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ArcCsc






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCsc[z] > Transformations > Transformations and argument simplifications > Argument involving basic arithmetic operations > Involving csc-1(1/2 z (1-z2)1/2) > Involving csc-1(1/2 z (1-z2)1/2) and csc-1(1/z)





http://functions.wolfram.com/01.17.16.0096.01









  


  










Input Form





ArcCsc[1/(2 z Sqrt[1 - z^2])] == (-((2 Sqrt[1 - 2 z^2] Sqrt[-z^2 + z^4])/(Sqrt[-z^2] Sqrt[-1 + z^2] Sqrt[-1 + 2 z^2]))) ArcCsc[1/z] + ((Pi Sqrt[1 - 2 z^2] Sqrt[-z^2 + z^4])/(2 Sqrt[-z^2] Sqrt[-1 + z^2] Sqrt[-1 + 2 z^2])) (Sqrt[-(1/z)] Sqrt[-z] Sqrt[1/(1 - Sqrt[2] z)] Sqrt[1 - Sqrt[2] z] - Sqrt[1/z] Sqrt[z] Sqrt[1/(1 + Sqrt[2] z)] Sqrt[1 + Sqrt[2] z] + Sqrt[z^2]/z + Sqrt[-z^2 + z^4]/(z Sqrt[-1 + z^2]))










Standard Form





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










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> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; 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</mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mtext> </mtext> </mrow> <mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsc /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <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> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <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> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </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 /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <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 /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <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 /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </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 /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <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> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <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> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccsc /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <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["ArcCsc", "[", FractionBox["1", RowBox[List["2", " ", "z_", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z_", "2"]]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"]]]]]], ")"]], " ", RowBox[List["ArcCsc", "[", FractionBox["1", "z"], "]"]]]], RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["-", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List[SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SqrtBox["2"], " ", "z"]]]]]]], "-", RowBox[List[SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", "z"]]]]]]], "+", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "+", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"]]]], RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]]]]]]]]










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





2003-08-21





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