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ArcSin






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSin[z] > Representations through equivalent functions > With related functions > Involving cot-1 > Involving sin-1(1/(a zc+1)1/2) > Involving sin-1(1/(a zc+1)1/2) and cot-1(a1/2 zc/2)





http://functions.wolfram.com/01.12.27.0083.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSin", "[", FractionBox["1", SqrtBox[RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "c"]]], "+", "1"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]], " "]], SqrtBox[RowBox[List["a", " ", SuperscriptBox["z", "c"]]]]], RowBox[List["ArcCot", "[", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " "]]]], "2"], RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "+", "1"]]]], "+", FractionBox[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]]]]]]]], ")"]]]]]]]]]]










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> <mn> 1 </mn> <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> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> c </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <msqrt> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> c </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mi> c </mi> </msup> </msqrt> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mi> c </mi> </msup> </msqrt> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> + </mo> <mfrac> <msqrt> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mi> c </mi> </msup> </msqrt> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> a </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mi> c </mi> </msup> </msqrt> </mrow> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsin /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <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> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccot /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> c </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </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["1", SqrtBox[RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "c_"]]], "+", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], ")"]], " ", RowBox[List["ArcCot", "[", RowBox[List[SqrtBox["a"], " ", SuperscriptBox["z", RowBox[List["c", "/", "2"]]]]], "]"]]]], SqrtBox[RowBox[List["a", " ", SuperscriptBox["z", "c"]]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "+", "1"]]]], "+", FractionBox[SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]], "-", "1"]]], SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["a"], " ", SqrtBox[SuperscriptBox["z", "c"]]]]]]]]]], ")"]]]]]]]]]]










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





2001-10-29