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ArcSin




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSin[z] > Representations through equivalent functions > With related functions > Involving coth-1 > Involving sin-1(a zc/(a2 z2 c-1)1/2) > Involving sin-1(a zc/(a2 z2 c-1)1/2) and coth-1(1/a zc)





http://functions.wolfram.com/01.12.27.0268.02









  


  








Input Form





ArcSin[(a z^c)/Sqrt[a^2 z^(2 c) - 1]] == (Pi/2) (Sqrt[1/(a z^c + 1)] Sqrt[a z^c + 1] - Sqrt[1/(1 - a z^c)] Sqrt[1 - a z^c]) - Sqrt[1/(1 - a^2 z^(2 c))] Sqrt[a^2 z^(2 c) - 1] ArcCoth[1/(a z^c)]








Standard Form





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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> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <msqrt> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mo> - </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsin /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arccoth /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </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[RowBox[List["a_", " ", SuperscriptBox["z_", "c_"]]], SqrtBox[RowBox[List[RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "c_"]]]]], "-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "c"]]], "+", "1"]]]], " ", SqrtBox[RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "c"]]], "+", "1"]]]]], "-", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List["a", " ", SuperscriptBox["z", "c"]]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["a", " ", SuperscriptBox["z", "c"]]]]]]]]]], ")"]]]], "-", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "c"]]]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "c"]]]]], "-", "1"]]], " ", RowBox[List["ArcCoth", "[", FractionBox["1", RowBox[List["a", " ", SuperscriptBox["z", "c"]]]], "]"]]]]]]]]]]


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





2001-10-29