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EllipticNomeQ






Mathematica Notation

Traditional Notation









Elliptic Functions > EllipticNomeQ[m] > Series representations > Generalized power series > Expansions at m==infinity





http://functions.wolfram.com/09.53.06.0006.01









  


  










Input Form





EllipticNomeQ[m] \[Proportional] Exp[-((Sqrt[-m] Pi Log[16 m])/(Sqrt[m] Log[-16 m]))] (1 + (Sqrt[-m^2] Pi (Log[-m] - Log[m]))/(2 Log[-16 m]^2 m^2) + ((Pi Sqrt[-m^2] (Log[-m] - Log[m]))/(64 Log[-16 m]^4 m^(7/2))) (8 (2 Sqrt[m] + Sqrt[-m] Pi) Log[-16 m] + 13 Sqrt[m] Log[-16 m]^2 - 8 Sqrt[-m] Pi Log[16 m]) + \[Ellipsis]) /; (Abs[m] -> Infinity)










Standard Form





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MathML Form







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</mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 16 </mn> </mrow> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 64 </mn> <mo> &#8290; 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</mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> m </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <exp /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <apply> <ln /> <apply> <times /> <cn type='integer'> 16 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> -16 </cn> <ci> m </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <apply> <plus /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> -16 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> -16 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 13 </cn> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> -16 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> -16 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <apply> <ln /> <apply> <times /> <cn type='integer'> 16 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> m </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticNomeQ", "[", "m_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", "m"]]], " ", "\[Pi]", " ", RowBox[List["Log", "[", RowBox[List["16", " ", "m"]], "]"]]]], RowBox[List[SqrtBox["m"], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "]"]]]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["m", "2"]]]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "m"]], "]"]], "-", RowBox[List["Log", "[", "m", "]"]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "]"]], "2"], " ", SuperscriptBox["m", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "m"]], "]"]], "-", RowBox[List["Log", "[", "m", "]"]]]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["m"]]], "+", RowBox[List[SqrtBox[RowBox[List["-", "m"]]], " ", "\[Pi]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "]"]]]], "+", RowBox[List["13", " ", SqrtBox["m"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "]"]], "2"]]], "-", RowBox[List["8", " ", SqrtBox[RowBox[List["-", "m"]]], " ", "\[Pi]", " ", RowBox[List["Log", "[", RowBox[List["16", " ", "m"]], "]"]]]]]], ")"]]]], RowBox[List["64", " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "]"]], "4"], " ", SuperscriptBox["m", RowBox[List["7", "/", "2"]]]]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2001-10-29





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