Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Sin






Mathematica Notation

Traditional Notation









Elementary Functions > Sin[z] > Differentiation > Fractional integro-differentiation





http://functions.wolfram.com/01.06.20.0010.01









  


  










Input Form





D[Sin[z]^n, {z, \[Alpha]}] == (Binomial[n, n/2] (1 - Mod[n, 2]) (1/(z^\[Alpha] Gamma[1 - \[Alpha]])))/ 2^n - (2^(\[Alpha] - n) Sqrt[Pi] Sum[(-1)^(k + n - 1) Binomial[n, k] (2 Cos[(n Pi)/2] HypergeometricPFQRegularized[{1}, {1/2 - \[Alpha]/2, 1 - \[Alpha]/2}, (-(1/4)) (-2 k + n)^2 z^2] - (-2 k + n) z HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, 3/2 - \[Alpha]/2}, (-(1/4)) (-2 k + n)^2 z^2] Sin[(n Pi)/2]), {k, 0, Floor[(n - 1)/2]}])/z^\[Alpha] /; Element[n, Integers] && n > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "n"]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", FractionBox["n", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["n", ",", "2"]], "]"]]]], ")"]], FractionBox[SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "n"]]], SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "n", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], " ", "z", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]










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> <msup> <mo> &#8706; </mo> <mi> &#945; </mi> </msup> <mrow> <msup> <mi> sin </mi> <mi> n </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <semantics> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 2 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`n </ci> <cn type='integer'> 2 </cn> </apply> </annotation-xml> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]], List[TagBox[FractionBox[&quot;n&quot;, &quot;2&quot;], Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> &#945; </mi> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 1 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> ; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> &#945; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> &#945; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;1&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;-&quot;, FractionBox[&quot;\[Alpha]&quot;, &quot;2&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[&quot;\[Alpha]&quot;, &quot;2&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[RowBox[List[&quot;-&quot;, FractionBox[&quot;1&quot;, &quot;4&quot;]]], &quot; &quot;, SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;n&quot;, &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;k&quot;]]]], &quot;)&quot;]], &quot;2&quot;], &quot; &quot;, SuperscriptBox[&quot;z&quot;, &quot;2&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 1 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> ; </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> &#945; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> &#945; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;1&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[&quot;\[Alpha]&quot;, &quot;2&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;3&quot;, &quot;2&quot;], &quot;-&quot;, FractionBox[&quot;\[Alpha]&quot;, &quot;2&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[RowBox[List[&quot;-&quot;, FractionBox[&quot;1&quot;, &quot;4&quot;]]], &quot; &quot;, SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;n&quot;, &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;k&quot;]]]], &quot;)&quot;]], &quot;2&quot;], &quot; &quot;, SuperscriptBox[&quot;z&quot;, &quot;2&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <ci> n </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#945; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <times /> <ci> n </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <cn type='integer'> 1 </cn> </list> <list> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#945; </ci> <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> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <ci> z </ci> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <cn type='integer'> 1 </cn> </list> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#945; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> n </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </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_", ",", "\[Alpha]_"]], "}"]]]]], SuperscriptBox[RowBox[List["Sin", "[", "z_", "]"]], "n_"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", FractionBox["n", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["n", ",", "2"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "n"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "n", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], " ", "z", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "n"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2002-12-18





© 1998- Wolfram Research, Inc.