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variants of this functions
LegendreQ




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,z] > General characteristics > Symmetries and periodicities > Mirror symmetry





http://functions.wolfram.com/07.10.04.0002.01









  


  








Input Form





LegendreQ[Conjugate[\[Nu]], Conjugate[z]] == Conjugate[LegendreQ[\[Nu], z]] /; !IntervalMemberQ[Interval[{-Infinity, -1}], z] && !IntervalMemberQ[Interval[{1, Infinity}], z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["Conjugate", "[", "\[Nu]", "]"]], ",", RowBox[List["Conjugate", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List["Conjugate", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "]"]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mstyle scriptlevel='0'> <msub> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mover> <mi> &#957; </mi> <mo> _ </mo> </mover> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mover> <mi> z </mi> <mo> _ </mo> </mover> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> &#10869; </mo> <mover> <mrow> <msub> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mstyle scriptlevel='0'> <msub> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mover> <mi> &#957; </mi> <mo> _ </mo> </mover> </msub> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mover> <mi> z </mi> <mo> _ </mo> </mover> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> <mo> &#10869; </mo> <mover> <mrow> <msub> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox[&quot;Q&quot;, LegendreQ] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> _ </mo> </mover> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["Conjugate", "[", "\[Nu]_", "]"]], ",", RowBox[List["Conjugate", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Conjugate", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "]"]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]]


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





2001-10-29