Elliptic integrals

Part of the Fungrim Identities reference — 304 identities for elliptic integrals.

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Arithmetic-geometric mean (28)
Carlson symmetric elliptic integrals (183)
Legendre elliptic integrals (80)
Weierstrass elliptic functions (13)

Arithmetic-geometric mean

$\mathrm{AGM}(0, b)=0$

Holds when $b\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. 08329d · Fungrim entry ↗

$\mathrm{AGM}(1, \sqrt{2})=\frac{2\sqrt{2}\sqrt{\pi}^{3}}{\Gamma(1/4)^2}$

Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. 0d9352 · Fungrim entry ↗

$a\mapsto\mathrm{AGM}(a, b)^{\prime}(a)=\frac{(\pi a-2\mathrm{AGM}(a, b)\mathrm{EllipticE}((a-b)/(a+b)^2))\mathrm{AGM}(a, b)}{\pi a(a-b)}$

Holds when $b\ne0\land a\ne b\land\frac{a}{b}\notin\lparen-\infty, 0\rbrack\land a\in\C\land b\in\C$ . Symbols: AGM — Arithmetic-geometric mean; EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. Reference: functions.wolfram.com 20828c · Fungrim entry ↗

$\mathrm{AGM}(z)=\mathrm{AGM}(1, z)=\mathrm{AGM}(z, 1)$

Holds when $z\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for expansion. 21f412 · Fungrim entry ↗

$\ln(\frac{1}{q})=(\pi)(\mathrm{AGM}(\mathrm{JacobiThetaQ}(2, 0, q)^2, \mathrm{JacobiThetaQ}(3, 0, q)^2))^{-1}$

Holds when $q\in\lparen0, 1\rparen$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. 26fd1b · Fungrim entry ↗

$\mathrm{AGM}(1, 3+2\sqrt{2})=\frac{2(2+\sqrt{2})\sqrt{\pi}^{3}}{\Gamma(1/4)^2}$

Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. 361801 · Fungrim entry ↗

$\mathrm{AGM}(a, -a)=0$

Holds when $a\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. 3e1398 · Fungrim entry ↗

$x\mapsto\mathrm{AGM}(1, x)^{\prime}(1)=\frac{n!\mathrm{SloaneA}(60\,691, n)\times(-1)^{n}}{8^{n}}$

Holds when $n\in\N$ . Symbols: AGM — Arithmetic-geometric mean; SloaneA — Sequence X in Sloane's OEIS. Used by the Compute Engine for expansion. 447541 · Fungrim entry ↗

$\mathrm{AGM}(1, b)=\frac{1}{2}((b+1)\mathrm{AGM}(1, \frac{2b^{1/2}}{b+1}))$

Holds when $b\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. 46c021 · Fungrim entry ↗

$\mathrm{AGM}(1, -\imaginaryI)=\frac{\sqrt{2}(1-\imaginaryI)\sqrt{\pi}^{3}}{\Gamma(1/4)^2}$

Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. 5174ea · Fungrim entry ↗

$\mathrm{AGM}(a, b)=\mathrm{AGM}(b, a)$

Holds when $a\in\C\land b\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for expansion. 59fab1 · Fungrim entry ↗

$\mathrm{AGM}(1, \imaginaryI)=\frac{\sqrt{2}(1+\imaginaryI)\sqrt{\pi}^{3}}{\Gamma(1/4)^2}$

Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. 69d0a3 · Fungrim entry ↗

$\mathrm{AGM}(a, b)=\frac{\pi(a+b)}{4\mathrm{EllipticK}((a-b)/(a+b)^2)}$

Holds when $b\ne0\land\frac{a}{b}\notin\lparen-\infty, 0\rbrack\land a\in\C\land b\in\C$ . Symbols: AGM — Arithmetic-geometric mean; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 71a0ff · Fungrim entry ↗

$\mathrm{AGM}(1, \sqrt{2})=(\mathrm{JacobiTheta}(4, 0, \imaginaryI)^2)^{-1}$

Symbols: AGM — Arithmetic-geometric mean; JacobiTheta — Jacobi theta function. Used by the Compute Engine for simplification. 7b362f · Fungrim entry ↗

$\mathrm{AGMSequence}(0, a, b)=(a,b)$

Holds when $a\in\C\land b\in\C$ . Symbols: AGMSequence — Convergents in AGM iteration. Used by the Compute Engine for expansion. 84b888 · Fungrim entry ↗

$\mathrm{AGM}(1, b)=b\mathrm{AGM}(1, \frac{1}{b})$

Holds when $b\notin\lparen-\infty, 0\rbrack\land b\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. 8e80c6 · Fungrim entry ↗

$\mathrm{AGM}(a, 0)=0$

Holds when $a\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. 8f176c · Fungrim entry ↗

$\mathrm{AGM}(b+1, 1-b)=\mathrm{AGM}(1, \sqrt{1-b^2})$

Holds when $b\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for expansion. 9d84d8 · Fungrim entry ↗

$(a(a^2-b^2)a\mapsto\mathrm{AGM}(a, b)^{\doubleprime}(a)+(3a^2-b^2)a\mapsto\mathrm{AGM}(a, b)^{\prime}(a))\mathrm{AGM}(a, b)+2a(b^2-a^2)a\mapsto\mathrm{AGM}(a, b)^{\prime}(a)^2-a\mathrm{AGM}(a, b)^2=0$

Holds when $b\ne0\land\frac{a}{b}\notin\lparen-\infty, 0\rbrack\land a\in\C\land b\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. Reference: functions.wolfram.com a4cc5a · Fungrim entry ↗

$\mathrm{AGM}(a, a)=a$

Holds when $a\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. b41bdd · Fungrim entry ↗

$\mathrm{AGM}(a, b)=a\mathrm{AGM}(1, \frac{b}{a})$

Holds when $a\ne0\land\frac{b}{a}\notin\lparen-\infty, 0\rbrack\land a\in\C\land b\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. ce2395 · Fungrim entry ↗

$\mathrm{AGM}(a, b)=\frac{a+b}{2\mathrm{Hypergeometric2F_1}(\frac{1}{2}, \frac{1}{2}, 1, (a-b)/(a+b)^2)}$

Holds when $b\ne0\land\frac{a}{b}\notin\lparen-\infty, 0\rbrack\land a\in\C\land b\in\C$ . Symbols: AGM — Arithmetic-geometric mean; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. d6d836 · Fungrim entry ↗

$\mathrm{EllipticK}(m)=(\pi)(2\mathrm{AGM}(1, \sqrt{1-m}))^{-1}$

Holds when $m\in\C$ . Symbols: AGM — Arithmetic-geometric mean; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. e15f43 · Fungrim entry ↗

$\mathrm{AGM}(1, \frac{\sqrt{2}}{2})=\frac{2\sqrt{\pi}^{3}}{\Gamma(1/4)^2}$

Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. e3896e · Fungrim entry ↗

$\mathrm{AGM}(a, b)=b\mathrm{AGM}(1, \frac{a}{b})$

$\mathrm{AGM}(1, 1)=1$

Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. eb0661 · Fungrim entry ↗

$\mathrm{AGM}(1, 3-2\sqrt{2})=\frac{2(2-\sqrt{2})\sqrt{\pi}^{3}}{\Gamma(1/4)^2}$

Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. f9190b · Fungrim entry ↗

$\mathrm{AGM}(a, b)=\mathrm{AGM}(\frac{a+b}{2}, \begin{cases}1&0\le\Re((a+b)/(2(ab)^{1/2}))\lor(ab)^{1/2}=0\\-1&\top\end{cases}\sqrt{ab})$

Holds when $a\in\C\land b\in\C$ . Symbols: AGM — Arithmetic-geometric mean. Used by the Compute Engine for simplification. fa6ff7 · Fungrim entry ↗

Carlson symmetric elliptic integrals

$\mathrm{CarlsonRD}(0, y, z)=\frac{1}{4}(3\pi\mathrm{CarlsonHypergeometricR}(\frac{-3}{2}, \bigl\lbrack1/2, 3/2\bigr\rbrack, \bigl\lbrack y, z\bigr\rbrack))$

Holds when $y\in\C\setminus\lparen-\infty, 0\rbrack\land z\in\C\setminus\lparen-\infty, 0\rbrack$ . Symbols: CarlsonHypergeometricR — Carlson multivariate hypergeometric function; CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 00c331 · Fungrim entry ↗

$\mathrm{CarlsonRC}(x, -y)=\frac{\mathrm{artanh}(\sqrt{x/(x+y)})-\frac{\imaginaryI\pi}{2}}{\sqrt{x+y}}$

Holds when $x\in\lparen0, \infty\rparen\land y\in\lparen0, \infty\rparen$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 00cdb7 · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, 1, 2)=\frac{3\sqrt{2}\Gamma(1/4)^2}{16\sqrt{\pi}}-\frac{3\sqrt{2}\sqrt{\pi}^{3}}{2\Gamma(1/4)^2}$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 060366 · Fungrim entry ↗

$\mathrm{CarlsonRG}(-x, -y, -z)=\imaginaryI\mathrm{CarlsonRG}(x, y, z)$

Holds when $x\in\lbrack0, \infty\rparen\land y\in\lbrack0, \infty\rparen\land z\in\lbrack0, \infty\rparen$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. 092716 · Fungrim entry ↗

$\mathrm{CarlsonRC}(x, 2x)=\frac{\pi}{4\sqrt{x}}$

Holds when $x\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 09a494 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(x, w, w, w)=\begin{cases}\frac{3(\mathrm{CarlsonRC}(x, w)-\frac{x^{1/2}}{w})}{2(w-x)}&x\ne w\\\sqrt{x}^{-3}&x=w\end{cases}$

