EllipticPi

Generated from the engine's own definition: compute-engine's symbol, which we neither extend nor document by hand.

Elliptic integral of the third kind: complete Π(n|m) with two arguments, incomplete Π(n; φ|m) with three (characteristic first, amplitude second, parameter convention m = k², as in Mathematica).

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaComplete elliptic integral of the third kind↗MathWorldCompleteEllipticIntegraloftheThirdKind↗WikidataQ109753363↗DLMF19.2.E8↗FungrimEllipticPi✓↗Fungrim entryWolfram LanguageEllipticPi!↗

EllipticPi(complex | infinity, complex | infinity, (complex | infinity)?) -> numberElliptic integral of the third kind: complete Π(n|m) with two arguments, incomplete Π(n; φ|m) with three (characteristic first, amplitude second, parameter convention m = k², as in Mathematica).

Typed signatures

Type	Package	Applies
(complex | infinity, complex | infinity, (complex | infinity)?) -> number	analyticreplaces compute-engine	always
(complex | infinity, complex | infinity, (complex | infinity)?) -> number	compute-engine	always

Domain: Compute engine

Plot

the complex plot cannot lower EllipticPi

Examples

Basic #

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In$\mathrm{FullSimplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{EllipticPi}(n, m), m, ∞, 2)))$Out$\operatorname{Series}(\mathrm{EllipticPi}(n, m), m, ∞, 2)$

Scope #

Threads over a matrix; the zero characteristic is not reduced

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In$\mathrm{FunctionExpand}(\mathrm{EllipticPi}(\bigl\lbrack\bigl\lbrack-1, 0\bigr\rbrack, \bigl\lbrack0, 1/2\bigr\rbrack\bigr\rbrack, 0))$Out$\bigl\lbrack\bigl\lbrack\mathrm{EllipticPi}(-1, 0), \mathrm{EllipticPi}(0, 0)\bigr\rbrack, \bigl\lbrack\mathrm{EllipticPi}(0, 0), \mathrm{EllipticPi}(\frac{1}{2}, 0)\bigr\rbrack\bigr\rbrack$

The matrix-function form stays entrywise on a diagonal matrix

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In$\mathrm{FunctionExpand}(\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{EllipticPi}(\operatorname{\_1}, 0), \bigl\lbrack\bigl\lbrack-1, 0\bigr\rbrack, \bigl\lbrack0, 1/2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack\mathrm{EllipticPi}(-1, 0), 0\bigr\rbrack, \bigl\lbrack0, \mathrm{EllipticPi}(\frac{1}{2}, 0)\bigr\rbrack\bigr\rbrack$

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In$N(\mathrm{EllipticPi}(\frac{3}{10}, \frac{2}{5}), 50)$Out$2.148\,795\,415\,866\,456\,635\,936\,022\,786\,375\,450\,350\,243\,300\,732\,270\,1$

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In$N(\mathrm{EllipticPi}(\frac{3}{10}, 2, \frac{2}{5}), 50)$Out$2.906\,928\,215\,899\,509\,476\,566\,159\,157\,785\,877\,416\,281\,562\,249\,610\,2$

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In$\mathrm{Simplify}(\int\!\mathrm{EllipticPi}(n, m)\, \mathrm{d}m)$Out$\int\!\mathrm{EllipticPi}(n, m)\, \mathrm{d}m$

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In$\mathrm{Simplify}(\operatorname{Series}(\mathrm{EllipticPi}(n, m), m, 0, 3))$Out$\mathrm{EllipticPi}(n, 0)+m\mathrm{EllipticPi}^{(0, 1)}(n, 0)+\frac{1}{2}\mathrm{EllipticPi}^{(0, 2)}(n, 0)m^2+\frac{1}{6}\mathrm{EllipticPi}^{(0, 3)}(n, 0)m^3+O\left(m^4\right)$

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In$\mathrm{Simplify}(\operatorname{Series}(\mathrm{EllipticPi}(n, m), n, 1, 1))$Out$\mathrm{EllipticPi}(1, m)+(n-1)\mathrm{EllipticPi}^{(1, 0)}(1, m)+O\left((n-1)^2\right)$

Properties #

Zero amplitude gives 0

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In$\mathrm{EllipticPi}(0, 0, 1)$Out$0$

Known value · Wolfram MathematicalFunctionData, EllipticPi

Zero amplitude gives 0 at $m=0$ too

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In$\mathrm{EllipticPi}(0, 0, 0)$Out$0$

Known value · Wolfram MathematicalFunctionData, EllipticPi

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In$N(\mathrm{EllipticPi}(\frac{1}{2}, i\mathrm{arsinh}(\tan(1/2)), \frac{1}{2}))$Out$0.488\,330\,039\,028\,240\,2i$

Known value · Wolfram MathematicalFunctionData, EllipticPi, FunctionalEquations

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In$N((\mathrm{EllipticPi}(\frac{1}{2}\sin(1/2)^2, \frac{1}{2}, \frac{1}{2})-\mathrm{EllipticF}(\frac{1}{2}, \frac{1}{2}))\cot(\frac{1}{2})\sqrt{1-\frac{1}{2}\sin(1/2)^2}+\mathrm{EllipticF}(\frac{1}{2}, \frac{1}{2})\mathrm{EllipticE}(\frac{1}{2}, \frac{1}{2}))$Out$0.258\,354\,379\,144\,597\,1$

Known value · Wolfram MathematicalFunctionData, EllipticPi, FunctionalEquations

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In$N(\mathrm{EllipticPi}(\frac{1}{2}, i\mathrm{arsinh}(\tan(1/2)), -1))$Out$0.525\,899\,989\,634\,695\,4i$

Known value · Wolfram MathematicalFunctionData, EllipticPi, FunctionalEquations

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In$N((\mathrm{EllipticPi}(\frac{1}{2}\sin(1/2)^2, 2, \frac{1}{2})-\mathrm{EllipticF}(2, \frac{1}{2}))\cot(\frac{1}{2})\sqrt{1-\frac{1}{2}\sin(1/2)^2}+\mathrm{EllipticF}(\frac{1}{2}, \frac{1}{2})\mathrm{EllipticE}(2, \frac{1}{2}))$Out$1.191\,199\,484\,399\,090\,3$

Known value · Wolfram MathematicalFunctionData, EllipticPi, FunctionalEquations

Possible issues #

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In$N(\mathrm{EllipticPi}(2, i\bigl\lbrack-1, 1\bigr\rbrack×10^{-6}+\frac{π}{2}))$Out$\bigl\lbrack-0.757\,569\,671\,407\,859\,9-0.618\,112\,885\,269\,955\,8i, -0.757\,561\,771\,177\,805\,3-6.163\,502\,500\,952\,997i\bigr\rbrack$

About

Examples	15
Checked against	Wolfram Language 9 agree3 differ3 unevaluatedSymPy 8 agree1 differ5 errormpmath 8 agree1 errorSageMath 7 agree
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×13Wolfram LanguageSageMathSymPy ×2mpmath ×2

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts