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ArcSin




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSin[z] > Representations through equivalent functions > With related functions > Involving tanh-1 > Involving sin-1(z/z-a1/2) > Involving sin-1(z/z-a1/2) and tanh-1(1/z1/2)





http://functions.wolfram.com/01.12.27.1620.01









  


  








Input Form





ArcSin[Sqrt[z]/Sqrt[z - a]] == (Sqrt[z - a]/(2 Sqrt[z])) Sqrt[z/(z - a)] ((1/z) Sqrt[1 - z/a] Sqrt[a/(a - z)] (2 a Sqrt[-(z^2/a^2)] ArcTanh[Sqrt[z/a]] - Pi z) + Pi) /; a > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSin", "[", FractionBox[SqrtBox["z"], SqrtBox[RowBox[List["z", "-", "a"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[" ", SqrtBox[RowBox[List["z", "-", "a"]]]]], RowBox[List["2", " ", SqrtBox["z"]]]], SqrtBox[FractionBox["z", RowBox[List["z", "-", "a"]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "z"], SqrtBox[RowBox[List["1", "-", FractionBox["z", "a"]]]], " ", SqrtBox[FractionBox["a", RowBox[List["a", "-", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], SuperscriptBox["a", "2"]]]]], RowBox[List["ArcTanh", "[", SqrtBox[FractionBox["z", "a"]], "]"]]]], "-", RowBox[List["\[Pi]", " ", "z"]]]], ")"]]]], "+", "\[Pi]"]], ")"]]]]]], "/;", RowBox[List["a", ">", "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> <mi> z </mi> </msqrt> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mi> a </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mi> a </mi> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> - </mo> <mi> a </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> z </mi> <mi> a </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> a </mi> <mrow> <mi> a </mi> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mi> z </mi> <mi> a </mi> </mfrac> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> a </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsin /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arctanh /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <pi /> </apply> </apply> </apply> <apply> <gt /> <ci> a </ci> <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["z_"], SqrtBox[RowBox[List["z_", "-", "a_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["z", "-", "a"]]], " ", SqrtBox[FractionBox["z", RowBox[List["z", "-", "a"]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["z", "a"]]]], " ", SqrtBox[FractionBox["a", RowBox[List["a", "-", "z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], SuperscriptBox["a", "2"]]]]], " ", RowBox[List["ArcTanh", "[", SqrtBox[FractionBox["z", "a"]], "]"]]]], "-", RowBox[List["\[Pi]", " ", "z"]]]], ")"]]]], "z"], "+", "\[Pi]"]], ")"]]]], RowBox[List["2", " ", SqrtBox["z"]]]], "/;", RowBox[List["a", ">", "0"]]]]]]]]


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





2003-08-21