Elementary functions

Part of the Fungrim Identities reference — 205 identities for elementary functions.

Generated reference

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Contents

• Exponential function (16)
• Golden ratio (5)
• Inverse tangent (44)
• Lambert W-function (8)
• Natural logarithm (11)
• Pi (4)
• Powers (8)
• Sinc function (24)
• Sine (59)
• 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 ↗

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\vert\exponentialE^{z}\vert=\exp(\Re(z))

Holds when z\in\C. Used by the Compute Engine for simplification. 1b3014 · Fungrim entry ↗

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\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 ↗

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\exp(\ln(z))=z

Holds when z\in\C. Used by the Compute Engine for simplification and equation solving. 296627 · Fungrim entry ↗

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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 ↗

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\exp(z^\star)=\exponentialE^{z}^\star

Holds when z\in\C. Used by the Compute Engine for expansion. 52d827 · Fungrim entry ↗

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\exp(\pi\imaginaryI)=-1

Used by the Compute Engine for simplification. 54aaf1 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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z\mapsto\exponentialE^{z}^{\prime}(z)=\exponentialE^{z}

Holds when z\in\C. Used by the Compute Engine for simplification. 96af56 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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

Used by the Compute Engine for simplification. a90f35 · Fungrim entry ↗

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\Re(\exponentialE^{z})=\exp(\Re(z))\cos(\Im(z))

Holds when z\in\C. Used by the Compute Engine for simplification. b7d62b · Fungrim entry ↗

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\mathrm{sgn}(\exponentialE^{z})=\exp(\Im(z)\imaginaryI)

Holds when z\in\C. Used by the Compute Engine for simplification. caf706 · Fungrim entry ↗

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\Im(\exponentialE^{z})=\exp(\Re(z))\sin(\Im(z))

Holds when z\in\C. Used by the Compute Engine for simplification. e2fac7 · Fungrim entry ↗

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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 ↗

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\frac{1}{\varphi}=\varphi-1

Used by the Compute Engine for simplification. 31f52c · Fungrim entry ↗

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\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 ↗

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\varphi^2-\varphi-1=0

Used by the Compute Engine for simplification. b464d3 · Fungrim entry ↗

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\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 ↗

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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 ↗

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\arctan(\imaginaryI z)=\imaginaryI\mathrm{artanh}(z)

Holds when z\in\C. Used by the Compute Engine for expansion. 072166 · Fungrim entry ↗

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\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 ↗

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\arctan(-z)=-\arctan(z)

Holds when z\in\C\cup\lbrace-\infty, \infty\rbrace. Used by the Compute Engine for expansion. 0ee626 · Fungrim entry ↗

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\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 ↗

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\arctan(1)=\frac{\pi}{4}

Used by the Compute Engine for simplification. 157c6c · Fungrim entry ↗

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\tan(\arctan(z))=z

Holds when z\in\C. Used by the Compute Engine for simplification and equation solving. 1f026d · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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. Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 34ff28 · Fungrim entry ↗

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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 ↗

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\arctan(\frac{1}{\sqrt{3}})=\frac{\pi}{6}

Used by the Compute Engine for simplification. 3c1021 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\arctan(0)=0

Used by the Compute Engine for simplification. 645e30 · Fungrim entry ↗

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\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 ↗

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\arctan(\sqrt{3})=\frac{\pi}{3}

Used by the Compute Engine for simplification. 706783 · Fungrim entry ↗

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\arctan(-\infty)=-(\frac{\pi}{2})

Used by the Compute Engine for simplification. 7295b5 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\arctan(2-\sqrt{3})=\frac{\pi}{12}

Used by the Compute Engine for simplification. 7dd050 · Fungrim entry ↗

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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 ↗

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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, Theorem 9 90631b · Fungrim entry ↗

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\arctan(-\imaginaryI)=-\imaginaryI\infty

Used by the Compute Engine for simplification. 9b0994 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\arctan(\imaginaryI)=\imaginaryI\infty

Used by the Compute Engine for simplification. a2d208 · Fungrim entry ↗

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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 ↗

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\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 ↗

