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http://functions.wolfram.com/09.07.06.0012.01
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EllipticThetaPrime[3, z, q] \[Proportional]
(((2 Sqrt[Pi] I)/(q - 1)^(3/2)) (1 + O[q - 1]) E^(z^2/Log[q])
(z + 2 E^(Pi^2/Log[q]) (z Cosh[(2 Pi z)/Log[q]] +
Pi Sinh[(2 Pi z)/Log[q]]) + O[E^((4 Pi^2)/Log[q])
(z Cosh[(4 Pi z)/Log[q]] + 2 Pi Sinh[(4 Pi z)/Log[q]])]))/
E^(3 I Pi Floor[3/4 - Arg[q - 1]/(2 Pi)]) /; Abs[q] < 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticThetaPrime", "[", RowBox[List["3", ",", "z", ",", "q"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox["\[Pi]"], "\[ImaginaryI]", " "]], SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", "1"]], ")"]], RowBox[List["3", "/", "2"]]]], 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["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["q", "-", "1"]], "]"]]]], ")"]], SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], RowBox[List["Log", "[", "q", "]"]]]], RowBox[List["(", RowBox[List["z", "+", RowBox[List["2", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["\[Pi]", "2"], RowBox[List["Log", "[", "q", "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]]]], ")"]]]], "+", RowBox[List["O", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["4", " ", SuperscriptBox["\[Pi]", "2"]]], RowBox[List["Log", "[", "q", "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["4", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]], "+", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["4", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]]]], ")"]]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "q", "]"]], "<", "1"]]]]]]
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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> <msubsup> <semantics> <mi> ϑ </mi> <annotation encoding='Mathematica'> TagBox["\[CurlyTheta]", EllipticThetaPrime] </annotation> </semantics> <mn> 3 </mn> <mo> ′ </mo> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> q </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> EllipticThetaPrime </ci> <cn type='integer'> 3 </cn> <ci> z </ci> <ci> q </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> z </ci> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> z </ci> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <ci> z </ci> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <sinh /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <ci> z </ci> <apply> <power /> <apply> <ln /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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["2", " ", 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["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["q", "-", "1"]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], RowBox[List["Log", "[", "q", "]"]]]], " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["\[Pi]", "2"], RowBox[List["Log", "[", "q", "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]]]], ")"]]]], "+", RowBox[List["O", "[", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["4", " ", SuperscriptBox["\[Pi]", "2"]]], RowBox[List["Log", "[", "q", "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["4", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]], "+", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["4", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]]]], ")"]]]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", "1"]], ")"]], RowBox[List["3", "/", "2"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "q", "]"]], "<", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