Holds when $x\in\C\land w\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind; CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 0aa9ac · Fungrim entry ↗

$\mathrm{CarlsonRF}(1, 2, 2)=\frac{\pi}{4}$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 0bf328 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, x, 2x)=\frac{\sqrt{2}\Gamma(1/4)^2}{8\sqrt{\pi}\sqrt{x}}$

Holds when $x\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 0ed5e2 · Fungrim entry ↗

$\mathrm{CarlsonRG}(x, x, y)=\frac{1}{2}(\begin{cases}x\mathrm{CarlsonRC}(y, x)+\sqrt{y}&x\ne0\\\sqrt{y}&x=0\end{cases})$

Holds when $x\in\C\land y\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind; CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. 120284 · Fungrim entry ↗

$\mathrm{WeierstrassP}(\mathrm{CarlsonRF}(z-\mathrm{EllipticRootE}(1, \tau), z-\mathrm{EllipticRootE}(2, \tau), z-\mathrm{EllipticRootE}(3, \tau)), \tau)=z$

Holds when $z\in\C\land\tau\in\mathrm{HH}$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind; WeierstrassP — Weierstrass elliptic function. Used by the Compute Engine for simplification. 124339 · Fungrim entry ↗

$\mathrm{CarlsonRD}(-x, -y, z)=-(\imaginaryI\mathrm{CarlsonRD}(x, y, -z))^\star$

Holds when $x\in\lparen0, \infty\rbrack\land y\in\lparen0, \infty\rbrack\land z\in\lparen0, \infty\rbrack$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 12b1d0 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, 0, x)=\begin{cases}\infty&x\ne0\\\tilde\infty&x=0\end{cases}$

Holds when $x\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 13a092 · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, 0, -1)=\infty\imaginaryI$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 14a365 · Fungrim entry ↗

$\mathrm{CarlsonRC}(1, y+1)=\mathrm{Hypergeometric2F_1}(1, \frac{1}{2}, \frac{3}{2}, -y)$

Holds when $y\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 157ebb · Fungrim entry ↗

$\mathrm{CarlsonRD}(\mathrm{lamda}x, \mathrm{lamda}y, \mathrm{lamda}z)=\frac{\mathrm{CarlsonRD}(x, y, z)}{\sqrt{\mathrm{lamda}}^{3}}$

Holds when $x\in\C\land y\in\C\land z\in\C\land\mathrm{lamda}\in\lparen0, \infty\rparen$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 197a91 · Fungrim entry ↗

$\mathrm{CarlsonRC}(1, 0)=\infty$

Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 1acb07 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 0, 1, -1)=-\infty$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 1b6362 · Fungrim entry ↗

$\mathrm{CarlsonRD}(1, 1, 1)=1$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 1c0fee · Fungrim entry ↗

$\mathrm{CarlsonRD}(x, y, z)=\mathrm{CarlsonRD}(y, x, z)$

Holds when $x\in\C\land y\in\C\land z\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for expansion. 1e8061 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, \imaginaryI, -\imaginaryI, 1)=\frac{3\Gamma(1/4)^2}{8\sqrt{\pi}}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 1eaaed · Fungrim entry ↗

$\mathrm{CarlsonRJ}(x, x, x, w)=\begin{cases}\frac{3(\mathrm{CarlsonRC}(x, w)-\frac{1}{x^{1/2}})}{x-w}&x\ne w\\\sqrt{w}^{-3}&x=w\end{cases}$

$\mathrm{CarlsonRF}(-x, -y, z)=(\imaginaryI\mathrm{CarlsonRF}(x, y, -z))^\star$

Holds when $x\in\lbrack0, \infty\rparen\land y\in\lbrack0, \infty\rparen\land z\in\lbrack0, \infty\rparen$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 23e0a7 · Fungrim entry ↗

$\mathrm{CarlsonRF}(x, y, z)=2\mathrm{CarlsonRF}(x+\sqrt{x}\sqrt{y}+\sqrt{x}\sqrt{z}+\sqrt{y}\sqrt{z}, y+\sqrt{x}\sqrt{y}+\sqrt{x}\sqrt{z}+\sqrt{y}\sqrt{z}, z+\sqrt{x}\sqrt{y}+\sqrt{x}\sqrt{z}+\sqrt{y}\sqrt{z})$

Holds when $x\in\C\land y\in\C\land z\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 2499cd · Fungrim entry ↗

$\mathrm{CarlsonRG}(1, 1, 1)=1$

Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. 250ff1 · Fungrim entry ↗

$\mathrm{CarlsonRC}(1, -1)=\frac{1}{4}(-\imaginaryI\pi\sqrt{2})+\frac{1}{2}(\sqrt{2}\ln(1+\sqrt{2}))$

Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 25435b · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, x, -(cx))=\frac{1}{\sqrt{x}}(\begin{cases}\mathrm{EllipticK}(c+1)&0\le\Re(x)\land\Im(x)=0\lor\Im(x)\lt0\\2\imaginaryI\mathrm{EllipticK}(-c)+\mathrm{EllipticK}(c+1)&\top\end{cases})$

Holds when $x\in\C\land c\in\lbrack0, \infty\rparen$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 271b73 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, 1, 2)=\frac{\sqrt{2}\Gamma(1/4)^2}{8\sqrt{\pi}}$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 28237a · Fungrim entry ↗

$\mathrm{artanh}(\frac{x}{y})=x\mathrm{CarlsonRC}(y^2, y^2-x^2)$

Holds when $y\in\lparen0, \infty\rparen\land x\in\lparen-y, y\rparen$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 2cdd2f · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, -1, 1)=\frac{3\sqrt{2}(1-\imaginaryI)\Gamma(1/4)^2}{16\sqrt{\pi}}-\frac{3\sqrt{2}(1+\imaginaryI)\pi^{1/2}^{3}}{2\Gamma(1/4)^2}$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 2dcf0c · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, -1, -1, -1)=\frac{-3}{4}-\frac{1}{8}(3\sqrt{2}\ln(1+\sqrt{2}))+\frac{1}{16}(3\sqrt{2}\imaginaryI\pi)$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 303827 · Fungrim entry ↗

$\mathrm{CarlsonRD}(1, -1, -1)=\frac{-3}{4}-\frac{1}{8}(3\sqrt{2}\ln(1+\sqrt{2}))+\frac{1}{16}(3\sqrt{2}\imaginaryI\pi)$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 3047b1 · Fungrim entry ↗

$\mathrm{CarlsonRD}(x, y, z)=2\mathrm{CarlsonRD}(x+\sqrt{x}\sqrt{y}+\sqrt{x}\sqrt{z}+\sqrt{y}\sqrt{z}, y+\sqrt{x}\sqrt{y}+\sqrt{x}\sqrt{z}+\sqrt{y}\sqrt{z}, z+\sqrt{x}\sqrt{y}+\sqrt{x}\sqrt{z}+\sqrt{y}\sqrt{z})+(3)((z+\sqrt{x}\sqrt{y}+\sqrt{x}\sqrt{z}+\sqrt{y}\sqrt{z})\sqrt{z})^{-1}$

Holds when $z\ne0\land x\in\C\land y\in\C\land z\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 31a3ba · Fungrim entry ↗

$\arccos(\frac{x}{y})=\mathrm{CarlsonRC}(x^2, y^2)\sqrt{y^2-x^2}$

Holds when $x\in\lbrack0, y\rbrack\land y\in\lparen0, \infty\rparen$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 33e034 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 0, -1, -1)=\infty\imaginaryI$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 3567c5 · Fungrim entry ↗

$\mathrm{CarlsonRC}(0, -1)=-(\frac{\imaginaryI\pi}{2})$

Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 35cb93 · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, y, z)=\frac{1}{4}(3\pi\mathrm{CarlsonHypergeometricR}(\frac{-3}{2}, \bigl\lbrack1/2, 1/2, 1/2, 1/2\bigr\rbrack, \bigl\lbrack y, z, z, z\bigr\rbrack))$

$\mathrm{CarlsonRF}(\mathrm{lamda}+x, \mathrm{lamda}+y, \mathrm{lamda})+\mathrm{CarlsonRF}(\frac{xy}{\mathrm{lamda}}+x, \frac{xy}{\mathrm{lamda}}+y, \frac{xy}{\mathrm{lamda}})=\mathrm{CarlsonRF}(x, y, 0)$

Holds when $x\in\lparen0, \infty\rparen\land y\in\lparen0, \infty\rparen\land\mathrm{lamda}\in\C\setminus\lparen-\infty, 0\rbrack$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 38fa65 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, 2, 2, 2)=\frac{3\pi}{8}-\frac{3}{4}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 397051 · Fungrim entry ↗

$\ln(\frac{x}{y})=(x-y)\mathrm{CarlsonRC}(\frac{(x+y)^2}{4}, xy)$

Holds when $y\in\lparen0, \infty\rparen\land x\in\lparen0, \infty\rparen$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 398bb7 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, -1, -1)=\frac{-(\imaginaryI\pi)}{2}$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 3a84d6 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, y, z, \sqrt{y}\sqrt{z})=\frac{3\mathrm{CarlsonRF}(0, y, z)}{2\sqrt{y}\sqrt{z}}$

Holds when $y\in\C\land z\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind; CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 3b6175 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(x, y, z, z)=\mathrm{CarlsonRD}(x, y, z)$

Holds when $x\in\C\land y\in\C\land z\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind; CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for expansion. 3dd30a · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, y, 1)=\begin{cases}\frac{3(\mathrm{EllipticK}(1-y)-\mathrm{EllipticE}(1-y))}{1-y}&y\ne1\\\frac{3\pi}{4}&y=1\end{cases}$