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\arctan(\sqrt{2}-1)=\frac{\pi}{8}

Used by the Compute Engine for simplification. a9ecff · Fungrim entry ↗

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\arctan(2+\sqrt{3})=\frac{5\pi}{12}

Used by the Compute Engine for simplification. b0049f · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\arctan(z)=\mathrm{arcctg}(\frac{1}{z})

Holds when z\in\C. Used by the Compute Engine for simplification. c580f4 · Fungrim entry ↗

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\arctan(\sqrt{2}+1)=\frac{3\pi}{8}

Used by the Compute Engine for simplification. c6c92a · Fungrim entry ↗

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\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 ↗

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\arctan(\infty)=\frac{\pi}{2}

Used by the Compute Engine for simplification. d418d3 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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Lambert W-function

\operatorname{W}(0)=0

Used by the Compute Engine for simplification. 0be17d · Fungrim entry ↗

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\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 ↗

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\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 ↗

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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 ↗

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\operatorname{W}(-(\frac{1}{\exponentialE}))=-1

Used by the Compute Engine for simplification. b93d09 · Fungrim entry ↗

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\operatorname{W}(\exponentialE)=1

Used by the Compute Engine for simplification. c95c4f · Fungrim entry ↗

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\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 ↗

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\operatorname{W}(-(\frac{\pi}{2}))=\frac{\imaginaryI\pi}{2}

Used by the Compute Engine for simplification. e1dd64 · Fungrim entry ↗

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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 ↗

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\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 ↗

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\ln(-1)=\pi\imaginaryI

Used by the Compute Engine for simplification. 2f1f7b · Fungrim entry ↗

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\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 ↗

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\ln(\exponentialE)=1

Used by the Compute Engine for simplification. 699c83 · Fungrim entry ↗

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\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 ↗

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

Used by the Compute Engine for simplification. c331da · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\Im(\ln(z))=\arg(z)

Holds when z\in\C\setminus\lbrace0\rbrace. Used by the Compute Engine for simplification. fbfb81 · Fungrim entry ↗

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Pi

\exp(\pi\imaginaryI)+1=0

Used by the Compute Engine for simplification. 271314 · Fungrim entry ↗

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\frac{1}{\pi}=\frac{1}{9}(2\sqrt{3}\mathrm{Hypergeometric2F_1}(-(1/3), \frac{1}{3}, 1, 1))

Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 68b73d · Fungrim entry ↗

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\frac{1}{\pi}=\frac{1}{2}(\mathrm{Hypergeometric2F_1}(\frac{1}{2}, -(1/2), 1, 1))

Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. a7095f · Fungrim entry ↗

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\frac{1}{\pi}=\frac{1}{4}(\mathrm{Hypergeometric2F_1}(-(1/2), -(1/2), 1, 1))

Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. c6c108 · Fungrim entry ↗

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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 ↗

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\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 ↗

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(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 ↗

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z^0=1

Holds when z\in\C. Used by the Compute Engine for expansion. 310f36 · Fungrim entry ↗

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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 ↗

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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 ↗

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\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 ↗

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\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 ↗

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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 ↗

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\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 ↗

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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 ↗

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\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 ↗

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\mathrm{sinc}(\frac{\pi}{3})=\frac{3\sqrt{3}}{2\pi}

Used by the Compute Engine for simplification. 340936 · Fungrim entry ↗

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\mathrm{sinc}(z^\star)=\mathrm{sinc}(z)^\star

Holds when z\in\C. Used by the Compute Engine for expansion. 3a428f · Fungrim entry ↗

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\mathrm{sinc}(\frac{\pi}{6})=\frac{3}{\pi}

Used by the Compute Engine for simplification. 45740a · Fungrim entry ↗

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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 ↗

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\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 ↗

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\max(\lbrace\mathrm{sinc}(x), x\in\R\rbrace)=1

Used by the Compute Engine for simplification. 632d1c · Fungrim entry ↗

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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 ↗

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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 ↗

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\mathrm{sinc}(0)=1

Used by the Compute Engine for simplification. b18020 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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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 ↗

