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ArcSin




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSin[z] > Representations through equivalent functions > With related functions > Involving csch-1 > Involving sin-1((z2+1)1/2/z) > Involving sin-1((z2+1)1/2/z) and csch-1(z)





http://functions.wolfram.com/01.12.27.0288.01









  


  








Input Form





ArcSin[Sqrt[z^2 + 1]/z] == (Pi Sqrt[z^2])/(2 z) + (Sqrt[-z^4]/z^2) ArcCsch[z] /; Re[z] Im[z] != 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]], "z"], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[SuperscriptBox["z", "2"]]]], RowBox[List["2", " ", "z"]]], "+", RowBox[List[FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "4"]]]], SuperscriptBox["z", "2"]], " ", RowBox[List["ArcCsch", "[", "z", "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], " ", RowBox[List["Im", "[", "z", "]"]]]], "\[NotEqual]", "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> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </msqrt> <mtext> </mtext> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsin /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccsch /> <ci> z </ci> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <neq /> <apply> <times /> <apply> <real /> <ci> z </ci> </apply> <apply> <imaginary /> <ci> z </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]], "z_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[SuperscriptBox["z", "2"]]]], RowBox[List["2", " ", "z"]]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "4"]]]], " ", RowBox[List["ArcCsch", "[", "z", "]"]]]], SuperscriptBox["z", "2"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], " ", RowBox[List["Im", "[", "z", "]"]]]], "\[NotEqual]", "0"]]]]]]]]


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





2001-10-29