Holds when $y\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind; EllipticE — Legendre complete elliptic integral of the second kind; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 3e05c6 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, x, 2x)=(\frac{\sqrt{2}\sqrt{\pi}^{3}}{2\Gamma(1/4)^2}+\frac{\sqrt{2}\Gamma(1/4)^2}{16\sqrt{\pi}})\sqrt{x}$

Holds when $x\in\C$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. 3f1547 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, 1, x)=\frac{\mathrm{EllipticE}(1-x)}{2}$

Holds when $x\in\C$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind; EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. 3f6d40 · Fungrim entry ↗

$\mathrm{CarlsonRG}(-x, -y, z)=-(\imaginaryI\mathrm{CarlsonRG}(x, y, -z))^\star$

$\mathrm{CarlsonRF}(0, x, y)=\frac{1}{\sqrt{x}}(\mathrm{EllipticK}(1-\frac{y}{x}))$

Holds when $\vert\arg(x)-\arg(y)\vert\lt\pi\land x\in\C\land y\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 415ff0 · Fungrim entry ↗

$\mathrm{arsinh}(\frac{x}{y})=x\mathrm{CarlsonRC}(x^2+y^2, y^2)$

Holds when $x\in\R\land y\in\lparen0, \infty\rparen$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 423b36 · Fungrim entry ↗

$\mathrm{CarlsonRC}(x, y)=\mathrm{CarlsonHypergeometricR}(\frac{-1}{2}, \bigl\lbrack\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\bigr\rbrack, \bigl\lbrack x, y, y\bigr\rbrack)$

Holds when $y\in\C\setminus\lparen-\infty, 0\rbrack\land x\in\C\setminus\lparen-\infty, 0\rparen$ . Symbols: CarlsonHypergeometricR — Carlson multivariate hypergeometric function; CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 42c7f1 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, 2, 2, 4)=\frac{1}{24}(\pi(9-4\sqrt{3}))$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 44d300 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, x, -(cx))=\frac{1}{2}(\begin{cases}\mathrm{EllipticE}(c+1)&0\le\Re(x)\land\Im(x)=0\lor\Im(x)\lt0\\\mathrm{EllipticE}(c+1)+2\imaginaryI(\mathrm{EllipticK}(-c)-\mathrm{EllipticE}(-c))&\top\end{cases}\sqrt{x})$

Holds when $x\in\C\land c\in\lbrack0, \infty\rparen$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind; EllipticE — Legendre complete elliptic integral of the second kind; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 48333c · Fungrim entry ↗

$\mathrm{CarlsonRD}(-1, -1, -1)=\imaginaryI$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 4a2403 · Fungrim entry ↗

$\mathrm{CarlsonRC}(-x, y)=(\imaginaryI\mathrm{CarlsonRC}(x, -y))^\star$

$\mathrm{CarlsonRF}(0, 2, 4)=\frac{\Gamma(1/4)^2}{8\sqrt{\pi}}$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 4c1988 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, -1, -1, 1)=\frac{-3}{2}-\frac{1}{8}(3\sqrt{2}\imaginaryI\pi)+\frac{1}{4}(3\sqrt{2}\ln(1+\sqrt{2}))$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 4c1db8 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(x, x, x, x)=\sqrt{x}^{-3}$

Holds when $x\in\C$ . Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 4c882a · Fungrim entry ↗

$\mathrm{CarlsonRF}(1, 1, 2)=\ln(1+\sqrt{2})$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 4cd504 · Fungrim entry ↗

$\mathrm{CarlsonRD}(1, 2, 2)=\frac{3\pi}{8}-\frac{3}{4}$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 4d2c10 · Fungrim entry ↗

$\mathrm{CarlsonRG}(1, 1, 2)=\frac{\ln(1+\sqrt{2})}{2}+\frac{\sqrt{2}}{2}$

Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. 4d7098 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(\mathrm{lamda}x, \mathrm{lamda}y, \mathrm{lamda}z, \mathrm{lamda}w)=\frac{\mathrm{CarlsonRJ}(x, y, z, w)}{\sqrt{\mathrm{lamda}}^{3}}$

Holds when $x\in\C\land y\in\C\land z\in\C\land w\in\C\land\mathrm{lamda}\in\lparen0, \infty\rparen$ . Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 4e21c7 · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, y, z)=\frac{\begin{cases}\frac{3(\mathrm{EllipticE}(1-z/y)-(z\mathrm{EllipticK}(1-z/y))/y)}{(z(1-z/y))/y}&z\ne0\land z\ne y\\\frac{3\pi}{4}&z=y\\\tilde\infty&z=0\end{cases}}{\sqrt{y}^{3}}$

Holds when $\vert\arg(y)-\arg(z)\vert\lt\pi\land z\in\C\land y\in\C\setminus\lbrace0\rbrace$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind; EllipticE — Legendre complete elliptic integral of the second kind; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 4e4380 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(\mathrm{lamda}+x, \mathrm{lamda}+y, \mathrm{lamda}, \mathrm{lamda}+w)+\mathrm{CarlsonRJ}(\frac{xy}{\mathrm{lamda}}+x, \frac{xy}{\mathrm{lamda}}+y, \frac{xy}{\mathrm{lamda}}, \frac{xy}{\mathrm{lamda}}+w)=\mathrm{CarlsonRJ}(x, y, 0, w)-3\mathrm{CarlsonRC}((\frac{xy}{\mathrm{lamda}}+\mathrm{lamda}+x+y)w^2, w(\mathrm{lamda}+w)(\frac{xy}{\mathrm{lamda}}+w))$

Holds when $x\in\lparen0, \infty\rparen\land y\in\lparen0, \infty\rparen\land w\in\lparen0, \infty\rparen\land\mathrm{lamda}\in\C\setminus\lparen-\infty, 0\rbrack$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind; CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 4eac3f · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 1, 1, 2)=\frac{3\pi}{2(2+\sqrt{2})}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 522f54 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, 1, -1, -1)=\imaginaryI(\frac{1}{4}(3\times2^{1/2}\ln(1+2^{1/2}))-\frac{3}{2})-\frac{3\sqrt{2}\pi}{8}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 534335 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, x, cx)=\frac{\mathrm{EllipticK}(1-c)}{\sqrt{x}}$

$\mathrm{CarlsonRF}(0, 1, x)=\mathrm{EllipticK}(1-x)$

Holds when $x\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 53d869 · Fungrim entry ↗

$\mathrm{CarlsonRD}(1, 1, -1)=\imaginaryI(\frac{1}{4}(3\times2^{1/2}\ln(1+2^{1/2}))-\frac{3}{2})-\frac{3\sqrt{2}\pi}{8}$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 545e8b · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 0, 0, 0)=\tilde\infty$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 55cd70 · Fungrim entry ↗

$\mathrm{CarlsonRC}(-1, 0)=-(\infty\imaginaryI)$

Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 56d1bc · Fungrim entry ↗

$\arcsin(\frac{x}{y})=x\mathrm{CarlsonRC}(y^2-x^2, y^2)$

Holds when $x\in\lbrack-y, y\rbrack\land y\in\lparen0, \infty\rparen$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 584a61 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, y, z, w)=\frac{1}{4}(3\pi\mathrm{CarlsonHypergeometricR}(\frac{-3}{2}, \bigl\lbrack1/2, 1/2, 1\bigr\rbrack, \bigl\lbrack y, z, w\bigr\rbrack))$

Holds when $y\in\C\setminus\lparen-\infty, 0\rbrack\land z\in\C\setminus\lparen-\infty, 0\rbrack\land w\in\C\setminus\lparen-\infty, 0\rbrack$ . Symbols: CarlsonHypergeometricR — Carlson multivariate hypergeometric function; CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 5a8f57 · Fungrim entry ↗

$\mathrm{CarlsonRF}(-x, -y, -z)=-(\imaginaryI\mathrm{CarlsonRF}(x, y, z))$

$\mathrm{CarlsonRC}(x, y)=\begin{cases}\frac{\arctan((y/x-1)^{1/2})}{\sqrt{y-x}}&x\lt y\\\frac{1}{\sqrt{x}}&x=y\\\frac{\mathrm{artanh}((1-y/x)^{1/2})}{\sqrt{x-y}}&y\lt x\end{cases}$

$\mathrm{CarlsonRF}(0, -1, -2)=-(\frac{\sqrt{2}\imaginaryI\Gamma(1/4)^2}{8\sqrt{\pi}})$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 5c178f · Fungrim entry ↗

$\mathrm{CarlsonRC}(0, 0)=\tilde\infty$

Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 5c2b08 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(x, x, x, w)=\mathrm{CarlsonRD}(w, w, x)$

Holds when $x\in\C\land w\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind; CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for expansion. 5c6f10 · Fungrim entry ↗

$\mathrm{CarlsonRG}(x, y, y)=\frac{1}{2}(\begin{cases}y\mathrm{CarlsonRC}(x, y)+\sqrt{x}&y\ne0\\\sqrt{x}&y=0\end{cases})$

$\mathrm{CarlsonRD}(0, 1, z)=\begin{cases}\frac{3(\mathrm{EllipticE}(1-z)-z\mathrm{EllipticK}(1-z))}{z(1-z)}&z\ne0\land z\ne1\\\frac{3\pi}{4}&z=1\\\tilde\infty&z=0\end{cases}$

Holds when $z\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind; EllipticE — Legendre complete elliptic integral of the second kind; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 61c002 · Fungrim entry ↗