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\mathrm{sinc}(\frac{\pi}{4})=\frac{2\sqrt{2}}{\pi}

Used by the Compute Engine for simplification. c9ead2 · Fungrim entry ↗

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\mathrm{sinc}(2z)=\mathrm{sinc}(z)\cos(z)

Holds when z\in\C. Used by the Compute Engine for simplification. d5000a · Fungrim entry ↗

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\min(\lbrace\mathrm{sinc}(x), x\in\R\rbrace)=\mathrm{sinc}(\mathrm{BesselJZero}(\frac{3}{2}, 1))

Used by the Compute Engine for simplification. da7fb1 · Fungrim entry ↗

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\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 ↗

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\mathrm{sinc}(-z)=\mathrm{sinc}(z)

Holds when z\in\C. Used by the Compute Engine for simplification. f19e0a · Fungrim entry ↗

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\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 ↗

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\mathrm{sinc}(\frac{\pi}{2})=\frac{2}{\pi}

Used by the Compute Engine for simplification. fdc94c · Fungrim entry ↗

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\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 ↗

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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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\sin(2z)=2\sin(z)\cos(z)

Holds when z\in\C. Used by the Compute Engine for simplification. 1b11be · Fungrim entry ↗

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\sin(\pi+z)=-\sin(z)

Holds when z\in\C. Used by the Compute Engine for expansion. 1c22f1 · Fungrim entry ↗

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z\mapsto\sin(z)^{\doubleprime}(z)+\sin(z)=0

Holds when z\in\C. Used by the Compute Engine for simplification. 21f156 · Fungrim entry ↗

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\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 ↗

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\sin(z)^2=1-\cos(z)^2

Holds when z\in\C. Used by the Compute Engine for simplification. 244127 · Fungrim entry ↗

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\min(\lbrace\sin(x), x\in\R\rbrace)=-1

Used by the Compute Engine for simplification. 27766c · Fungrim entry ↗

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z\mapsto\sin(z)^{\doubleprime}(z)=-\sin(z)

Holds when z\in\C. Used by the Compute Engine for simplification. 297b3c · Fungrim entry ↗

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\sin(x)=\Im(\exp(\imaginaryI x))

Holds when x\in\R. Used by the Compute Engine for simplification. 299209 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}

Used by the Compute Engine for simplification. 3c833f · Fungrim entry ↗

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\sin(z)^2+\cos(z)^2=1

Holds when z\in\C. Used by the Compute Engine for simplification. 4948ea · Fungrim entry ↗

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\sin(\frac{\pi}{2}+\pi k)=(-1)^{k}

Holds when k\in\Z. Used by the Compute Engine for simplification. 506d0c · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\sin(\frac{3\pi}{2})=-1

Used by the Compute Engine for simplification. 56667c · Fungrim entry ↗

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\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}

Used by the Compute Engine for simplification. 5fc688 · Fungrim entry ↗

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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 ↗

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\sin(\frac{\pi}{2})=1

Used by the Compute Engine for simplification. 69c5ef · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\sin(3z)=3\sin(z)-4\sin(z)^3

Holds when z\in\C. Used by the Compute Engine for simplification. 729215 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\sin(\imaginaryI z)=\imaginaryI\sinh(z)

Holds when z\in\C. Used by the Compute Engine for expansion. 755655 · Fungrim entry ↗

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\sin(z^\star)=\sin(z)^\star

Holds when z\in\C. Used by the Compute Engine for expansion. 82c83f · Fungrim entry ↗

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\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 ↗

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\sin(z)^2-\cos(z)^2=-\cos(2z)

Holds when z\in\C. Used by the Compute Engine for simplification. 954066 · Fungrim entry ↗

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\sin(\pi-z)=\sin(z)

Holds when z\in\C. Used by the Compute Engine for simplification. 9cc0f2 · Fungrim entry ↗

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\sin(-z)=-\sin(z)

Holds when z\in\C. Used by the Compute Engine for expansion. a2a30d · Fungrim entry ↗

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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 ↗

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\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 ↗

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\mathrm{ArgMin}(x\mapsto\sin(x), \R)=\lbrace\pi(2n-\frac{1}{2}), n\in\Z\rbrace

