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variants of this functions
ChebyshevU




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.05.20.0004.01









  


  








Input Form





D[ChebyshevU[\[Nu], z], {z, 2}] == (1/(z^2 - 1)^2) ((z^2 (3 + 2 \[Nu] + \[Nu]^2) - \[Nu] (2 + \[Nu])) ChebyshevU[\[Nu], z] - 3 z (1 + \[Nu]) ChebyshevT[1 + \[Nu], z])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "2"]], "}"]]], RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], ")"]], "2"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]], "+", SuperscriptBox["\[Nu]", "2"]]], ")"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]]]], ")"]], " ", RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "-", RowBox[List["3", " ", "z", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mn> 2 </mn> </msup> <mrow> <msub> <mi> U </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> U </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> T </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> ChebyshevU </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> ChebyshevU </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> z </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> ChebyshevT </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "2"]], "}"]]]]], RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]], "+", SuperscriptBox["\[Nu]", "2"]]], ")"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]]]], ")"]], " ", RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "-", RowBox[List["3", " ", "z", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], ")"]], "2"]]]]]]


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





2001-10-29