$\mathrm{CarlsonRC}(x, y)=\mathrm{CarlsonRF}(x, y, y)$

Holds when $x\in\C\land y\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind; CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for expansion. 61f98d · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, -1, 1, 1)=\frac{3\sqrt{2}(1-\imaginaryI)\Gamma(1/4)^2}{16\sqrt{\pi}}-\frac{3\sqrt{2}(1+\imaginaryI)\pi^{1/2}^{3}}{2\Gamma(1/4)^2}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 62b0c4 · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, 2, 1)=\frac{3\sqrt{2}\sqrt{\pi}^{3}}{\Gamma(1/4)^2}$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 63644d · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 1, 1, 1)=\frac{3\pi}{4}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 64a808 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(-x, -y, -z, -w)=\imaginaryI\mathrm{CarlsonRJ}(x, y, z, w)$

Holds when $x\in\lparen0, \infty\rbrack\land y\in\lparen0, \infty\rbrack\land z\in\lparen0, \infty\rbrack\land w\in\lparen0, \infty\rbrack$ . Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 64d87a · Fungrim entry ↗

$\mathrm{CarlsonRJ}(x, y, z, w)=\mathrm{CarlsonRJ}(x, z, y, w)=\mathrm{CarlsonRJ}(y, x, z, w)=\mathrm{CarlsonRJ}(y, z, x, w)=\mathrm{CarlsonRJ}(z, x, y, w)=\mathrm{CarlsonRJ}(z, y, x, w)$

Holds when $x\in\C\land y\in\C\land z\in\C\land w\in\C$ . Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for expansion. 655a2b · Fungrim entry ↗

$\mathrm{CarlsonRF}(-1, -1, -1)=-\imaginaryI$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 6674bb · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, \frac{\Gamma(1/4)^4}{16\pi}, \frac{\Gamma(1/4)^4}{32\pi})=1$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 67e015 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, \frac{1}{2}, 1)=\frac{\Gamma(1/4)^2}{4\sqrt{\pi}}$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 6c4567 · Fungrim entry ↗

$\mathrm{CarlsonRD}(y, z, x)+\mathrm{CarlsonRD}(z, x, y)+\mathrm{CarlsonRD}(x, y, z)=(3)(\sqrt{x}\sqrt{y}\sqrt{z})^{-1}$

Holds when $x\ne0\land y\ne0\land x\in\C\land y\in\C\land z\in\C$ or $z\ne0$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 6dda7a · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, 1, 2, 4)=-\pi\frac{\sqrt{2}}{8}+\ln(1+\sqrt{2})$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 6e9544 · Fungrim entry ↗

$\mathrm{CarlsonRC}(x, y)=\begin{cases}\frac{\arccos(x/y^{1/2})}{\sqrt{y-x}}&x\lt y\\\frac{1}{\sqrt{x}}&x=y\\\frac{\mathrm{arcosh}(x/y^{1/2})}{\sqrt{x-y}}&y\lt x\end{cases}$

$\mathrm{CarlsonRC}(1, x)=\mathrm{Hypergeometric2F_1}(1, \frac{1}{2}, \frac{3}{2}, 1-x)$

Holds when $x\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 72b5bd · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, y, z, w)=\frac{1}{4}(3\pi\mathrm{CarlsonHypergeometricR}(\frac{-3}{2}, \bigl\lbrack1/2, 1/2, 1/2, 1/2\bigr\rbrack, \bigl\lbrack y, z, w, w\bigr\rbrack))$

$\mathrm{CarlsonRC}(x, -(cx))=\frac{\begin{cases}\mathrm{artanh}((c+1)^{1/2})&0\le\Re(x)\land\Im(x)=0\lor\Im(x)\lt0\\\mathrm{artanh}((c+1)^{1/2})+\imaginaryI\pi&\top\end{cases}}{\sqrt{x}\sqrt{c+1}}$

Holds when $x\in\C\land c\in\lparen0, \infty\rparen$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 7348e3 · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, 0, 1)=\infty$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 748131 · Fungrim entry ↗

$\mathrm{CarlsonRD}(x, x, y)=\begin{cases}\frac{3(\mathrm{CarlsonRC}(y, x)-\frac{1}{y^{1/2}})}{y-x}&x\ne y\\\sqrt{x}^{-3}&x=y\end{cases}$

Holds when $x\in\C\land y\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind; CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 771801 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, -1, -1, 1)=-(\frac{1}{4}(3\pi(1+\imaginaryI)))$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 78131f · Fungrim entry ↗

$\mathrm{CarlsonRF}(\mathrm{lamda}x, \mathrm{lamda}y, \mathrm{lamda}z)=\frac{\mathrm{CarlsonRF}(x, y, z)}{\sqrt{\mathrm{lamda}}}$

Holds when $x\in\C\land y\in\C\land z\in\C\land\mathrm{lamda}\in\lparen0, \infty\rparen$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for expansion. 7a168a · Fungrim entry ↗

$\arctan(\frac{x}{y})=x\mathrm{CarlsonRC}(y^2, x^2+y^2)$

$\mathrm{CarlsonRG}(0, x, -x)=\frac{\sqrt{2}\begin{cases}1+\imaginaryI&0\le\Re(x)\land\Im(x)=0\lor\Im(x)\lt0\\1-\imaginaryI&\top\end{cases}\sqrt{\pi}^{3}\sqrt{x}}{2\Gamma(1/4)^2}$

Holds when $x\in\C$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. 7c50d1 · Fungrim entry ↗

$\mathrm{CarlsonRC}(x, 0)=\begin{cases}\infty\mathrm{sgn}(\frac{1}{x^{1/2}})&x\ne0\\\tilde\infty&x=0\end{cases}$

Holds when $x\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 7cbe17 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, x, y)=\frac{1}{2}(\mathrm{EllipticE}(1-\frac{y}{x})\sqrt{x})$

Holds when $\vert\arg(x)-\arg(y)\vert\lt\pi\land x\in\C\land y\in\C$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind; EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. 7cddc6 · Fungrim entry ↗

$\mathrm{CarlsonRC}(x, y)=\mathrm{CarlsonHypergeometricR}(\frac{-1}{2}, \bigl\lbrack\frac{1}{2}, 1\bigr\rbrack, \bigl\lbrack x, y\bigr\rbrack)$

$\mathrm{CarlsonRC}(-1, 1)=\frac{1}{2}(-\imaginaryI\sqrt{2}\ln(1+\sqrt{2}))+\frac{\pi\sqrt{2}}{4}$

Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 7ea1ad · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 1, 2, \sqrt{2})=\frac{3\Gamma(1/4)^2}{16\sqrt{\pi}}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 7f8a58 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(-x, -y, -z, w)=-(\imaginaryI\mathrm{CarlsonRJ}(x, y, z, -w))^\star$

$\mathrm{CarlsonRD}(0, 1, 1)=\frac{3\pi}{4}$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 84ea08 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, 1, 2)=\frac{\sqrt{2}\sqrt{\pi}^{3}}{2\Gamma(1/4)^2}+\frac{\sqrt{2}\Gamma(1/4)^2}{16\sqrt{\pi}}$

Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. 84f403 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, \frac{\Gamma(1/4)^4}{32\pi}, \frac{-\Gamma(1/4)^4}{32\pi})=1-\imaginaryI$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 8519dd · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, 1, 1)=\frac{\pi}{2}$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 8bb972 · Fungrim entry ↗

$\mathrm{CarlsonRC}(x, cx)=\begin{cases}\frac{\arctan((c-1)^{1/2})}{\sqrt{x(c-1)}}&1\lt c\\\frac{1}{\sqrt{x}}&c=1\\\frac{\mathrm{artanh}((1-c)^{1/2})}{\sqrt{x(1-c)}}&c\lt1\end{cases}$

$\mathrm{CarlsonRD}(0, y, z)=\frac{\begin{cases}\frac{3(\mathrm{EllipticK}(1-y/z)-\mathrm{EllipticE}(1-y/z))}{1-y/z}&y\ne z\\\frac{3\pi}{4}&y=z\end{cases}}{\sqrt{z}^{3}}$

Holds when $\vert\arg(y)-\arg(z)\vert\lt\pi\land y\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind; EllipticE — Legendre complete elliptic integral of the second kind; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 8d0629 · Fungrim entry ↗

$\mathrm{CarlsonRF}(x, y, z)=\mathrm{CarlsonRF}(\frac{1}{4}(x+\sqrt{x}\sqrt{y}+\sqrt{x}\sqrt{z}+\sqrt{y}\sqrt{z}), \frac{1}{4}(y+\sqrt{x}\sqrt{y}+\sqrt{x}\sqrt{z}+\sqrt{y}\sqrt{z}), \frac{1}{4}(z+\sqrt{x}\sqrt{y}+\sqrt{x}\sqrt{z}+\sqrt{y}\sqrt{z}))$

$\mathrm{CarlsonRC}(x, y)=2\mathrm{CarlsonRC}(x+y+2\sqrt{x}\sqrt{y}, y+y+2\sqrt{x}\sqrt{y})$

Holds when $x\in\C\land y\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 8f5d76 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, 0, -1)=-(\infty\imaginaryI)$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 90af98 · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, 0, 0)=\tilde\infty$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 980014 · Fungrim entry ↗

$\mathrm{CarlsonRG}(x, x, x)=\sqrt{x}$

Holds when $x\in\C$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. 990145 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, 0, 1)=\infty$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. 9a95a5 · Fungrim entry ↗

$\mathrm{CarlsonRF}(x, x, x)=\frac{1}{\sqrt{x}}$

Holds when $x\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for expansion. 9b0388 · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, x, 1)=\frac{1}{4}(3\pi\mathrm{Hypergeometric2F_1}(\frac{1}{2}, \frac{3}{2}, 2, 1-x))$

Holds when $x\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 9bfd88 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, 1, -1)=\frac{\sqrt{2}(1+\imaginaryI)\sqrt{\pi}^{3}}{2\Gamma(1/4)^2}$

Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. 9e30e7 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 1, 2, 1)=\frac{3\sqrt{2}\sqrt{\pi}^{3}}{\Gamma(1/4)^2}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. 9f2b18 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(-1, -1, -1, -1)=\imaginaryI$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. a091d1 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, 2, 2, 1)=3-\frac{3\pi}{4}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. a1414f · Fungrim entry ↗

$\mathrm{CarlsonRC}(2, 1)=\ln(1+\sqrt{2})$

Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. a15c03 · Fungrim entry ↗

$\mathrm{CarlsonRD}(\mathrm{lamda}, \mathrm{lamda}+x, \mathrm{lamda}+y)+\mathrm{CarlsonRD}(\frac{xy}{\mathrm{lamda}}, \frac{xy}{\mathrm{lamda}}+x, \frac{xy}{\mathrm{lamda}}+y)=\mathrm{CarlsonRD}(0, x, y)-(3)(y\sqrt{\frac{xy}{\mathrm{lamda}}+\mathrm{lamda}+x+y})^{-1}$

Holds when $x\in\lparen0, \infty\rparen\land y\in\lparen0, \infty\rparen\land\mathrm{lamda}\in\C\setminus\lparen-\infty, 0\rbrack$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. a203e9 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, x, cx)=\frac{1}{2}(\mathrm{EllipticE}(1-c)\sqrt{x})$

Holds when $x\in\C\land c\in\lbrack0, \infty\rparen$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind; EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. a2e9dd · Fungrim entry ↗

$\mathrm{CarlsonRC}(\mathrm{lamda}x, \mathrm{lamda}y)=\frac{\mathrm{CarlsonRC}(x, y)}{\sqrt{\mathrm{lamda}}}$

Holds when $x\in\C\land y\in\C\land\mathrm{lamda}\in\lparen0, \infty\rparen$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. a839d5 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, 1, 2, 2)=3\ln(1+\sqrt{2})-\frac{3\sqrt{2}}{2}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. a9f190 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, x, x)=\frac{\pi}{2\sqrt{x}}$

Holds when $x\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. ab5af3 · Fungrim entry ↗

$\mathrm{CarlsonRC}(x, x)=\frac{1}{\sqrt{x}}$

Holds when $x\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. ad96f4 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, 1, 1, 0)=\infty$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. b07652 · Fungrim entry ↗

$\mathrm{CarlsonRC}(2x, x)=\frac{\ln(1+\sqrt{2})}{\sqrt{x}}$

Holds when $x\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. b136bd · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, 1, 1, 2)=3-\frac{3\pi}{4}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. b1c84e · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, x, 1)=\frac{1}{2}(\pi\mathrm{Hypergeometric2F_1}(\frac{1}{2}, \frac{1}{2}, 1, 1-x))$

Holds when $x\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. b2fdfe · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 1, 1, -1)=-(\frac{1}{4}(3\pi(1+\imaginaryI)))$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. b468f3 · Fungrim entry ↗

$\mathrm{CarlsonRG}(x, y, z)=\mathrm{CarlsonRG}(x, z, y)=\mathrm{CarlsonRG}(y, x, z)=\mathrm{CarlsonRG}(y, z, x)=\mathrm{CarlsonRG}(z, x, y)=\mathrm{CarlsonRG}(z, y, x)$

Holds when $x\in\C\land y\in\C\land z\in\C$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for expansion. b478a1 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, y, z)=\frac{1}{4}(\pi\mathrm{CarlsonHypergeometricR}(\frac{1}{2}, \bigl\lbrack1/2, 1/2\bigr\rbrack, \bigl\lbrack y, z\bigr\rbrack))$

Holds when $y\in\C\setminus\lparen-\infty, 0\rbrack\land z\in\C\setminus\lparen-\infty, 0\rbrack$ . Symbols: CarlsonHypergeometricR — Carlson multivariate hypergeometric function; CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. b4a735 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 0, 1, 1)=\infty$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. b891d1 · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, 1, x)=\frac{3\pi\mathrm{Hypergeometric2F_1}(\frac{1}{2}, \frac{1}{2}, 2, 1-x)}{4x}$

$\mathrm{CarlsonRC}(-x, y)=\frac{\frac{\pi}{2}-\imaginaryI\mathrm{artanh}(x/(x+y)^{1/2})}{\sqrt{x+y}}$

$\mathrm{CarlsonRG}(0, 0, 0)=0$

Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. bcc121 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 1, 2, 2)=\frac{3\sqrt{2}\Gamma(1/4)^2}{16\sqrt{\pi}}-\frac{3\sqrt{2}\sqrt{\pi}^{3}}{2\Gamma(1/4)^2}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. c05ed8 · Fungrim entry ↗

$\mathrm{CarlsonRF}(1, 1, 1)=1$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. c166ca · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, 16, 16)=\pi$

Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. c5a9cf · Fungrim entry ↗

$\mathrm{CarlsonRD}(x, y, y)=\begin{cases}\frac{3(\mathrm{CarlsonRC}(x, y)-\frac{x^{1/2}}{y})}{2(y-x)}&x\ne y\\\sqrt{x}^{-3}&x=y\end{cases}$

$\mathrm{CarlsonRD}(x, x, x)=\sqrt{x}^{-3}$

Holds when $x\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. ccb4d1 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, 1, 1)=\frac{\pi}{4}$

Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. cd55cf · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, x, x)=\frac{\pi\sqrt{x}}{4}$

Holds when $x\in\C$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. cdb587 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, -1, -1, -1)=\frac{3\imaginaryI\pi}{4}$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. cdee01 · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, \imaginaryI, -\imaginaryI)=\frac{\Gamma(1/4)^2}{4\sqrt{\pi}}$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. cf5caa · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, y, z)=\frac{1}{2}(\pi\mathrm{CarlsonHypergeometricR}(\frac{-1}{2}, \bigl\lbrack1/2, 1/2\bigr\rbrack, \bigl\lbrack y, z\bigr\rbrack))$

Holds when $y\in\C\setminus\lparen-\infty, 0\rbrack\land z\in\C\setminus\lparen-\infty, 0\rbrack$ . Symbols: CarlsonHypergeometricR — Carlson multivariate hypergeometric function; CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. d0c9ff · Fungrim entry ↗

$\mathrm{CarlsonRC}(1, 1)=1$

Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. d38c27 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(x, y, y, w)=\begin{cases}\frac{3(\mathrm{CarlsonRC}(x, y)-\mathrm{CarlsonRC}(x, w))}{w-y}&y\ne w\\\frac{3(\mathrm{CarlsonRC}(x, y)-\frac{x^{1/2}}{y})}{2(y-x)}&x\ne y\land y=w\\\sqrt{x}^{-3}&x=y=w\end{cases}$

Holds when $x\in\C\land y\in\C\land w\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind; CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. d4b12e · Fungrim entry ↗

$\mathrm{CarlsonRG}(1, 2, 2)=\frac{1}{2}+\frac{\pi}{4}$

Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. d51efc · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, -1, -1)=\frac{3\imaginaryI\pi}{4}$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. d52bda · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, 0, 1)=\frac{1}{2}$

Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. d5ff09 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, 0, x)=\frac{\sqrt{x}}{2}$

Holds when $x\in\C$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. d829be · Fungrim entry ↗

$\mathrm{arcosh}(\frac{x}{y})=\mathrm{CarlsonRC}(x^2, y^2)\sqrt{x^2-y^2}$

Holds when $x\in\lbrack y, \infty\rparen\land y\in\lparen0, \infty\rparen$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. d9765b · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, 1, 0)=\tilde\infty$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. dbe634 · Fungrim entry ↗

$\mathrm{CarlsonRC}(-x, -y)=-(\imaginaryI\mathrm{CarlsonRC}(x, y))$

$\mathrm{CarlsonRJ}(1, 1, 1, -1)=\frac{-3}{2}-\frac{1}{8}(3\sqrt{2}\imaginaryI\pi)+\frac{1}{4}(3\sqrt{2}\ln(1+\sqrt{2}))$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. e04867 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 0, z, w)=\begin{cases}\infty\mathrm{sgn}(\frac{1}{wz^{1/2}})&z\ne0\land w\ne0\\\tilde\infty&\top\end{cases}$

Holds when $z\in\C\land w\in\C$ . Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. e1a3cb · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, 1, 12\sqrt{2}-16)=\frac{(2+\sqrt{2})\Gamma(1/4)^2}{16\sqrt{\pi}}$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. e30d7e · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, 0, 0)=\tilde\infty$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. e39456 · Fungrim entry ↗

$\mathrm{CarlsonRC}(0, 1)=\frac{\pi}{2}$

Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. e464ec · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, x, -x)=\frac{\sqrt{2}\begin{cases}1-\imaginaryI&0\le\Re(x)\land\Im(x)=0\lor\Im(x)\lt0\\1+\imaginaryI&\top\end{cases}\Gamma(1/4)^2}{8\sqrt{\pi}\sqrt{x}}$

Holds when $x\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. e54e61 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 1, 1, 0)=\tilde\infty$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. e60205 · Fungrim entry ↗

$\mathrm{CarlsonRG}(0, x, 1)=\frac{1}{4}(\pi\mathrm{Hypergeometric2F_1}(\frac{-1}{2}, \frac{1}{2}, 1, 1-x))$

Holds when $x\in\C$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. e98dd0 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(1, 1, 1, 1)=1$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. e9d5a9 · Fungrim entry ↗

$\mathrm{CarlsonRC}(1, 2)=\frac{\pi}{4}$

Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. eac389 · Fungrim entry ↗

$\mathrm{CarlsonRC}(1, y+1)=\begin{cases}\frac{\arctan(y^{1/2})}{\sqrt{y}}&y\ne0\\1&y=0\end{cases}$

Holds when $y\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. eb1d4f · Fungrim entry ↗

$\mathrm{CarlsonRF}(x, x, y)=\mathrm{CarlsonRC}(y, x)$

$\mathrm{CarlsonRD}(2, 2, 1)=3-\frac{3\pi}{4}$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. eda57d · Fungrim entry ↗

$\mathrm{CarlsonRD}(0, 0, z)=\begin{cases}\infty\mathrm{sgn}(z^{1/2}^{-3})&z\ne0\\\tilde\infty&\top\end{cases}$

Holds when $z\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. f07e9d · Fungrim entry ↗

$\mathrm{CarlsonRF}(0, 1, -1)=\frac{\sqrt{2}(1-\imaginaryI)\Gamma(1/4)^2}{8\sqrt{\pi}}$

Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. f1dd8a · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 0, 0, 1)=\tilde\infty$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. f1fd51 · Fungrim entry ↗

$\mathrm{CarlsonRF}(x, y, z)=\mathrm{CarlsonRF}(x, z, y)=\mathrm{CarlsonRF}(y, x, z)=\mathrm{CarlsonRF}(y, z, x)=\mathrm{CarlsonRF}(z, x, y)=\mathrm{CarlsonRF}(z, y, x)$

Holds when $x\in\C\land y\in\C\land z\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for expansion. f29729 · Fungrim entry ↗

$\mathrm{CarlsonRD}(1, 1, 2)=3\ln(1+\sqrt{2})-\frac{3\sqrt{2}}{2}$

Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. f47947 · Fungrim entry ↗

$\mathrm{CarlsonRD}(-x, -y, -z)=\imaginaryI\mathrm{CarlsonRD}(x, y, z)$

$\mathrm{CarlsonRG}(\mathrm{lamda}x, \mathrm{lamda}y, \mathrm{lamda}z)=\mathrm{CarlsonRG}(x, y, z)\sqrt{\mathrm{lamda}}$

Holds when $x\in\C\land y\in\C\land z\in\C\land\mathrm{lamda}\in\lparen0, \infty\rparen$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. f9ca94 · Fungrim entry ↗

$\mathrm{CarlsonRJ}(0, 0, -1, 1)=-(\infty\imaginaryI)$

Symbols: CarlsonRJ — Carlson symmetric elliptic integral of the third kind. Used by the Compute Engine for simplification. fd3017 · Fungrim entry ↗

$\mathrm{CarlsonRG}(x, y, z)=\mathrm{CarlsonHypergeometricR}(\frac{1}{2}, \bigl\lbrack\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\bigr\rbrack, \bigl\lbrack x, y, z\bigr\rbrack)$

Holds when $x\in\C\setminus\lparen-\infty, 0\rparen\land y\in\C\setminus\lparen-\infty, 0\rparen\land z\in\C\setminus\lparen-\infty, 0\rparen$ . Symbols: CarlsonHypergeometricR — Carlson multivariate hypergeometric function; CarlsonRG — Carlson symmetric elliptic integral of the second kind. Used by the Compute Engine for simplification. fda084 · Fungrim entry ↗

$\mathrm{CarlsonRC}(0, y)=\begin{cases}\frac{\pi}{2\sqrt{y}}&y\ne0\\\tilde\infty&y=0\end{cases}$

Holds when $y\in\C$ . Symbols: CarlsonRC — Degenerate Carlson symmetric elliptic integral of the first kind. Used by the Compute Engine for simplification. ff58cf · Fungrim entry ↗

Legendre elliptic integrals

$\mathrm{IncompleteEllipticF}(\frac{-\pi}{2}, 1)=-\infty$

Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. 04c829 · Fungrim entry ↗

$\mathrm{EllipticPi}(1, 0)=\tilde\infty$

Symbols: EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. 061c49 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\arcsin(\frac{1}{m^{1/2}}), m)=\frac{\mathrm{EllipticK}(\frac{1}{m})}{\sqrt{m}}$

Holds when $m\in\C\setminus\lbrace0\rbrace$ . Symbols: EllipticK — Legendre complete elliptic integral of the first kind; IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for expansion. 087a7c · Fungrim entry ↗

$\mathrm{EllipticK}(\frac{1}{2}(1+\sqrt{3}\imaginaryI))=\frac{\sqrt[4]{3}\exp(\frac{\imaginaryI\pi}{12})\Gamma(1/3)^3}{\pi\times2^{\frac{7}{3}}}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 0abbe1 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\frac{\pi}{2}, m)=\mathrm{EllipticK}(m)$

Holds when $m\in\C$ . Symbols: EllipticK — Legendre complete elliptic integral of the first kind; IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. 0b8fd6 · Fungrim entry ↗

$\mathrm{EllipticK}(m)=\mathrm{CarlsonRF}(0, 1-m, 1)$

Holds when $m\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 0cc11f · Fungrim entry ↗

$\mathrm{EllipticPi}(\frac{1}{2}, 0)=\frac{\pi\sqrt{2}}{2}$

Symbols: EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. 124d02 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\frac{\pi}{2}, 1)=\infty$

Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. 16612f · Fungrim entry ↗

$\mathrm{EllipticE}(m)=\frac{1}{2}(\pi\mathrm{Hypergeometric2F_1}(\frac{-1}{2}, \frac{1}{2}, 1, m))$

Holds when $m\in\C$ . Symbols: EllipticE — Legendre complete elliptic integral of the second kind; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 16d2e1 · Fungrim entry ↗

$\mathrm{EllipticK}(\frac{1}{2}(1-\imaginaryI\sqrt{3}))=\frac{\sqrt[4]{3}\exp(-((\imaginaryI\pi)/12))\Gamma(1/3)^3}{\pi\times2^{\frac{7}{3}}}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 175b7a · Fungrim entry ↗

$\mathrm{EllipticPi}(0, 1)=\infty$

Symbols: EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. 18e226 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\frac{\pi}{2}, m)=\mathrm{EllipticE}(m)$

Holds when $m\in\C$ . Symbols: EllipticE — Legendre complete elliptic integral of the second kind; IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. 1b881e · Fungrim entry ↗

$\mathrm{EllipticE}(0)=\frac{\pi}{2}$

Symbols: EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. 1d62a7 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\frac{\pi k}{2}, 1)=k$

Holds when $k\in\Z$ . Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. 2245df · Fungrim entry ↗

$\mathrm{IncompleteEllipticPi}(n, -\phi, m)=-\mathrm{IncompleteEllipticPi}(n, \phi, m)$

Holds when $n\in\C\land\phi\in\C\land m\in\C$ . Symbols: IncompleteEllipticPi — Legendre incomplete elliptic integral of the third kind. Used by the Compute Engine for expansion. 255d81 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\frac{\pi}{2}, -1)=\sqrt{2}(\frac{\sqrt{\pi}^{3}}{\Gamma(1/4)^2}+\frac{\Gamma(1/4)^2}{8\sqrt{\pi}})$

Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. 2573ba · Fungrim entry ↗

$\mathrm{EllipticK}((3-2\sqrt{2})^2)=\frac{(2+\sqrt{2})\Gamma(1/4)^2}{16\sqrt{\pi}}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 2991b5 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\frac{-\pi}{2}, m)=-\mathrm{EllipticE}(m)$

$\mathrm{IncompleteEllipticE}(\frac{\pi}{3}, 1)=\frac{\sqrt{3}}{2}$

Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. 3aed02 · Fungrim entry ↗

$\mathrm{EllipticE}(\frac{1}{2})=\frac{\sqrt{\pi}^{3}}{\Gamma(1/4)^2}+\frac{\Gamma(1/4)^2}{8\sqrt{\pi}}$

Symbols: EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. 3b272e · Fungrim entry ↗

$\mathrm{EllipticPi}(0, \frac{1}{2})=\frac{\Gamma(1/4)^2}{4\sqrt{\pi}}$

Symbols: EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. 3c4979 · Fungrim entry ↗

$\mathrm{EllipticK}(\frac{1}{2}-\frac{3^{1/2}}{4})=\frac{\sqrt[4]{3}\Gamma(1/3)^3}{4\pi\sqrt[3]{2}}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 40a376 · Fungrim entry ↗

$\mathrm{EllipticE}(m)=\frac{1}{3}((1-m)(\mathrm{CarlsonRD}(0, 1-m, 1)+\mathrm{CarlsonRD}(0, 1, 1-m)))$

Holds when $m\ne1\land m\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind; EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. 41cf8e · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(0, m)=0$

Holds when $m\in\C$ . Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. 4268fc · Fungrim entry ↗

$\mathrm{EllipticK}(1)=\infty$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 45b157 · Fungrim entry ↗

$\mathrm{EllipticK}(\frac{1}{8}(4-3\sqrt{2}))=\frac{\Gamma(1/4)^2}{4\sqrt[4]{2}\sqrt{\pi}}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 4b040d · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\frac{\pi}{4}, 2)=\frac{\sqrt{2}\sqrt{\pi}^{3}}{\Gamma(1/4)^2}$

Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. 4dabda · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\frac{\pi}{2}, 0)=\frac{\pi}{2}$

Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. 51a946 · Fungrim entry ↗

$\mathrm{EllipticE}(m)-(1-m)\mathrm{EllipticK}(m)=\frac{1}{3}(m(1-m)\mathrm{CarlsonRD}(0, 1, 1-m))$

Holds when $m\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind; EllipticE — Legendre complete elliptic integral of the second kind; EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 55d23d · Fungrim entry ↗

$\mathrm{EllipticE}(2)=\frac{\sqrt{2}(1+\imaginaryI)\sqrt{\pi}^{3}}{\Gamma(1/4)^2}$

Symbols: EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. 5d2c01 · Fungrim entry ↗

$\mathrm{EllipticPi}(n, 0)=(\pi)(2\sqrt{1-n})^{-1}$

Holds when $n\in\C$ . Symbols: EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. 5d8804 · Fungrim entry ↗

$\mathrm{IncompleteEllipticPi}(n, \pi k+\phi, m)=2k\mathrm{EllipticPi}(n, m)+\mathrm{IncompleteEllipticPi}(n, \phi, m)$

Holds when $n\ne1\land m\ne1\land n\in\C\land\phi\in\C\land m\in\C\land k\in\Z$ . Symbols: EllipticPi — Legendre complete elliptic integral of the third kind; IncompleteEllipticPi — Legendre incomplete elliptic integral of the third kind. Used by the Compute Engine for simplification. 5f84d9 · Fungrim entry ↗

$\mathrm{EllipticPi}(0, 0)=\frac{\pi}{2}$

Symbols: EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. 618a54 · Fungrim entry ↗

$\mathrm{EllipticK}(2)=\frac{\sqrt{2}(1-\imaginaryI)\Gamma(1/4)^2}{8\sqrt{\pi}}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. 630eca · Fungrim entry ↗

$\mathrm{EllipticE}(m)=2\mathrm{CarlsonRG}(0, 1-m, 1)$

Holds when $m\in\C$ . Symbols: CarlsonRG — Carlson symmetric elliptic integral of the second kind; EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. 6520e7 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\pi k+\phi, m)=2k\mathrm{EllipticK}(m)+\mathrm{IncompleteEllipticF}(\phi, m)$

Holds when $m\ne1\land\phi\in\C\land m\in\C\land k\in\Z$ . Symbols: EllipticK — Legendre complete elliptic integral of the first kind; IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. 685126 · Fungrim entry ↗

$\mathrm{EllipticK}(m^\star)=\mathrm{EllipticK}(m)^\star$

Holds when $m\in\C\setminus\lparen1, \infty\rparen$ . Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for expansion. 713966 · Fungrim entry ↗

$2\mathrm{EllipticE}(m)-\mathrm{EllipticK}(m)=\frac{1}{2}(\pi\mathrm{Hypergeometric2F_1}(\frac{-1}{2}, \frac{3}{2}, 1, m))$

Holds when $m\in\C$ . Symbols: EllipticE — Legendre complete elliptic integral of the second kind; EllipticK — Legendre complete elliptic integral of the first kind; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 752619 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\frac{-\pi}{2}, m)=-\mathrm{EllipticK}(m)$

$\mathrm{IncompleteEllipticF}(\frac{\pi}{4}, 2)=\frac{\sqrt{2}\Gamma(1/4)^2}{8\sqrt{\pi}}$

Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. 8b4be6 · Fungrim entry ↗

$\mathrm{EllipticE}(m^\star)=\mathrm{EllipticE}(m)^\star$

Holds when $m\in\C\setminus\lparen1, \infty\rparen$ . Symbols: EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for expansion. 8e5c81 · Fungrim entry ↗

$\mathrm{IncompleteEllipticPi}(n, \phi, m)=\frac{1}{3}(n\mathrm{CarlsonRJ}(\cos(\phi)^2, 1-m\sin(\phi)^2, 1, 1-n\sin(\phi)^2)\sin(\phi)^3)+\sin(\phi)\mathrm{CarlsonRF}(\cos(\phi)^2, 1-m\sin(\phi)^2, 1)$

Holds when $\frac{-\pi}{2}\le\Re(\phi)\le\frac{\pi}{2}\land n\in\C\land\phi\in\C\land m\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind; CarlsonRJ — Carlson symmetric elliptic integral of the third kind; IncompleteEllipticPi — Legendre incomplete elliptic integral of the third kind. Used by the Compute Engine for simplification. 8f4e31 · Fungrim entry ↗

$\mathrm{EllipticPi}(m, m)=\frac{\mathrm{EllipticE}(m)}{1-m}$

Holds when $m\in\C$ . Symbols: EllipticE — Legendre complete elliptic integral of the second kind; EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. 9227bf · Fungrim entry ↗

$\mathrm{EllipticK}(m)-\mathrm{EllipticE}(m)=\frac{1}{3}(m\mathrm{CarlsonRD}(0, 1-m, 1))$

$\mathrm{EllipticE}(1)=1$

Symbols: EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. 958a3f · Fungrim entry ↗

$\mathrm{EllipticPi}(\frac{1}{2}, \frac{1}{2})=\frac{2\sqrt{\pi}^{3}}{\Gamma(1/4)^2}+\frac{\Gamma(1/4)^2}{4\sqrt{\pi}}$

Symbols: EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. 9b0385 · Fungrim entry ↗

$\mathrm{EllipticPi}(n, m)=\frac{1}{3}(n\mathrm{CarlsonRJ}(0, 1-m, 1, 1-n))+\mathrm{CarlsonRF}(0, 1-m, 1)$

Holds when $m\ne1\land n\in\C\land m\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind; CarlsonRJ — Carlson symmetric elliptic integral of the third kind; EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. 9ccaef · Fungrim entry ↗

$\mathrm{EllipticE}(-1)=\sqrt{2}(\frac{\sqrt{\pi}^{3}}{\Gamma(1/4)^2}+\frac{\Gamma(1/4)^2}{8\sqrt{\pi}})$

Symbols: EllipticE — Legendre complete elliptic integral of the second kind. Used by the Compute Engine for simplification. 9f3474 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\frac{\pi k}{2}, m)=k\mathrm{EllipticE}(m)$

Holds when $m\in\C\land k\in\Z$ . Symbols: EllipticE — Legendre complete elliptic integral of the second kind; IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. a14442 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(0, 0)=0$

Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. a6c07e · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\frac{\pi}{6}, 1)=\frac{\ln(3)}{2}$

Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. a91f8d · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(-\phi, m)=-\mathrm{IncompleteEllipticE}(\phi, m)$

Holds when $\phi\in\C\land m\in\C$ . Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for expansion. aa1b8e · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\frac{\pi}{6}, 4)=\frac{\mathrm{EllipticK}(\frac{1}{4})}{2}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind; IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. aac129 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\frac{\pi}{2}, -1)=\frac{\sqrt{2}\Gamma(1/4)^2}{8\sqrt{\pi}}$

Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. ace837 · Fungrim entry ↗

$\mathrm{EllipticK}(-1)=\frac{\sqrt{2}\Gamma(1/4)^2}{8\sqrt{\pi}}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. afb22a · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(-\phi, m)=-\mathrm{IncompleteEllipticF}(\phi, m)$

Holds when $\phi\in\C\land m\in\C$ . Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for expansion. b0eb37 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\frac{\pi}{2}, 1)=1$

Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. b62aae · Fungrim entry ↗

$\mathrm{EllipticK}(m)=\frac{1}{2}(\pi\mathrm{Hypergeometric2F_1}(\frac{1}{2}, \frac{1}{2}, 1, m))$

Holds when $m\in\C$ . Symbols: EllipticK — Legendre complete elliptic integral of the first kind; Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. b760d1 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\phi, 1)=\begin{cases}\ln(\frac{\sin(\phi)+1}{\cos(\phi)})&(-\pi)/2\le\Re(\phi)\le\pi/2\land\phi\notin\lbrace(-\pi)/2, \pi/2\rbrace\\\infty\mathrm{sgn}(\phi)&\phi\in\lbrace(-\pi)/2, \pi/2\rbrace\\\tilde\infty&\top\end{cases}$

Holds when $\phi\in\C$ . Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. b7cfb3 · Fungrim entry ↗

$\mathrm{EllipticK}(4\sqrt{3}-7)=\frac{\sqrt{3+2\sqrt{3}}\Gamma(1/3)^3}{\pi\times2^{\frac{10}{3}}}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. b95ffa · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(0, 0)=0$

Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. ba1965 · Fungrim entry ↗

$\mathrm{EllipticK}(0)=\frac{\pi}{2}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. bb4501 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(0, m)=0$

Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. be3e09 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\frac{\pi}{2}, 0)=\frac{\pi}{2}$

Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. c0ad12 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\pi k+\phi, m)=2k\mathrm{EllipticE}(m)+\mathrm{IncompleteEllipticE}(\phi, m)$

Holds when $\phi\in\C\land m\in\C\land k\in\Z$ . Symbols: EllipticE — Legendre complete elliptic integral of the second kind; IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. c28288 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\frac{\pi}{3}, 1)=\ln(2+\sqrt{3})$

Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. c584c3 · Fungrim entry ↗

$\mathrm{EllipticK}(\frac{1}{2})=\frac{\Gamma(1/4)^2}{4\sqrt{\pi}}$

Symbols: EllipticK — Legendre complete elliptic integral of the first kind. Used by the Compute Engine for simplification. cc22bf · Fungrim entry ↗

$\mathrm{EllipticPi}(1, m)=\tilde\infty$

Holds when $m\in\C$ . Symbols: EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. ce4df4 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\phi, 0)=\phi$

