EllipticK

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

Complete elliptic integral of the first kind K(m), parameter convention m = k².

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaElliptic integral↗Complete elliptic integral of the first kind↗MathWorldCompleteEllipticIntegraloftheFirstKind↗WikidataQ109752514↗DLMF19.2.E8↗FungrimEllipticK✓↗Fungrim entryWolfram LanguageEllipticK!↗

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

SageMathelliptic_kc(complex | infinity)SymPyelliptic_k(complex | infinity)mpmathellipk(complex | infinity)Juliaenumeratio_ball(elliptic_k, complex | infinity)

Typed signatures

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

Domain: Compute engine

Plot

ComplexPlot

real slice

Examples

Basic #

live ↗

In$\operatorname{Series}(\mathrm{EllipticK}(x), x, 0, 5)$Out$\frac{π}{2}+\frac{πx}{8}+\frac{1}{128}×9πx^2+\frac{1}{512}×25πx^3+\frac{1}{32\,768}×1\,225πx^4+\frac{1}{131\,072}×3\,969πx^5+O\left(x^6\right)$

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\mathrm{EllipticK}(x), x, ∞, 2))$Out$\frac{-i\ln(x)+2i+π-2i\ln(4)}{8\sqrt{x}^{3}}+\frac{-i\ln(x)+π-2i\ln(4)}{2\sqrt{x}}$

Scope #

live ↗

In$N(\mathrm{EllipticK}(\frac{4}{5}), 50)$Out$2.257\,205\,326\,820\,853\,655\,083\,256\,004\,523\,387\,397\,235\,419\,281\,74$

live ↗

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

live ↗

In$\mathrm{EllipticK}(∞)$Out$0$

live ↗

In$\mathrm{FunctionExpand}(\mathrm{EllipticK}(\bigl\lbrack\bigl\lbrack-1, 0\bigr\rbrack, \bigl\lbrack1/2, -1\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack\frac{\sqrt{2}Γ(1/4)^2}{8\sqrt{π}}, \frac{π}{2}\bigr\rbrack, \bigl\lbrack\frac{Γ(1/4)^2}{4\sqrt{π}}, \frac{\sqrt{2}Γ(1/4)^2}{8\sqrt{π}}\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{FunctionExpand}(\mathrm{MatrixFunction}(\mathrm{EllipticK}, \bigl\lbrack\bigl\lbrack-1, 0\bigr\rbrack, \bigl\lbrack1/2, -1\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack\frac{\sqrt{2}Γ(1/4)^2}{8\sqrt{π}}, 0\bigr\rbrack, \bigl\lbrack-\frac{1}{8}(-\frac{2^{1/2}Γ(1/4)^2}{4π^{1/2}}+\frac{2^{1/2}π^{3/2}}{Γ(1/4)^2}+\frac{2^{1/2}Γ(1/4)^2}{8π^{1/2}}), \frac{\sqrt{2}Γ(1/4)^2}{8\sqrt{π}}\bigr\rbrack\bigr\rbrack$

live ↗

In$\operatorname{Series}(\mathrm{EllipticK}(m), m, 0, 3)$Out$\frac{π}{2}+\frac{πm}{8}+\frac{1}{128}×9πm^2+\frac{1}{512}×25πm^3+O\left(m^4\right)$

Properties #

A singular-value-style argument; only the radical's form changes, K stays

live ↗

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

Known value · Wolfram MathematicalFunctionData, EllipticK

live ↗

In$\mathrm{EllipticK}(1)$Out$∞$

Known value · Wolfram MathematicalFunctionData, EllipticK

Possible issues #

live ↗

In$N(\mathrm{EllipticK}(2-i(-1-2π-e^{-π^4}-π^2+(1+π)^2)))$Out$1.311\,028\,777\,146\,059\,8-1.311\,028\,777\,146\,059\,8i$

About

Examples	11
Checked against	Wolfram Language 9 agree2 differSageMath 5 agree1 differ1 unevaluated2 errorSymPy 5 agree3 differ3 errormpmath 6 agreeJulia 2 agree1 differ3 error
Crosswalk	Wikipedia ×2MathWorldWikidataDLMFFungrimFungrim entry ×37Wolfram LanguageSageMathSymPympmathJulia

Implementation

mappedmpmathpackages/oracle/src/mappings.ts

The parameter m, not the modulus k, as in Wolfram.

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts