EllipticE

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

Elliptic integral of the second kind: complete E(m) with one argument, incomplete E(φ|m) with two (amplitude first, parameter convention m = k², as in Mathematica).

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaElliptic integral↗Complete elliptic integral of the second kind↗MathWorldCompleteEllipticIntegraloftheSecondKind↗WikidataQ109753012↗DLMF19.2.E8↗FungrimEllipticE!↗Fungrim entryWolfram LanguageEllipticE✓↗

EllipticE(complex | infinity, (complex | infinity)?) -> numberπ/2 at an exact 0 (the values-at-zero patch).

SageMathelliptic_ec(complex | infinity)SymPyelliptic_e(complex | infinity)mpmathellipe(complex | infinity)Juliaenumeratio_ball(elliptic_e, 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

Basic #

live ↗

In$\operatorname{Series}(\mathrm{EllipticE}(m), m, 0, 6)$Out$\frac{π}{2}-\frac{πm}{8}-\frac{1}{128}×3πm^2-\frac{1}{512}×5πm^3-\frac{1}{32\,768}×175πm^4-\frac{1}{131\,072}×441πm^5-\frac{1}{2\,097\,152}×4\,851πm^6+O\left(m^7\right)$

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\mathrm{EllipticE}(x), x, ∞, 2))$Out$\operatorname{Series}(\mathrm{EllipticE}(x), x, ∞, 2)$

Scope #

Fifty digits of the complete integral at $m=1/3$

live ↗

In$N(\mathrm{EllipticE}(\frac{1}{3}), 50)$Out$1.430\,315\,257\,172\,219\,723\,922\,528\,414\,532\,283\,730\,294\,450\,907\,408\,9$

Threads over a matrix; ellipticE(1) folds to 1 and ellipticE(0) to $\pi/2$

live ↗

In$\mathrm{EllipticE}(\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, \frac{π}{2}\bigr\rbrack, \bigl\lbrack\frac{π}{2}, 1\bigr\rbrack\bigr\rbrack$

As a matrix function on the identity, the result is the identity

live ↗

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

$E(1)=1$ and $E(0)=\pi/2$ both evaluate

live ↗

In$\bigl\lbrack\mathrm{EllipticE}(0), \mathrm{EllipticE}(1)\bigr\rbrack$Out$\bigl\lbrack\frac{π}{2}, 1\bigr\rbrack$

Both directions at infinity give complexInfinity

live ↗

In$\bigl\lbrack\mathrm{EllipticE}(∞), \mathrm{EllipticE}(\tilde∞)\bigr\rbrack$Out$\bigl\lbrack\tilde∞, \tilde∞\bigr\rbrack$

live ↗

In$N(\mathrm{EllipticE}(\frac{2}{3}, 2), 50)$Out$0.560\,575\,729\,788\,691\,552\,433\,437\,180\,131\,530\,783\,282\,087\,458\,609\,75$

live ↗

In$\operatorname{Series}(\mathrm{EllipticE}(m), m, 0, 3)$Out$\frac{π}{2}-\frac{πm}{8}-\frac{1}{128}×3πm^2-\frac{1}{512}×5πm^3+O\left(m^4\right)$

live ↗

In$\operatorname{Series}(\mathrm{EllipticE}(m), m, 1, 1)$Out$1-\frac{1}{8}π(m-1)\mathrm{Hypergeometric2F1}(\frac{1}{2}, \frac{3}{2}, 2, 1)+O\left((m-1)^2\right)$

Properties #

Incomplete integral with zero amplitude vanishes

live ↗

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

Known value · Wolfram MathematicalFunctionData, EllipticE

Zero amplitude gives 0 for any parameter

live ↗

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

Known value · Wolfram MathematicalFunctionData, EllipticE

At $m=1$ the complete integral is 1

live ↗

In$\mathrm{EllipticE}(1)$Out$1$

Known value · Wolfram MathematicalFunctionData, EllipticE:Complete

live ↗

In$\mathrm{EllipticE}(0)$Out$\frac{π}{2}$

Known value · Wolfram MathematicalFunctionData, EllipticE:Complete

live ↗

In$\mathrm{EllipticE}((1+\sqrt{2}-\sqrt{2(1+\sqrt{2})})^4)$Out$\mathrm{EllipticE}((1+\sqrt{2}-\sqrt{2}\sqrt{1+\sqrt{2}})^4)$

Known value · Wolfram MathematicalFunctionData, EllipticE:Complete

live ↗

In$\mathrm{EllipticE}(1-(1+\sqrt{2}-\sqrt{2(1+2^{1/2})})^4)$Out$\mathrm{EllipticE}(1-(1+\sqrt{2}-\sqrt{2}\sqrt{1+2^{1/2}})^4)$

Known value · Wolfram MathematicalFunctionData, EllipticE:Complete

About

Examples	16
Checked against	Wolfram Language 14 agree2 unevaluatedSageMath 10 agree1 differ2 unevaluated2 errorSymPy 12 agree2 differ1 errormpmath 10 agreeJulia 7 agree
Crosswalk	Wikipedia ×2MathWorldWikidataDLMFFungrimFungrim entry ×28Wolfram LanguageSageMath ×2SymPy ×2mpmath ×2Julia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts