Sine: Representations through more general functions (formula 01.06.26.0054)

Sin

$Mathematica Notation$

Elementary Functions

Sin[z]

Representations through more general functions

Through Meijer G

Generalized cases involving tan^-1 in the arguments

http://functions.wolfram.com/01.06.26.0054.01

Input Form

(1 + z^2)^(\[Nu]/2) Sin[\[Nu] ArcTan[z]] == (-(2^(-1 - \[Nu])/Gamma[-\[Nu]])) MeijerG[{{1 + \[Nu]/2, (1 + \[Nu])/2}, {}}, {{1/2}, {0}}, z, 1/2]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["\[Nu]", "/", "2"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", RowBox[List["ArcTan", "[", "z", "]"]]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " "]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", "0", "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ν </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["1", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <ci> sin </ci> <apply> <times /> <ci> ν </ci> <apply> <times /> <apply> <power /> <ci> tan </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> Γ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list /> </list> <list> <list> <cn type='rational'> 1 <sep /> 2 </cn> </list> <list> <cn type='integer'> 0 </cn> </list> </list> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["z_", "2"]]], ")"]], FractionBox["\[Nu]_", "2"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]_", " ", RowBox[List["ArcTan", "[", "z_", "]"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", "0", "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]]]]]]

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

2001-10-29