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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,3,z] > Differentiation > Low-order differentiation > With respect to mu





http://functions.wolfram.com/07.09.20.0006.01









  


  










Input Form





D[LegendreP[\[Nu], \[Mu], 3, z], \[Mu]] == (1/2) (Log[z + 1] - Log[z - 1]) LegendreP[\[Nu], \[Mu], 3, z] + ((z + 1)^(\[Mu]/2)/(z - 1)^(\[Mu]/2)) Sum[((Pochhammer[-\[Nu], k] Pochhammer[\[Nu] + 1, k])/ (Gamma[1 - \[Mu] + k] k!)) PolyGamma[1 + k - \[Mu]] ((1 - z)/2)^k, {k, 0, Infinity}] /; Abs[(1 - z)/2] < 1 && !(Element[\[Mu], Integers] && \[Mu] > 0)










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> &#956; </mi> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mi> z </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </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> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["\[Mu]_"]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["z", "+", "1"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], 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"]], "]"]]]], ")"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "-", "\[Mu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Mu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Mu]", ">", "0"]]]], ")"]]]]]]]]]]]]










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





2001-10-29