Symbols: ArgMin — Locations of minimum value. Used by the Compute Engine for simplification. ad04bd · Fungrim entry ↗

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\sin(\frac{\pi}{6})=\frac{1}{2}

Used by the Compute Engine for simplification. ad6b74 · Fungrim entry ↗

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\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 ↗

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\cos(z)+\imaginaryI\sin(z)=\exp(\imaginaryI z)

Holds when z\in\C. Used by the Compute Engine for simplification. adbc1a · Fungrim entry ↗

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

Holds when z\in\C. Used by the Compute Engine for simplification. bae475 · Fungrim entry ↗

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\max(\lbrace\sin(x), x\in\R\rbrace)=1

Used by the Compute Engine for simplification. bfe28b · Fungrim entry ↗

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\sin(0)=0

Used by the Compute Engine for simplification. c52772 · Fungrim entry ↗

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\mathrm{ArgMax}(x\mapsto\sin(x), \R)=\lbrace\pi(2n+\frac{1}{2}), n\in\Z\rbrace

Symbols: ArgMax — Locations of maximum value. Used by the Compute Engine for simplification. c5bdcc · Fungrim entry ↗

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\sin(\pi k)=0

Holds when k\in\Z. Used by the Compute Engine for simplification. c62afa · Fungrim entry ↗

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\sin(z)^2=\frac{1}{2}(1-\cos(2z))

Holds when z\in\C. Used by the Compute Engine for simplification. cf6e35 · Fungrim entry ↗

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\sin(z)=-\imaginaryI\sinh(\imaginaryI z)

Holds when z\in\C. Used by the Compute Engine for simplification. cfc5c3 · Fungrim entry ↗

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(\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 ↗

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\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 ↗

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\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 ↗

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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 ↗

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\sin(\frac{\pi}{2}-z)=\cos(z)

Holds when z\in\C. Used by the Compute Engine for simplification. da58f7 · Fungrim entry ↗

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\sin(\pi)=0

Used by the Compute Engine for simplification. e2161b · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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z\mapsto\sin(z)^{\prime}(z)=\cos(z)

Holds when z\in\C. Used by the Compute Engine for simplification. f7ab32 · Fungrim entry ↗

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Square roots

\sqrt{z}^2=z

Holds when z\in\C. Used by the Compute Engine for expansion. 0984ef · Fungrim entry ↗

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\sqrt{\imaginaryI}=\frac{1+\imaginaryI}{\sqrt{2}}

Used by the Compute Engine for simplification. 0ad836 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\arg(\sqrt{z})=\frac{\arg(z)}{2}

Holds when z\in\C. Used by the Compute Engine for simplification. 22e0be · Fungrim entry ↗

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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 ↗

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\sqrt{-1}=\imaginaryI

Used by the Compute Engine for simplification. 2eb54a · Fungrim entry ↗

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\sqrt{\tilde\infty}=\tilde\infty

Used by the Compute Engine for simplification. 31a8ca · Fungrim entry ↗

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\sqrt{x^2}=\vert x\vert

Holds when x\in\R. Used by the Compute Engine for simplification. 3cc884 · Fungrim entry ↗

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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 ↗

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\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 ↗

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\sqrt{\frac{z}{2}}=\frac{\sqrt{z}}{\sqrt{2}}

Holds when z\in\C. Used by the Compute Engine for simplification. 616bcb · Fungrim entry ↗

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\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 ↗

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\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 ↗

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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 ↗

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\mathrm{sgn}(\sqrt{z})=\sqrt{\mathrm{sgn}(z)}

Holds when z\in\C. Used by the Compute Engine for expansion. 8c1ee5 · Fungrim entry ↗

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\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 ↗

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\sqrt{\infty}=\infty

Used by the Compute Engine for simplification. 9dec73 · Fungrim entry ↗

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\vert\sqrt{z}\vert=\sqrt{\vert z\vert}

Holds when z\in\C. Used by the Compute Engine for expansion. ac54c7 · Fungrim entry ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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\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 ↗

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