ↗](https://fungrim.org/entry/8c1ee5) --- $$\sqrt{z-cz^2}=\sqrt{z}\sqrt{1-cz}$$ **Holds when** $z\in\C\land c\in\lbrack0, \infty\rparen$. Used by the Compute Engine for simplification. [`99c0b3` · Fungrim entry ↗](https://fungrim.org/entry/99c0b3) --- $$\sqrt{\infty}=\infty$$ Used by the Compute Engine for simplification. [`9dec73` · Fungrim entry ↗](https://fungrim.org/entry/9dec73) --- $$\vert\sqrt{z}\vert=\sqrt{\vert z\vert}$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`ac54c7` · Fungrim entry ↗](https://fungrim.org/entry/ac54c7) --- $$\sqrt{z^\star}=\sqrt{z}^\star$$ **Holds when** $z\in\C\setminus\lparen-\infty, 0\rparen$. Used by the Compute Engine for expansion. [`c58f46` · Fungrim entry ↗](https://fungrim.org/entry/c58f46) --- $$\sqrt{\frac{1}{z}}=\frac{1}{\sqrt{z}}$$ **Holds when** $z\in\C\setminus\lparen-\infty, 0\rbrack$. Used by the Compute Engine for expansion. [`d0a331` · Fungrim entry ↗](https://fungrim.org/entry/d0a331) --- $$\sqrt{\frac{z}{z+c}}=\frac{\sqrt{z}}{\sqrt{z+c}}$$ **Holds when** $z\in\C\land c\in\lbrack0, \infty\rparen\land z+c\ne0$. Used by the Compute Engine for simplification. [`d40229` · Fungrim entry ↗](https://fungrim.org/entry/d40229) --- $$\sqrt{z^2}=z$$ **Holds when** $z\in\C\land\arg(z)\in\lparen\frac{-\pi}{2}, \frac{\pi}{2}\rbrack$. Used by the Compute Engine for simplification. [`d8791e` · Fungrim entry ↗](https://fungrim.org/entry/d8791e) --- $$\Im(\sqrt{z})=\mathrm{sgn}(\Im(z))\sqrt{\frac{\vert z\vert-\Re(z)}{2}}$$ **Holds when** $z\in\C\setminus\lparen-\infty, 0\rparen$. Used by the Compute Engine for simplification. [`e722ca` · Fungrim entry ↗](https://fungrim.org/entry/e722ca) --- $$\sqrt{\exp(\imaginaryI\theta)\infty}=\exp(\frac{\imaginaryI\theta}{2})\infty$$ **Holds when** $\theta\in\lparen-\pi, \pi\rbrack$. Used by the Compute Engine for expansion. [`f9f31d` · Fungrim entry ↗](https://fungrim.org/entry/f9f31d) ---