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 EllipticTheta

 http://functions.wolfram.com/09.03.06.0030.01

 Input Form

 EllipticTheta[3, z, q] == (((I Sqrt[Pi])/Sqrt[q - 1]) Sum[Binomial[k + 1/2, k] Sum[(((-1)^j Binomial[k, j])/(2 j + 1)) Subscript[p, j, k] (q - 1)^k E^(z^2/Log[q]) (1 + 2 Sum[E^((k^2 Pi^2)/Log[q]) Cosh[(2 k Pi z)/Log[q]], {k, 1, Infinity}]), {j, 0, k}], {k, 0, Infinity}])/ E^(I Pi Floor[3/4 - Arg[q - 1]/(2 Pi)]) /; (Abs[q] < 1 && Abs[q - 1] < 1) && Subscript[c, k] == (-1)^k/(k + 1) && Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/k) Sum[(j m - k + m) Subscript[c, m] Subscript[p, j, k - m], {m, 1, k}] && Element[k, Integers] && k > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "z", ",", "q"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["\[Pi]"]]], SqrtBox[RowBox[List["q", "-", "1"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["q", "-", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "+", FractionBox["1", "2"]]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], RowBox[List[RowBox[List["2", "j"]], "+", "1"]]], SubscriptBox["p", RowBox[List["j", ",", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", "1"]], ")"]], "k"], SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], RowBox[List["Log", "[", "q", "]"]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["k", "2"], SuperscriptBox["\[Pi]", "2"]]], RowBox[List["Log", "[", "q", "]"]]]], RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", "k", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "q", "]"]], "<", "1"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["q", "-", "1"]], "]"]], "<", "1"]]]], ")"]], "\[And]", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["k", "+", "1"]]]]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List[FractionBox["1", "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], SubscriptBox["c", "m"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]]]]]]]]

 MathML Form

 ϑ 3 ( z , q ) π q - 1 - π 3 4 - arg ( q - 1 ) 2 π k = 0 ( k + 1 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", FractionBox["1", "2"]]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] j = 0 k ( - 1 ) j 2 j + 1 ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] p j , k ( q - 1 ) k z 2 log ( q ) ( 1 + 2 k = 1 k 2 π 2 log ( q ) cosh ( 2 k π z log ( q ) ) ) /; ( "\[LeftBracketingBar]" q "\[RightBracketingBar]" < 1 "\[LeftBracketingBar]" q - 1 "\[RightBracketingBar]" < 1 ) c k ( - 1 ) k k + 1 p j , 0 1 p j , k 1 k m = 1 k ( j m - k + m ) c m p j , k - m k TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition EllipticTheta 3 z q 1 2 q -1 1 2 -1 -1 3 4 -1 q -1 2 -1 k 0 Binomial k 1 2 k j 0 k -1 j 2 j 1 -1 Binomial k j Subscript p j k q -1 k z 2 q -1 1 2 k 1 k 2 2 q -1 2 k z q -1 q 1 q -1 1 Subscript c k -1 k k 1 -1 Subscript p j 0 1 Subscript p j k 1 k -1 m 1 k j m -1 k m Subscript c m Subscript p j k -1 m k SuperPlus [/itex]

 Rule Form

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

 2007-05-02