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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,2,z] > Series representations > Generalized power series > Expansions at z==1





http://functions.wolfram.com/07.11.06.0005.01









  


  










Input Form





LegendreQ[\[Nu], \[Mu], 2, z] == (Pi/2) Csc[\[Mu] Pi] ((Cos[\[Mu] Pi]/Gamma[1 - \[Mu]]) ((1 + z)^(\[Mu]/2)/(1 - z)^(\[Mu]/2)) Sum[((Pochhammer[-\[Nu], k] Pochhammer[\[Nu] + 1, k])/ (Pochhammer[1 - \[Mu], k] k!)) ((1 - z)/2)^k, {k, 0, Infinity}] - (Pochhammer[\[Nu] - \[Mu] + 1, 2 \[Mu]]/Gamma[1 + \[Mu]]) ((1 - z)^(\[Mu]/2)/(1 + z)^(\[Mu]/2)) Sum[((Pochhammer[-\[Nu], k] Pochhammer[\[Nu] + 1, k])/ (Pochhammer[1 + \[Mu], k] k!)) ((1 - z)/2)^k, {k, 0, Infinity}]) /; Abs[(1 - z)/2] < 1 && !Element[\[Mu], Integers]










Standard Form





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MathML Form







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</mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;\[Nu]&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Mu]&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Nu]&quot;, &quot;-&quot;, &quot;\[Mu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[Mu]&quot;]]], Pochhammer] </annotation> </semantics> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;\[Nu]&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mtext> </mtext> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Mu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#956; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LegendreQ </ci> <ci> &#957; </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <csc /> <apply> <times /> <ci> &#956; </ci> <pi /> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <cos /> <apply> <times /> <ci> &#956; </ci> <pi /> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <notin /> <ci> &#956; </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Mu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Mu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], "-", FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "\[Mu]", "+", "1"]], ",", RowBox[List["2", " ", "\[Mu]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Mu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List["\[Mu]", "\[Element]", "Integers"]]]]]]]]]]]]










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





2001-10-29