Modular lambda function: Differentiation (formula 09.51.20.0004)

ModularLambda

$Mathematica Notation$

Elliptic Functions

ModularLambda[z]

Differentiation

Low-order differentiation

http://functions.wolfram.com/09.51.20.0004.01

Input Form

D[ModularLambda[z], {z, 4}] == ((128 EllipticK[ModularLambda[z]]^5 (-1 + ModularLambda[z]))/Pi^4) ModularLambda[z] (-6 EllipticE[ModularLambda[z]]^3 + 18 EllipticE[ModularLambda[z]]^2 EllipticK[ModularLambda[z]] ModularLambda[z] - 6 EllipticE[ModularLambda[z]] EllipticK[ModularLambda[z]]^2 (1 + ModularLambda[z]) (-1 + 2 ModularLambda[z]) + EllipticK[ModularLambda[z]]^3 (1 + ModularLambda[z]) (-2 + ModularLambda[z] + 2 ModularLambda[z]^2))

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "4"]], "}"]]], RowBox[List["ModularLambda", "[", "z", "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["128", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "5"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["ModularLambda", "[", "z", "]"]]]], ")"]], " "]], SuperscriptBox["\[Pi]", "4"]], RowBox[List["ModularLambda", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "6"]], " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "3"]]], "+", RowBox[List["18", " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], " ", RowBox[List["ModularLambda", "[", "z", "]"]]]], "-", RowBox[List["6", " ", RowBox[List["EllipticE", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["ModularLambda", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["ModularLambda", "[", "z", "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "3"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["ModularLambda", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["ModularLambda", "[", "z", "]"]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "2"]]]]], ")"]]]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mn> 4 </mn> </msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo>  </mo> <mrow> <mfrac> <mrow> <mn> 128 </mn> <mo> ⁢ </mo> <mi> K </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 5 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <msup> <mi> π </mi> <mn> 4 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 18 </mn> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox["z", Identity, Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 4 </cn> </degree> </bvar> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 128 </cn> <ci> K </ci> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <plus /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -6 </cn> <apply> <power /> <apply> <ci> EllipticE </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> EllipticE </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "4"]], "}"]]]]], RowBox[List["ModularLambda", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["128", " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "5"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["ModularLambda", "[", "z", "]"]]]], ")"]]]], ")"]], " ", RowBox[List["ModularLambda", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "6"]], " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "3"]]], "+", RowBox[List["18", " ", SuperscriptBox[RowBox[List["EllipticE", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], " ", RowBox[List["ModularLambda", "[", "z", "]"]]]], "-", RowBox[List["6", " ", RowBox[List["EllipticE", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["ModularLambda", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["ModularLambda", "[", "z", "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["EllipticK", "[", RowBox[List["ModularLambda", "[", "z", "]"]], "]"]], "3"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["ModularLambda", "[", "z", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["ModularLambda", "[", "z", "]"]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["ModularLambda", "[", "z", "]"]], "2"]]]]], ")"]]]]]], ")"]]]], SuperscriptBox["\[Pi]", "4"]]]]]]

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

2007-05-02