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 WeierstrassInvariants

 http://functions.wolfram.com/09.19.20.0010.01

 Input Form

 D[WeierstrassInvariants[{Subscript[\[Omega], 1], Subscript[\[Omega], 3]}], {Subscript[\[Omega], 3], \[Alpha]}] == {(Pi^4/12) FDPowerConstant[Subscript[\[Omega], 3], -4, \[Alpha]] Subscript[\[Omega], 3]^(-4 - \[Alpha]) + (15/(Subscript[\[Omega], 3]^\[Alpha] (2 Subscript[\[Omega], 1]^4))) Sum[(1/m^4) Hypergeometric2F1Regularized[1, 4, 1 - \[Alpha], -((n Subscript[\[Omega], 3])/(m Subscript[\[Omega], 1]))], {n, -Infinity, Infinity}, {m, 1, Infinity}], (Pi^6/216) FDPowerConstant[Subscript[\[Omega], 3], -6, \[Alpha]] Subscript[\[Omega], 3]^(-6 - \[Alpha]) + (35/(Subscript[\[Omega], 3]^\[Alpha] (8 Subscript[\[Omega], 1]^6))) Sum[(1/m^6) Hypergeometric2F1Regularized[1, 6, 1 - \[Alpha], -((n Subscript[\[Omega], 3])/(m Subscript[\[Omega], 1]))], {n, -Infinity, Infinity}, {m, 1, Infinity}]}

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", "\[Alpha]"]], "}"]]], RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]", "1"], ",", SubscriptBox["\[Omega]", "3"]]], "}"]], "]"]]]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "4"], "12"], RowBox[List["FDPowerConstant", "[", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", RowBox[List["-", "4"]], ",", "\[Alpha]"]], "]"]], SubsuperscriptBox["\[Omega]", "3", RowBox[List[RowBox[List["-", "4"]], "-", "\[Alpha]"]]]]], "+", RowBox[List[FractionBox[RowBox[List["15", " ", SubsuperscriptBox["\[Omega]", "3", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["2", " ", SubsuperscriptBox["\[Omega]", "1", "4"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", SuperscriptBox["m", "4"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "4", ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["-", FractionBox[RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]], RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]]]]]]], "]"]]]]]]]]]]]], ",", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "6"], "216"], RowBox[List["FDPowerConstant", "[", RowBox[List[SubscriptBox["\[Omega]", "3"], ",", RowBox[List["-", "6"]], ",", "\[Alpha]"]], "]"]], SubsuperscriptBox["\[Omega]", "3", RowBox[List[RowBox[List["-", "6"]], "-", "\[Alpha]"]]]]], "+", RowBox[List[FractionBox[RowBox[List["35", " ", SubsuperscriptBox["\[Omega]", "3", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["8", SubsuperscriptBox["\[Omega]", "1", "6"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", SuperscriptBox["m", "6"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "6", ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["-", FractionBox[RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]], RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]]]]]]], "]"]]]]]]]]]]]]]], "}"]]]]]]

 MathML Form

 { α g 2 ( ω 1 , ω 3 ) ω 3 α , α g 3 ( ω 1 , ω 3 ) ω 3 α } { π 4 12 ℱ𝒞 exp ( α ) ( ω 3 , - 4 ) ω 3 - 4 - α + 15 ω 3 - α 2 ω 1 4 n = - m = 1 1 m 4 2 F ~ 1 ( 1 , 4 ; 1 - α ; - n ω 3 m ω 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["4", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Alpha]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]], RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]]]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] , π 6 216 ℱ𝒞 exp ( α ) ( ω 3 , - 6 ) ω 3 - 6 - α + 35 ω 3 - α 8 ω 1 6 n = - m = 1 1 m 6 2 F ~ 1 ( 1 , 6 ; 1 - α ; - n ω 3 m ω 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["6", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Alpha]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["n", " ", SubscriptBox["\[Omega]", "3"]]], RowBox[List["m", " ", SubscriptBox["\[Omega]", "1"]]]]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] } D Subscript g 2 Subscript ω 1 Subscript ω 3 Subscript ω 3 α D Subscript g 3 Subscript ω 1 Subscript ω 3 Subscript ω 3 α 4 12 -1 Subscript ℱ𝒞 exp α Subscript ω 3 -4 Subscript ω 3 -4 -1 α 15 Subscript ω 3 -1 α 2 Subscript ω 1 4 -1 m 1 n -1 1 m 4 -1 Hypergeometric2F1Regularized 1 4 1 -1 α -1 n Subscript ω 3 m Subscript ω 1 -1 6 216 -1 Subscript ℱ𝒞 exp α Subscript ω 3 -6 Subscript ω 3 -6 -1 α 35 Subscript ω 3 -1 α 8 Subscript ω 1 6 -1 m 1 n -1 1 m 6 -1 Hypergeometric2F1Regularized 1 6 1 -1 α -1 n Subscript ω 3 m Subscript ω 1 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]_", "3"], ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Omega]_", "1"], ",", SubscriptBox["\[Omega]_", "3"]]], "}"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "12"], " ", SuperscriptBox["\[Pi]", "4"], " ", RowBox[List["FDPowerConstant", "[", RowBox[List[SubscriptBox["\[Omega]\[Omega]", "3"], ",", RowBox[List["-", "4"]], ",", "\[Alpha]"]], "]"]], " ", SubsuperscriptBox["\[Omega]\[Omega]", "3", RowBox[List[RowBox[List["-", "4"]], "-", "\[Alpha]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["15", " ", SubsuperscriptBox["\[Omega]\[Omega]", "3", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "4", ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["-", FractionBox[RowBox[List["n", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]], RowBox[List["m", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]]]]]]], "]"]], SuperscriptBox["m", "4"]]]]]]]], RowBox[List["2", " ", SubsuperscriptBox["\[Omega]\[Omega]", "1", "4"]]]]]], ",", RowBox[List[RowBox[List[FractionBox["1", "216"], " ", SuperscriptBox["\[Pi]", "6"], " ", RowBox[List["FDPowerConstant", "[", RowBox[List[SubscriptBox["\[Omega]\[Omega]", "3"], ",", RowBox[List["-", "6"]], ",", "\[Alpha]"]], "]"]], " ", SubsuperscriptBox["\[Omega]\[Omega]", "3", RowBox[List[RowBox[List["-", "6"]], "-", "\[Alpha]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["35", " ", SubsuperscriptBox["\[Omega]\[Omega]", "3", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "6", ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List["-", FractionBox[RowBox[List["n", " ", SubscriptBox["\[Omega]\[Omega]", "3"]]], RowBox[List["m", " ", SubscriptBox["\[Omega]\[Omega]", "1"]]]]]]]], "]"]], SuperscriptBox["m", "6"]]]]]]]], RowBox[List["8", " ", SubsuperscriptBox["\[Omega]\[Omega]", "1", "6"]]]]]]]], "}"]]]]]]

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

 2001-10-29

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