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Sin




Mathematica Notation

Traditional Notation









Elementary Functions > Sin[z] > Transformations > Argument involving complex components





http://functions.wolfram.com/01.06.16.0082.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", RowBox[List["Arg", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Re", "[", "z", "]"]], SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], RowBox[List["-", "a"]]]]], RowBox[List["2", RowBox[List["Im", "[", "z", "]"]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"], SuperscriptBox[RowBox[List["Re", "[", "z", "]"]], "2"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "-", RowBox[List[RowBox[List["Re", "[", "z", "]"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"], SuperscriptBox[RowBox[List["Re", "[", "z", "]"]], "2"]]]]]]]]], ")"]], "a"], "-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "+", RowBox[List[RowBox[List["Re", "[", "z", "]"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"], SuperscriptBox[RowBox[List["Re", "[", "z", "]"]], "2"]]]]]]]]], ")"]], "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> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mtext> </mtext> <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 /> <apply> <real /> <ci> z </ci> </apply> <apply> <power /> <apply> <abs /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <real /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <real /> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </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 /> <apply> <real /> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sin", "[", RowBox[List["a_", " ", RowBox[List["Arg", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", "z", "]"]], RowBox[List["-", "a"]]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"], SuperscriptBox[RowBox[List["Re", "[", "z", "]"]], "2"]]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "-", RowBox[List[RowBox[List["Re", "[", "z", "]"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"], SuperscriptBox[RowBox[List["Re", "[", "z", "]"]], "2"]]]]]]]]], ")"]], "a"], "-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "+", RowBox[List[RowBox[List["Re", "[", "z", "]"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["Im", "[", "z", "]"]], "2"], SuperscriptBox[RowBox[List["Re", "[", "z", "]"]], "2"]]]]]]]]], ")"]], "a"]]], ")"]]]], RowBox[List["2", " ", RowBox[List["Im", "[", "z", "]"]]]]]]]]]


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





2001-10-29