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ModularLambda






Mathematica Notation

Traditional Notation









Elliptic Functions > ModularLambda[z] > Differentiation > Low-order differentiation





http://functions.wolfram.com/09.51.20.0005.01









  


  










Input Form





D[ModularLambda[z], {z, 5}] == (-(1/Pi^5)) 512 I EllipticK[ModularLambda[z]]^6 (-1 + ModularLambda[z]) ModularLambda[z] (15 EllipticE[ModularLambda[z]]^4 - 60 EllipticE[ModularLambda[z]]^3 EllipticK[ModularLambda[z]] ModularLambda[z] + 30 EllipticE[ModularLambda[z]]^2 EllipticK[ModularLambda[z]]^2 (1 + ModularLambda[z]) (-1 + 2 ModularLambda[z]) - 10 EllipticE[ModularLambda[z]] EllipticK[ModularLambda[z]]^3 (1 + ModularLambda[z]) (-2 + ModularLambda[z] + 2 ModularLambda[z]^2) + EllipticK[ModularLambda[z]]^4 (-3 - 4 ModularLambda[z] + ModularLambda[z]^2 + 6 ModularLambda[z]^3 + 2 ModularLambda[z]^4))










Standard Form





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MathML Form







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</mo> <msup> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Identity, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Identity, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; 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</mo> <mo> ( </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Identity, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Identity, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Identity, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <msup> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Identity, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#955; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Lambda]&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Identity, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 3 </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'> 5 </cn> </degree> </bvar> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 512 </cn> <imaginaryi /> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 6 </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'> 5 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <apply> <ci> EllipticE </ci> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 60 </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'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 30 </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> <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'> 10 </cn> <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> <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'> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> ModularLambda </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -3 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





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