EllipticNomeQ[m] \[Proportional] (16 m)^((Pi Sqrt[m])/(Sqrt[-m] Log[-16 m])) (1 + O[(Sqrt[-m^2] (Log[-m] - Log[m]))/(Log[-16 m]^2 m^2)]) /; (Abs[m] -> Infinity)

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "\[Proportional]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["16", "m"]], ")"]], FractionBox[RowBox[List["\[Pi]", " ", SqrtBox["m"]]], RowBox[List[SqrtBox[RowBox[List["-", "m"]]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "]"]]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["m", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "m"]], "]"]], "-", RowBox[List["Log", "[", "m", "]"]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "]"]], "2"], " ", SuperscriptBox["m", "2"]]]], "]"]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mi> m </mi> </msqrt> </mrow> <mrow> <msqrt> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 16 </mn> </mrow> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <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> <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> </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> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <ci> m </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <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> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <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> <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 /> <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> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> m </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticNomeQ", "[", "m_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["16", " ", "m"]], ")"]], FractionBox[RowBox[List["\[Pi]", " ", SqrtBox["m"]]], RowBox[List[SqrtBox[RowBox[List["-", "m"]]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "]"]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["m", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "m"]], "]"]], "-", RowBox[List["Log", "[", "m", "]"]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "16"]], " ", "m"]], "]"]], "2"], " ", SuperscriptBox["m", "2"]]]], "]"]]]], ")"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

2001-10-29