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variants of this functions
EllipticTheta






Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticTheta[3,z,q] > Differentiation > Low-order differentiation > With respect to q





http://functions.wolfram.com/09.03.20.0009.01









  


  










Input Form





D[EllipticTheta[3, z, q], q] == (-(1/(4 q))) ((EllipticThetaPrime[1, z, q]^2 EllipticTheta[3, z, q])/ EllipticTheta[1, z, q]^2) + (1/(2 q)) ((EllipticTheta[3, 0, q]^2 EllipticTheta[2, z, q] EllipticTheta[4, z, q])/EllipticTheta[1, z, q]^2) EllipticThetaPrime[1, z, q] - (1/(4 q)) EllipticTheta[2, 0, q]^2 EllipticTheta[3, 0, q]^2 (EllipticTheta[4, z, q]^2/ EllipticTheta[1, z, q]^2) EllipticTheta[3, z, q] + (1/(Pi^2 q)) EllipticTheta[3, z, q] (WeierstrassZeta[1, WeierstrassInvariants[{1, Log[q]/(Pi I)}]] + (Pi^2/12) (EllipticTheta[3, 0, q]^4 + EllipticTheta[4, 0, q]^4))










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["q_"]]], RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "z_", ",", "q_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]], "2"], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "z", ",", "q"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["4", " ", "q"]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]], "2"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", "q"]], "]"]], "2"], " ", RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "z", ",", "q"]], "]"]], " ", RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", "z", ",", "q"]], "]"]]]], ")"]], " ", RowBox[List["EllipticThetaPrime", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "q"]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]], "2"]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["2", ",", "0", ",", "q"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", "q"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", "z", ",", "q"]], "]"]], "2"], " ", RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "z", ",", "q"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["4", " ", "q"]], ")"]], " ", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["1", ",", "z", ",", "q"]], "]"]], "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "z", ",", "q"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["WeierstrassZeta", "[", RowBox[List["1", ",", RowBox[List["WeierstrassInvariants", "[", RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["Log", "[", "q", "]"]], RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]]]], "}"]], "]"]]]], "]"]], "+", RowBox[List[FractionBox["1", "12"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", "q"]], "]"]], "4"], "+", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", "0", ",", "q"]], "]"]], "4"]]], ")"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", "q"]]]]]]]]]










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





2007-05-02