Complete elliptic integral of the first kind: Representations through more general functions (formula 08.02.26.0138)

EllipticK

$Mathematica Notation$

Elliptic Integrals

EllipticK[z]

Representations through more general functions

Through other functions

Involving some elliptic-type functions

http://functions.wolfram.com/08.02.26.0138.01

Input Form

EllipticK[1 - z]/EllipticK[z] == -((I Subscript[\[Omega], 2])/ Subscript[\[Omega], 1]) /; z == InverseEllipticNomeQ[E^((I Pi Subscript[\[Omega], 2])/ Subscript[\[Omega], 1])] && {Subscript[\[Omega], 1], Subscript[\[Omega], 2]} == WeierstrassHalfPeriods[{Subscript[g, 2], Subscript[g, 3]}]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "z"]], "]"]]]], RowBox[List["EllipticK", "[", "z", "]"]]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SubscriptBox["\[Omega]", "2"]]], SubscriptBox["\[Omega]", "1"]]]]]], "/;", RowBox[List[RowBox[List["z", "\[Equal]", RowBox[List["InverseEllipticNomeQ", "[", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SubscriptBox["\[Omega]", "2"]]], SubscriptBox["\[Omega]", "1"]]], "]"]]]], "\[And]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "2"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "]"]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <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> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo fontweight='normal' fontstyle='italic'> ( </mo> <mrow> <mstyle fontweight='normal' fontstyle='italic'> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mstyle> <mo> , </mo> <mstyle fontweight='normal' fontstyle='italic'> <msub> <mi fontweight='normal' fontstyle='italic'> g </mi> <mn> 3 </mn> </msub> </mstyle> </mrow> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo fontweight='normal' fontstyle='italic'> ( </mo> <mrow> <mstyle fontweight='normal' fontstyle='italic'> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mstyle> <mo> , </mo> <mstyle fontweight='normal' fontstyle='italic'> <msub> <mi fontweight='normal' fontstyle='italic'> g </mi> <mn> 3 </mn> </msub> </mstyle> </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> <mfrac> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <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> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> { </mo> <mrow> <mrow> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo fontweight='normal' fontstyle='italic'> ( </mo> <mrow> <mstyle fontweight='normal' fontstyle='italic'> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mstyle> <mo> , </mo> <mstyle fontweight='normal' fontstyle='italic'> <msub> <mi fontweight='normal' fontstyle='italic'> g </mi> <mn> 3 </mn> </msub> </mstyle> </mrow> <mo fontweight='normal' fontstyle='normal'> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi> ω </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo fontweight='normal' fontstyle='italic'> ( </mo> <mrow> <mstyle fontweight='normal' fontstyle='italic'> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mstyle> <mo> , </mo> <mstyle fontweight='normal' fontstyle='italic'> <msub> <mi fontweight='normal' fontstyle='italic'> g </mi> <mn> 3 </mn> </msub> </mstyle> </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

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List["EllipticK", "[", RowBox[List["1", "-", "z_"]], "]"]], RowBox[List["EllipticK", "[", "z_", "]"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SubscriptBox["\[Omega]", "2"]]], SubscriptBox["\[Omega]", "1"]]]], "/;", RowBox[List[RowBox[List["z", "\[Equal]", RowBox[List["InverseEllipticNomeQ", "[", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", SubscriptBox["\[Omega]", "2"]]], SubscriptBox["\[Omega]", "1"]]], "]"]]]], "&&", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "2"]]], "}"]], "\[Equal]", RowBox[List["WeierstrassHalfPeriods", "[", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]], "]"]]]]]]]]]]]]

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

2001-10-29