Complete elliptic integral of the first kind: Representations through more general functions (formula 08.02.26.0137)

EllipticK

$Mathematica Notation$

Elliptic Integrals

EllipticK[z]

Representations through more general functions

Through other functions

Involving some elliptic-type functions

http://functions.wolfram.com/08.02.26.0137.01

Input Form

EllipticK[InverseEllipticNomeQ[E^((I Pi Subscript[\[Omega], 2])/ Subscript[\[Omega], 1])]] == Sqrt[Subscript[e, 1] - Subscript[e, 3]] Subscript[\[Omega], 1] /; {Subscript[e, 1], Subscript[e, 2], Subscript[e, 3]} == {WeierstrassP[Subscript[\[Omega], 1], {Subscript[g, 2], Subscript[g, 3]}], WeierstrassP[Subscript[\[Omega], 1] + Subscript[\[Omega], 2], {Subscript[g, 2], Subscript[g, 3]}], WeierstrassP[ Subscript[\[Omega], 2], {Subscript[g, 2], Subscript[g, 3]}]} && {Subscript[g, 2], Subscript[g, 3]} == WeierstrassInvariants[ {Subscript[\[Omega], 1], Subscript[\[Omega], 2]}]

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mrow> <msub> <mi> e </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mrow> </msqrt> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> e </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> e </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi fontweight='normal' fontstyle='italic'> g </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo fontweight='normal' fontstyle='italic'> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi fontweight='normal' fontstyle='italic'> g </mi> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo fontweight='normal' fontstyle='italic'> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> </mrow> </mrow> <mo fontweight='normal' fontstyle='normal'> } </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mrow> <msub> <mi> e </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mrow> </msqrt> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> e </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> e </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi fontweight='normal' fontstyle='italic'> g </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo fontweight='normal' fontstyle='italic'> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi fontweight='normal' fontstyle='italic'> g </mi> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo fontweight='normal' fontstyle='italic'> ( </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> </mrow> </mrow> <mo fontweight='normal' fontstyle='normal'> } </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29