RamificationIndex[LegendreQ[\[Nu], \[Mu], 2, z], z, -1] == s /; \[Mu] == (2 r)/s && Element[{r, s}, Integers] && s > 1 && GCD[r, s] == 1

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], ",", "z", ",", RowBox[List["-", "1"]]]], "]"]], "\[Equal]", "s"]], "/;", RowBox[List[RowBox[List["\[Mu]", "\[Equal]", FractionBox[RowBox[List["2", "r"]], "s"]]], "\[And]", RowBox[List["Element", "[", RowBox[List[RowBox[List["{", RowBox[List["r", ",", "s"]], "}"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List["s", ">", "1"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["r", ",", "s"]], "]"]], "\[Equal]", "1"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> &#8475; </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mi> s </mi> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#956; </mi> <mo> &#10869; </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> r </mi> </mrow> <mi> s </mi> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> s </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#8475; </ci> <ci> z </ci> </apply> <apply> <ci> LegendreQ </ci> <ci> &#957; </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> s </ci> </apply> <apply> <and /> <apply> <eq /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <apply> <power /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <list> <ci> r </ci> <ci> s </ci> </list> <integers /> </apply> <apply> <gt /> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <gcd /> <ci> r </ci> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], ",", "z_", ",", RowBox[List["-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["s", "/;", RowBox[List[RowBox[List["\[Mu]", "\[Equal]", FractionBox[RowBox[List["2", " ", "r"]], "s"]]], "&&", RowBox[List[RowBox[List["{", RowBox[List["r", ",", "s"]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List["s", ">", "1"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["r", ",", "s"]], "]"]], "\[Equal]", "1"]]]]]]]]]]

2001-10-29