|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.09.04.0014.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
RamificationIndex[LegendreP[\[Nu], \[Mu], 3, z], z, -1] == s /;
\[Mu] == r/s && Element[r, Integers] && Element[s - 1, Integers] &&
s - 1 > 0 && GCD[r, s] == 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RamificationIndex", "[", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], ",", "z", ",", RowBox[List["-", "1"]]]], "]"]], "\[Equal]", "s"]], "/;", RowBox[List[RowBox[List["\[Mu]", "\[Equal]", FractionBox["r", "s"]]], "\[And]", RowBox[List["Element", "[", RowBox[List["r", ",", "Integers"]], "]"]], "\[And]", RowBox[List["Element", "[", RowBox[List[RowBox[List["s", "-", "1"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["s", "-", "1"]], ">", "0"]], "\[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> ℛ </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> 𝔓 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalP]", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mi> μ </mi> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mi> s </mi> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> μ </mi> <mo> ⩵ </mo> <mfrac> <mi> r </mi> <mi> s </mi> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> r </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> , </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> ℛ </ci> <ci> z </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔓 </ci> </apply> <ci> ν </ci> </apply> <ci> μ </ci> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> s </ci> </apply> <apply> <and /> <apply> <eq /> <ci> μ </ci> <apply> <times /> <ci> r </ci> <apply> <power /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> r </ci> <integers /> </apply> <apply> <in /> <apply> <plus /> <ci> s </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </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["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], ",", "z_", ",", RowBox[List["-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["s", "/;", RowBox[List[RowBox[List["\[Mu]", "\[Equal]", FractionBox["r", "s"]]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["s", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["s", "-", "1"]], ">", "0"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["r", ",", "s"]], "]"]], "\[Equal]", "1"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
LegendreP[n,z] | LegendreP[nu,z] | LegendreP[nu,mu,z] | LegendreP[n,mu,2,z] | LegendreP[nu,mu,2,z] | |
|
|
|