EllipticF

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

Incomplete elliptic integral of the first kind F(φ|m) (amplitude first, parameter convention m = k², as in Mathematica). F(π/2|m) = K(m).

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaIncomplete elliptic integral of the first kind↗MathWorldEllipticIntegraloftheFirstKind↗WikidataQ109752309↗Wolfram LanguageEllipticF!↗

EllipticF(complex | infinity, complex | infinity) -> numberIncomplete elliptic integral of the first kind F(φ|m) (amplitude first, parameter convention m = k², as in Mathematica). F(π/2|m) = K(m).

SageMathelliptic_f(complex | infinity, complex | infinity)SymPyelliptic_f(complex | infinity, complex | infinity)mpmathellipf(complex | infinity, complex | infinity)

Typed signatures

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

Domain: Compute engine

Plot

ComplexPlot

real slice

Examples

Scope #

Threads over a matrix; the zero entries give 0

live ↗

In$\mathrm{EllipticF}(\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack, 0)$Out$\bigl\lbrack\bigl\lbrack\mathrm{EllipticF}(1, 0), 0\bigr\rbrack, \bigl\lbrack0, \mathrm{EllipticF}(1, 0)\bigr\rbrack\bigr\rbrack$

The matrix-function form agrees with entrywise evaluation

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{EllipticF}(\operatorname{\_1}, 0), \bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\mathrm{EllipticF}(1, 0), 0\bigr\rbrack, \bigl\lbrack0, \mathrm{EllipticF}(1, 0)\bigr\rbrack\bigr\rbrack$

live ↗

In$N(\mathrm{EllipticF}(\frac{12}{5}, 3), 50)$Out$1.001\,077\,380\,456\,106\,236\,079\,659\,586\,383\,835\,893\,149\,713\,590\,458\,5-1.923\,166\,189\,504\,625\,353\,184\,144\,409\,335\,278\,171\,570\,687\,870\,309\,4i$

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In$\bigl\lbrack\mathrm{EllipticF}(0, m), \mathrm{EllipticF}(\frac{π}{2}, m)\bigr\rbrack$Out$\bigl\lbrack0, \mathrm{EllipticF}(\frac{π}{2}, m)\bigr\rbrack$

Properties #

Zero amplitude gives 0

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

Known value · Wolfram MathematicalFunctionData, EllipticF

Zero amplitude gives 0 regardless of the parameter

live ↗

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

Known value · Wolfram MathematicalFunctionData, EllipticF

F undoes the Jacobi amplitude, and both start at 0

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In$\mathrm{EllipticF}(\mathrm{JacobiAmplitude}(0, 0), 0)$Out$0$

Known value · Wolfram MathematicalFunctionData, EllipticF

F inverts the amplitude, so the value at 0 is 0

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

Known value · Wolfram MathematicalFunctionData, EllipticF

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In$\mathrm{Solve}(\mathrm{EllipticF}(z, m)^3+\mathrm{EllipticF}(z, m)=x, z)$Out$\mathrm{Solve}(\mathrm{EllipticF}(z, m)^3+\mathrm{EllipticF}(z, m)=x, z)$

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In$N(2\mathrm{EllipticF}(\frac{1}{2}, \frac{1}{2}))$Out$1.020\,934\,271\,256\,009\,7$

Known value · Wolfram MathematicalFunctionData, EllipticF, Euler's addition theorem

live ↗

In$N(\mathrm{EllipticF}(\frac{1}{3}, \frac{1}{2})+\mathrm{EllipticF}(\frac{1}{2}, \frac{1}{2}))$Out$0.846\,894\,634\,165\,577\,8$

Known value · Wolfram MathematicalFunctionData, EllipticF, Euler's addition theorem

About

Examples	11
Checked against	Wolfram Language 8 agree3 differSageMath 8 agree1 differSymPy 7 agree1 differmpmath 6 agree
Crosswalk	WikipediaMathWorldWikidataWolfram LanguageSageMathSymPympmath

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

Amplitude first, then the parameter m, as in Wolfram.

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts