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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((z+c)1/2/z1/2) > Involving sin-1((z+a)1/2/z1/2) and csch-1(z/a1/2)





http://functions.wolfram.com/01.12.27.0281.01









  


  








Input Form





ArcSin[Sqrt[a + z]/Sqrt[z]] == (Sqrt[a + z]/(Sqrt[a/z + 1] Sqrt[z])) (Pi/2 - Sqrt[-(a/z)] Sqrt[z/a] ArcCsch[Sqrt[z/a]])








Standard Form





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










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["a_", "+", "z_"]]], SqrtBox["z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[RowBox[List["a", "+", "z"]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["a", "z"]]]], " ", SqrtBox[FractionBox["z", "a"]], " ", RowBox[List["ArcCsch", "[", SqrtBox[FractionBox["z", "a"]], "]"]]]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List[FractionBox["a", "z"], "+", "1"]]], " ", SqrtBox["z"]]]]]]]]


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





2001-10-29