Klein invariant modular function: Series representations (formula 09.50.06.0003)

KleinInvariantJ

$Mathematica Notation$

Elliptic Functions

KleinInvariantJ[z]

Series representations

Generalized power series

Expansions at generic point z==z₀

For the function itself

http://functions.wolfram.com/09.50.06.0003.01

Input Form

KleinInvariantJ[z] \[Proportional] KleinInvariantJ[Subscript[z, 0]] + ((8 I EllipticK[w]^2 (-1 + w) w (1 - ModularLambda[Subscript[z, 0]] + ModularLambda[Subscript[z, 0]]^2)^2 (2 - 3 ModularLambda[Subscript[z, 0]] - 3 ModularLambda[Subscript[z, 0]]^2 + 2 ModularLambda[Subscript[z, 0]]^ 3))/(27 ((-EllipticE[1 - w]) EllipticK[w] + EllipticK[1 - w] (-EllipticE[w] + EllipticK[w])) (-1 + ModularLambda[Subscript[z, 0]])^3 ModularLambda[Subscript[z, 0]]^ 3)) (z - Subscript[z, 0]) - 8 EllipticK[w]^3 (1 - w) w (1 - ModularLambda[Subscript[z, 0]] + ModularLambda[Subscript[z, 0]]^2) ((EllipticE[w] ModularLambda[Subscript[z, 0]] (2 - 7 ModularLambda[Subscript[z, 0]] + 7 ModularLambda[Subscript[z, 0]]^2 - 7 ModularLambda[Subscript[z, 0]]^4 + 7 ModularLambda[Subscript[z, 0]]^5 - 2 ModularLambda[Subscript[z, 0]]^6) + EllipticK[w] w (2 w (3 - 10 ModularLambda[Subscript[z, 0]] + 9 ModularLambda[Subscript[z, 0]]^2 + ModularLambda[Subscript[z, 0]]^3 + 2 ModularLambda[Subscript[z, 0]]^4 - 3 ModularLambda[Subscript[z, 0]]^5 + ModularLambda[Subscript[z, 0]]^6) - 6 + 18 ModularLambda[Subscript[z, 0]] - 11 ModularLambda[Subscript[z, 0]]^2 - 9 ModularLambda[Subscript[z, 0]]^3 - 4 ModularLambda[Subscript[z, 0]]^4 + 13 ModularLambda[Subscript[z, 0]]^5 - 9 ModularLambda[Subscript[z, 0]]^6 + 2 ModularLambda[Subscript[z, 0]]^7))/ (27 (EllipticK[1 - w] (EllipticE[w] - EllipticK[w]) + EllipticE[1 - w] EllipticK[w])^2 (ModularLambda[Subscript[z, 0]] - 1)^4 ModularLambda[Subscript[z, 0]]^4)) (z - Subscript[z, 0])^2 + \[Ellipsis] /; (z -> Subscript[z, 0]) && w = InverseEllipticNomeQ[E^(I Pi Subscript[z, 0])]

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> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox["z", Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <semantics> <mrow> <mi> J </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["J", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> w </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> - </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 27 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> w </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> - </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mn> 27 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> w </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <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> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 13 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 11 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 18 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> w </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 6 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 3 </mn> </msup> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <msup> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", TagBox[SubscriptBox["z", "0"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mtext> </mtext> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mi> w </mi> </mrow> </mrow> <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> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Set </ci> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> KleinInvariantJ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> KleinInvariantJ </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 8 </cn> <imaginaryi /> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <ci> w </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticE </ci> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> EllipticE </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> w </ci> <apply> <plus /> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> EllipticE </ci> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> EllipticE </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticE </ci> <ci> w </ci> </apply> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> w </ci> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 13 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 11 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> w </ci> <apply> <plus /> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <ci> ModularLambda </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -6 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> w </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> … </ci> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <ci> w </ci> </apply> </apply> <apply> <ci> InverseEllipticNomeQ </ci> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </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