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Sin




Mathematica Notation

Traditional Notation









Elementary Functions > Sin[z] > Transformations > Multiple arguments > Argument involving symbolic multiples of variable





http://functions.wolfram.com/01.06.16.0157.01









  


  








Input Form





Sin[a z] == (1/(2 I)) ((Cos[z] + I Sin[z])^a Exp[-2 I Pi a Floor[(Pi - Re[z])/(2 Pi)]] - (Cos[z] - I Sin[z])^a Exp[-2 I Pi a Floor[(Pi + Re[z])/(2 Pi)]])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", "\[ImaginaryI]"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]], "a"], " ", RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Re", "[", "z", "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]], "a"], " ", RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Re", "[", "z", "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]]]], ")"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> - </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mi> &#960; </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sin /> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> z </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <sin /> <ci> z </ci> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <pi /> <ci> a </ci> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <cos /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <sin /> <ci> z </ci> </apply> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <pi /> <ci> a </ci> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <real /> <ci> z </ci> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sin", "[", RowBox[List["a_", " ", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]], "a"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Re", "[", "z", "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", "z", "]"]]]]]], ")"]], "a"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "a", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Re", "[", "z", "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]], RowBox[List["2", " ", "\[ImaginaryI]"]]]]]]]


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





2007-05-02