Sine: Transformations (formula 01.06.16.0083)

Sin

$Mathematica Notation$

Elementary Functions

Sin[z]

Transformations

Argument involving complex components

http://functions.wolfram.com/01.06.16.0083.01

Input Form

Sin[a Arg[z]] == ((I/(2 Im[z] Re[z])) Abs[Im[z]] Abs[Re[z]] ((Re[z] - (I Abs[Im[z]] Abs[Re[z]])/Re[z])^a - (Re[z] + (I Abs[Im[z]] Abs[Re[z]])/Re[z])^a))/Abs[z]^a

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", RowBox[List["Arg", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]], " ", RowBox[List["Re", "[", "z", "]"]]]]], SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], RowBox[List["-", "a"]]], " ", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], RowBox[List["Re", "[", "z", "]"]]]]], ")"]], "a"], "-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], RowBox[List["Re", "[", "z", "]"]]]]], ")"]], "a"]]], ")"]]]]]]]]

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> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sin /> <apply> <times /> <ci> a </ci> <apply> <arg /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <abs /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <abs /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <abs /> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <real /> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <real /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <abs /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <abs /> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <real /> <ci> z </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <abs /> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <apply> <abs /> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> a </ci> </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