$\mathrm{IncompleteEllipticE}(\frac{\pi}{6}, 1)=\frac{1}{2}$

Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. d88dd1 · Fungrim entry ↗

$\mathrm{EllipticPi}(0, m)=\mathrm{EllipticK}(m)$

Holds when $m\in\C$ . Symbols: EllipticK — Legendre complete elliptic integral of the first kind; EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for expansion. dd67fb · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\frac{-\pi}{2}, 1)=-1$

Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. dec0d2 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\phi, m)=\sin(\phi)\mathrm{CarlsonRF}(\cos(\phi)^2, 1-m\sin(\phi)^2, 1)$

Holds when $\frac{-\pi}{2}\le\Re(\phi)\le\frac{\pi}{2}\land\phi\in\C\land m\in\C$ . Symbols: CarlsonRF — Carlson symmetric elliptic integral of the first kind; IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. e2445d · Fungrim entry ↗

$\mathrm{EllipticPi}(n, 1)=\begin{cases}\frac{\infty}{1-n}&n\ne1\\\tilde\infty&n=1\end{cases}$

Holds when $n\in\C$ . Symbols: EllipticPi — Legendre complete elliptic integral of the third kind. Used by the Compute Engine for simplification. e9c797 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\frac{\pi}{6}, 4)=2\mathrm{EllipticE}(\frac{1}{4})-\frac{1}{2}(3\mathrm{EllipticK}(\frac{1}{4}))$

Symbols: EllipticE — Legendre complete elliptic integral of the second kind; EllipticK — Legendre complete elliptic integral of the first kind; IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. eba27c · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\phi, 0)=\phi$

Holds when $\phi\in\C$ . Symbols: IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. efc7a4 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\arcsin(\frac{1}{m^{1/2}}), m)=(\mathrm{EllipticE}(\frac{1}{m})-(1-\frac{1}{m})\mathrm{EllipticK}(\frac{1}{m}))\sqrt{m}$

Holds when $m\in\C\setminus\lbrace0, 1\rbrace$ . Symbols: EllipticE — Legendre complete elliptic integral of the second kind; EllipticK — Legendre complete elliptic integral of the first kind; IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. f0bcb5 · Fungrim entry ↗

$\mathrm{IncompleteEllipticE}(\phi, 1)=2\lfloor\frac{\Re(\phi)}{\pi}+\frac{1}{2}\rfloor+\sin(\phi)\times(-1)^{\lfloor\frac{\Re(\phi)}{\pi}+\frac{1}{2}\rfloor}$

$\mathrm{IncompleteEllipticE}(\phi, m)=\sin(\phi)\mathrm{CarlsonRF}(\cos(\phi)^2, 1-m\sin(\phi)^2, 1)-\frac{1}{3}(m\mathrm{CarlsonRD}(\cos(\phi)^2, 1-m\sin(\phi)^2, 1)\sin(\phi)^3)$

Holds when $\frac{-\pi}{2}\le\Re(\phi)\le\frac{\pi}{2}\land\phi\in\C\land m\in\C$ . Symbols: CarlsonRD — Degenerate Carlson symmetric elliptic integral of the third kind; CarlsonRF — Carlson symmetric elliptic integral of the first kind; IncompleteEllipticE — Legendre incomplete elliptic integral of the second kind. Used by the Compute Engine for simplification. f48f54 · Fungrim entry ↗

$\mathrm{IncompleteEllipticF}(\frac{\pi}{4}, 1)=\ln(1+\sqrt{2})$

Symbols: IncompleteEllipticF — Legendre incomplete elliptic integral of the first kind. Used by the Compute Engine for simplification. f5d489 · Fungrim entry ↗

Weierstrass elliptic functions

$\mathrm{WeierstrassZeta}(z, \tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau, 1)}{\mathrm{JacobiTheta}(1, z, \tau)}-\frac{z\mathrm{JacobiTheta}(1, 0, \tau, 3)}{3\mathrm{JacobiTheta}(1, 0, \tau, 1)}$

Holds when $z\notin\mathrm{Lattice}(1, \tau)\land z\in\C\land\tau\in\mathrm{HH}$ . Symbols: JacobiTheta — Jacobi theta function; WeierstrassZeta — Weierstrass zeta function. Used by the Compute Engine for simplification. 0207dc · Fungrim entry ↗

$z\mapsto\mathrm{WeierstrassSigma}(z, \tau)^{\prime}(z)=\mathrm{WeierstrassZeta}(z, \tau)\mathrm{WeierstrassSigma}(z, \tau)$

Holds when $z\notin\mathrm{Lattice}(1, \tau)\land z\in\C\land\tau\in\mathrm{HH}$ . Symbols: WeierstrassSigma — Weierstrass sigma function; WeierstrassZeta — Weierstrass zeta function. Used by the Compute Engine for simplification. 0e649f · Fungrim entry ↗

$\mathrm{WeierstrassP}(-z, \tau)=\mathrm{WeierstrassP}(z, \tau)$

Holds when $z\notin\mathrm{Lattice}(1, \tau)\land z\in\C\land\tau\in\mathrm{HH}$ . Symbols: WeierstrassP — Weierstrass elliptic function. Used by the Compute Engine for simplification. 12a9e8 · Fungrim entry ↗

$\mathrm{WeierstrassSigma}(-z, \tau)=-\mathrm{WeierstrassSigma}(z, \tau)$

Holds when $z\in\C\land\tau\in\mathrm{HH}$ . Symbols: WeierstrassSigma — Weierstrass sigma function. Used by the Compute Engine for expansion. 23beb5 · Fungrim entry ↗

$\mathrm{WeierstrassSigma}(z+1, \tau)=-(\mathrm{WeierstrassSigma}(z, \tau)\exp(2(z+\frac{1}{2})\mathrm{WeierstrassZeta}(1/2, \tau)))$

Holds when $z\in\C\land\tau\in\mathrm{HH}$ . Symbols: WeierstrassSigma — Weierstrass sigma function; WeierstrassZeta — Weierstrass zeta function. Used by the Compute Engine for simplification. 35403b · Fungrim entry ↗

$\mathrm{WeierstrassZeta}(-z, \tau)=-\mathrm{WeierstrassZeta}(z, \tau)$

Holds when $z\notin\mathrm{Lattice}(1, \tau)\land z\in\C\land\tau\in\mathrm{HH}$ . Symbols: WeierstrassZeta — Weierstrass zeta function. Used by the Compute Engine for expansion. 72eb69 · Fungrim entry ↗

$\mathrm{WeierstrassZeta}(\tau+z, \tau)=\mathrm{WeierstrassZeta}(\frac{\tau}{2}, \tau)+\mathrm{WeierstrassZeta}(z, \tau)$

$\mathrm{WeierstrassP}(n\tau+m+z, \tau)=\mathrm{WeierstrassP}(z, \tau)$

Holds when $z\notin\mathrm{Lattice}(1, \tau)\land z\in\C\land\tau\in\mathrm{HH}\land m\in\Z\land n\in\Z$ . Symbols: WeierstrassP — Weierstrass elliptic function. Used by the Compute Engine for simplification. a95b7e · Fungrim entry ↗

$\mathrm{WeierstrassP}(z, \tau)=\frac{\pi\mathrm{JacobiTheta}(4, z, \tau)\mathrm{JacobiTheta}(2, 0, \tau)\mathrm{JacobiTheta}(3, 0, \tau)}{\mathrm{JacobiTheta}(1, z, \tau)}^2-\frac{1}{3}((\mathrm{JacobiTheta}(2, 0, \tau)^4+\mathrm{JacobiTheta}(3, 0, \tau)^4)\pi^2)$

Holds when $z\notin\mathrm{Lattice}(1, \tau)\land z\in\C\land\tau\in\mathrm{HH}$ . Symbols: JacobiTheta — Jacobi theta function; WeierstrassP — Weierstrass elliptic function. Used by the Compute Engine for simplification. af0dfc · Fungrim entry ↗

$\mathrm{WeierstrassSigma}(z, \tau)=\frac{\mathrm{JacobiTheta}(1, z, \tau)\exp(-((\mathrm{JacobiTheta}(1, 0, \tau, 3)z^2)/(6\mathrm{JacobiTheta}(1, 0, \tau, 1))))}{\mathrm{JacobiTheta}(1, 0, \tau, 1)}$

Holds when $z\in\C\land\tau\in\mathrm{HH}$ . Symbols: JacobiTheta — Jacobi theta function; WeierstrassSigma — Weierstrass sigma function. Used by the Compute Engine for simplification. b96c9d · Fungrim entry ↗

$\mathrm{WeierstrassSigma}(\tau+z, \tau)=-(\mathrm{WeierstrassSigma}(z, \tau)\exp(2(\frac{\tau}{2}+z)\mathrm{WeierstrassZeta}(\tau/2, \tau)))$

$z\mapsto\mathrm{WeierstrassZeta}(z, \tau)^{\prime}(z)=-\mathrm{WeierstrassP}(z, \tau)$

Holds when $z\notin\mathrm{Lattice}(1, \tau)\land z\in\C\land\tau\in\mathrm{HH}$ . Symbols: WeierstrassP — Weierstrass elliptic function; WeierstrassZeta — Weierstrass zeta function. Used by the Compute Engine for simplification. e677fb · Fungrim entry ↗

$\mathrm{WeierstrassZeta}(z+1, \tau)=\mathrm{WeierstrassZeta}(\frac{1}{2}, \tau)+\mathrm{WeierstrassZeta}(z, \tau)$

Contents​

Arithmetic-geometric mean​

Carlson symmetric elliptic integrals​

Legendre elliptic integrals​

Weierstrass elliptic functions​

Contents

Arithmetic-geometric mean

Carlson symmetric elliptic integrals

Legendre elliptic integrals

Weierstrass elliptic functions