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variants of this functions
ChebyshevT




Mathematica Notation

Traditional Notation









Hypergeometric Functions > ChebyshevT[nu,z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power function





http://functions.wolfram.com/07.04.21.0003.01









  


  








Input Form





Integrate[z^(\[Alpha] - 1) ChebyshevT[\[Nu], z], z] == 2^(-1 - \[Alpha]) z^(\[Alpha] - 1) (ChebyshevT[\[Alpha] - 1, z] + I Sqrt[1 - z^2] ChebyshevU[\[Alpha] - 2, z]) (z (z + I Sqrt[1 - z^2]))^(1 - \[Alpha]) ((1/(\[Alpha] - \[Nu])) ((ChebyshevT[\[Nu] - \[Alpha], z] + I Sqrt[1 - z^2] ChebyshevU[\[Nu] - \[Alpha] - 1, z]) Hypergeometric2F1[(\[Nu] - \[Alpha])/2, 1 - \[Alpha], 1 + (\[Nu] - \[Alpha])/2, 1 - 2 z^2 - 2 I z Sqrt[1 - z^2]]) + (1/(2 + \[Nu] - \[Alpha])) ((ChebyshevT[2 + \[Nu] - \[Alpha], z] + I Sqrt[1 - z^2] ChebyshevU[1 + \[Nu] - \[Alpha], z]) Hypergeometric2F1[1 + (\[Nu] - \[Alpha])/2, 1 - \[Alpha], 2 + (\[Nu] - \[Alpha])/2, 1 - 2 z^2 - 2 I z Sqrt[1 - z^2]]) + (1/(\[Nu] + \[Alpha])) ((ChebyshevT[\[Nu] + \[Alpha], z] - I Sqrt[1 - z^2] ChebyshevU[\[Nu] + \[Alpha] - 1, z]) Hypergeometric2F1[-((\[Nu] + \[Alpha])/2), 1 - \[Alpha], 1 - (\[Nu] + \[Alpha])/2, 1 - 2 z^2 - 2 I z Sqrt[1 - z^2]]) - (1/(\[Nu] + \[Alpha] - 2)) ((ChebyshevT[2 - \[Nu] - \[Alpha], z] + I Sqrt[1 - z^2] ChebyshevU[1 - \[Nu] - \[Alpha], z]) Hypergeometric2F1[1 - (\[Nu] + \[Alpha])/2, 1 - \[Alpha], 2 - (\[Nu] + \[Alpha])/2, 1 - 2 z^2 - 2 I z Sqrt[1 - z^2]]))








Standard Form





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MathML Form







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</mo> <mrow> <msub> <mi> U </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#945; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#945; </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[RowBox[List[&quot;\[Alpha]&quot;, &quot;+&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;]]], Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Alpha]&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, FractionBox[RowBox[List[&quot;\[Alpha]&quot;, &quot;+&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[RowBox[List[RowBox[List[&quot;-&quot;, &quot;2&quot;]], &quot; 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</ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> ChebyshevT </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> ChebyshevU </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; 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</ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#945; </ci> <ci> &#957; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; 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</ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> &#945; 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</ci> <ci> &#957; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#945; </ci> <ci> &#957; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29