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http://functions.wolfram.com/09.14.27.0007.01
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WeierstrassPPrime[z, {Subscript[g, 2], Subscript[g, 3]}] ==
-2 (Subscript[e, 1] - Subscript[e, 3])^(3/2)
((JacobiCN[Sqrt[Subscript[e, 1] - Subscript[e, 3]] z, m]
JacobiDN[Sqrt[Subscript[e, 1] - Subscript[e, 3]] z, m])/
JacobiSN[Sqrt[Subscript[e, 1] - Subscript[e, 3]] z, m]^3) /;
m == ModularLambda[Subscript[\[Omega], 3]/Subscript[\[Omega], 1]]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassPPrime", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "2"]], SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", FractionBox[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]]], " ", "z"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]]], " ", "z"]], ",", "m"]], "]"]]]], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]]], " ", "z"]], ",", "m"]], "]"]], "3"]]]]]], "/;", RowBox[List["m", "\[Equal]", RowBox[List["ModularLambda", "[", FractionBox[SubscriptBox["\[Omega]", "3"], SubscriptBox["\[Omega]", "1"]], "]"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> ℘ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <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> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> e </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <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> <mi> z </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <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> <mi> z </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <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> <mi> z </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> ⩵ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", FractionBox[SubscriptBox["\[Omega]", "3"], SubscriptBox["\[Omega]", "1"]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msup> <mi> ℘ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <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> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msub> <mi> e </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> e </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <mrow> <mi> cn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <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> <mi> z </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <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> <mi> z </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <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> <mi> z </mi> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> ⩵ </mo> <semantics> <mrow> <mi> λ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <msub> <mi> ω </mi> <mn> 3 </mn> </msub> <msub> <mi> ω </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Lambda]", "(", FractionBox[SubscriptBox["\[Omega]", "3"], SubscriptBox["\[Omega]", "1"]], ")"]], InterpretTemplate[Function[ModularLambda[Slot[1]]]]] </annotation> </semantics> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WeierstrassPPrime", "[", RowBox[List["z_", ",", RowBox[List["{", RowBox[List[SubscriptBox["g_", "2"], ",", SubscriptBox["g_", "3"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]], ")"]], RowBox[List["3", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]]], " ", "z"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]]], " ", "z"]], ",", "m"]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List[SubscriptBox["e", "1"], "-", SubscriptBox["e", "3"]]]], " ", "z"]], ",", "m"]], "]"]], "3"]]]], "/;", RowBox[List["m", "\[Equal]", RowBox[List["ModularLambda", "[", FractionBox[SubscriptBox["\[Omega]", "3"], SubscriptBox["\[Omega]", "1"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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