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ArcSin






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.12.27.1427.01









  


  










Input Form





ArcSin[Sqrt[(z - Sqrt[z^2 - 1])/(2 z)]] == (-(1/4)) Pi Sqrt[1/z] Sqrt[z] Sqrt[-(1/(1 + z)^2)] Sqrt[1 + z] Sqrt[-((1 + z)/z^2)] Sqrt[I z] Sqrt[-(I/z)] (z + Sqrt[I/z] Sqrt[(-I) z] z - Sqrt[z^2]) + ((Sqrt[-1 + z] Sqrt[1/z^2] z Sqrt[z^2] Sqrt[-(1/(1 + z)^2)] Sqrt[1 + z] Sqrt[-((1 + z)/z^2)])/(2 Sqrt[1 - z])) ArcCosh[1/z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSin", "[", SqrtBox[FractionBox[RowBox[List["z", "-", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]], RowBox[List["2", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[Pi]", " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"], " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", "z"]], SuperscriptBox["z", "2"]]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], RowBox[List["(", RowBox[List["z", "+", RowBox[List[SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", "z"]], "-", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], " ", SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z", " ", SqrtBox[SuperscriptBox["z", "2"]], " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", "z"]], SuperscriptBox["z", "2"]]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]], RowBox[List["ArcCosh", "[", FractionBox["1", "z"], "]"]]]]]]]]]]










MathML Form







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<mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <mtext> </mtext> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> 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<mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsin /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> 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</cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSin", "[", SqrtBox[FractionBox[RowBox[List["z_", "-", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]]]], RowBox[List["2", " ", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[Pi]", " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"], " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", "z"]], SuperscriptBox["z", "2"]]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List[SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", "z"]], "-", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], " ", SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z", " ", SqrtBox[SuperscriptBox["z", "2"]], " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", "z"]], SuperscriptBox["z", "2"]]]]]]], ")"]], " ", RowBox[List["ArcCosh", "[", FractionBox["1", "z"], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]]]










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





2003-08-21