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http://functions.wolfram.com/09.13.16.0019.01
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WeierstrassP[n z, {Subscript[g, 2], Subscript[g, 3]}] ==
WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] -
(Subscript[\[Psi], n + 1] Subscript[\[Psi], n - 1])/
Subscript[\[Psi], n]^2 /; Element[n, Integers] && n > 0 &&
Subscript[\[Psi], 1] == 1 && Subscript[\[Psi], 2] ==
-WeierstrassPPrime[z, {Subscript[g, 2], Subscript[g, 3]}] &&
Subscript[\[Psi], 3] ==
3 WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}]^4 -
(3/2) Subscript[g, 2] WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}]^
2 - 3 Subscript[g, 3] WeierstrassP[z, {Subscript[g, 2],
Subscript[g, 3]}] - Subscript[g, 2]^2/16 &&
Subscript[\[Psi], 4] == WeierstrassPPrime[z, {Subscript[g, 2],
Subscript[g, 3]}]
(-2 WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}]^6 +
(5/2) Subscript[g, 2] WeierstrassP[z, {Subscript[g, 2],
Subscript[g, 3]}]^4 + 10 Subscript[g, 3]
WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}]^3 +
(5/8) Subscript[g, 2]^2 WeierstrassP[z, {Subscript[g, 2],
Subscript[g, 3]}]^2 + (1/2) Subscript[g, 2] Subscript[g, 3]
WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] +
Subscript[g, 3]^2 - Subscript[g, 2]^3/32) &&
(Subscript[\[Psi], n] ==
(-(Subscript[\[Psi], n/2]/WeierstrassPPrime[z, {Subscript[g, 2],
Subscript[g, 3]}])) (Subscript[\[Psi], n/2 + 2]
Subscript[\[Psi], n/2 - 1]^2 - Subscript[\[Psi], n/2 - 2]
Subscript[\[Psi], n/2 + 1]^2) /; Element[n/2, Integers] && n >= 0) &&
(Subscript[\[Psi], n] == Subscript[\[Psi], (n - 1)/2 + 2]
Subscript[\[Psi], (n - 1)/2]^3 - Subscript[\[Psi], (n - 1)/2 - 1]
Subscript[\[Psi], (n - 1)/2 + 1]^3 /; Element[(n - 1)/2, Integers] &&
n >= 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List[RowBox[List["n", " ", "z"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["WeierstrassP", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "-", FractionBox[RowBox[List[SubscriptBox["\[Psi]", RowBox[List["n", "+", "1"]]], " ", SubscriptBox["\[Psi]", RowBox[List["n", "-", "1"]]]]], SubsuperscriptBox["\[Psi]", "n", "2"]]]]]], "/;", "\[IndentingNewLine]", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", "\[IndentingNewLine]", RowBox[List[SubscriptBox["\[Psi]", "1"], "\[Equal]", "1"]], "\[And]", "\n", RowBox[List[SubscriptBox["\[Psi]", "2"], "\[Equal]", RowBox[List["-", RowBox[List["WeierstrassPPrime", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], 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RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], "2"]]], "+", RowBox[List[FractionBox["1", "2"], " ", SubscriptBox["g", "2"], " ", SubscriptBox["g", "3"], " ", RowBox[List["WeierstrassP", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]]]], "+", SubsuperscriptBox["g", "3", "2"], "-", FractionBox[SubsuperscriptBox["g", "2", "3"], "32"]]], ")"]]]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "n"], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["\[Psi]", FractionBox["n", "2"]], RowBox[List[RowBox[List["WeierstrassPPrime", "[", RowBox[List["z", ",", RowBox[List["{", RowBox[List[SubscriptBox["g", "2"], ",", SubscriptBox["g", "3"]]], "}"]]]], "]"]], " "]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", RowBox[List[FractionBox["n", "2"], "+", "2"]]], " ", SubsuperscriptBox["\[Psi]", RowBox[List[FractionBox["n", "2"], "-", "1"]], "2"]]], "-", RowBox[List[SubscriptBox["\[Psi]", RowBox[List[FractionBox["n", "2"], "-", "2"]]], " ", SubsuperscriptBox["\[Psi]", RowBox[List[FractionBox["n", "2"], "+", "1"]], "2"]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[FractionBox["n", "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]], ")"]], "\[And]", "\n", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "n"], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "+", "2"]]], " ", SubsuperscriptBox["\[Psi]", FractionBox[RowBox[List["n", "-", "1"]], "2"], "3"]]], "-", RowBox[List[SubscriptBox["\[Psi]", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "-", "1"]]], " ", SubsuperscriptBox["\[Psi]", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "+", "1"]], "3"]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]], ")"]]]]]]]]
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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> <mi> ℘ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <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> <mi> ℘ </mi> <mo> ⁡ </mo> <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> <mfrac> <mrow> <msub> <mi> ψ </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <msub> <mi> ψ </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <msubsup> <mi> ψ </mi> <mi> n </mi> <mn> 2 </mn> </msubsup> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> ψ </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> ψ </mi> <mn> 2 </mn> </msub> <mo> ⩵ </mo> <mrow> <mo> - </mo> <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> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> ψ </mi> <mn> 3 </mn> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <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> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <msup> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <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> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <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> </mrow> <mo> - </mo> <mfrac> <msubsup> <mi> g </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> <mn> 16 </mn> </mfrac> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> ψ </mi> <mn> 4 </mn> </msub> <mo> ⩵ </mo> <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> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <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> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mtext> </mtext> <msup> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <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> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <msup> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <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> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msubsup> <mi> g </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <mtext> </mtext> <msup> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <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> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> ℘ </mi> <mo> ⁡ </mo> <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> </mrow> <mo> + </mo> <msubsup> <mi> g </mi> <mn> 3 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mfrac> <msubsup> <mi> g </mi> <mn> 2 </mn> <mn> 3 </mn> </msubsup> <mn> 32 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> ψ </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <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> </mfrac> </mrow> <mo> ⁢ </mo> <msub> <mi> ψ </mi> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> ψ </mi> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> ⁢ </mo> <msubsup> <mi> ψ </mi> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <msub> <mi> ψ </mi> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> ⁢ </mo> <msubsup> <mi> ψ </mi> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </msubsup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> ψ </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mrow> <mrow> <msub> <mi> ψ </mi> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> ⁢ </mo> <msubsup> <mi> ψ </mi> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mn> 3 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <msub> <mi> ψ </mi> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <msubsup> <mi> ψ </mi> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 3 </mn> </msubsup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> WeierstrassP </ci> <apply> <times /> <ci> n </ci> <ci> z </ci> </apply> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> <apply> <plus /> <apply> <ci> WeierstrassP </ci> <ci> z </ci> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> 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</ci> </apply> <apply> <ci> CompoundExpression </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> ψ </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> WeierstrassP </ci> <ci> z </ci> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> WeierstrassP </ci> <ci> z </ci> <list> <apply> <ci> Subscript </ci> <ci> g 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){kind=small}
