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variants of this functions
EllipticThetaPrime






Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticThetaPrime[3,z,q] > Series representations > q-series > Expansions at q==1





http://functions.wolfram.com/09.07.06.0011.01









  


  










Input Form





EllipticThetaPrime[3, z, q] == (((6 Sqrt[Pi] I)/(q - 1)^(3/2)) Sum[Binomial[k + 3/2, k] Sum[(((-1)^j Binomial[k, j])/(2 j + 3)) Subscript[p, j, k] (q - 1)^k E^(z^2/Log[q]) (z + 2 Sum[E^((m^2 Pi^2)/Log[q]) (z Cosh[(2 m Pi z)/Log[q]] + Pi m Sinh[(2 m Pi z)/Log[q]]), {m, 1, Infinity}]), {j, 0, k}], {k, 0, Infinity}])/E^(3 I Pi Floor[3/4 - Arg[q - 1]/(2 Pi)]) /; (Abs[q] < 1 && Abs[q - 1] < 1) && Subscript[c, k] == (-1)^(k - 1)/(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





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MathML Form







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</ci> <ci> p </ci> <ci> j </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticThetaPrime", "[", RowBox[List["3", ",", "z_", ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["6", " ", SqrtBox["\[Pi]"], " ", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "3"]], " ", "\[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["3", "2"]]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], ")"]], " ", 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["z", "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["m", "2"], " ", SuperscriptBox["\[Pi]", "2"]]], RowBox[List["Log", "[", "q", "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", "m", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", "m", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", "m", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]]]], ")"]]]]]]]]]], ")"]]]], RowBox[List[RowBox[List["2", " ", "j"]], "+", "3"]]]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", "1"]], ")"]], RowBox[List["3", "/", "2"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "q", "]"]], "<", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["q", "-", "1"]], "]"]], "<", "1"]]]], ")"]], "&&", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], RowBox[List["k", "+", "1"]]]]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List["-", FractionBox[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"]]]]]]]]], "k"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]










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





2007-05-02