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ArcCsc






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCsc[z] > Representations through equivalent functions > With related functions > Involving cosh-1 > Involving csc-1((2z)1/2/(z-(z2-1)1/2)1/2) > Involving csc-1((2z)1/2/(z-(z2-1)1/2)1/2) and cosh-1(1/z)





http://functions.wolfram.com/01.17.27.1452.01









  


  










Input Form





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










Standard Form





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










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> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mtext> </mtext> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> z </mi> <mo> - </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsc /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <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> </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 /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </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'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <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> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arccosh /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <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> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </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["ArcCsc", "[", FractionBox[SqrtBox[RowBox[List["2", " ", "z_"]]], SqrtBox[RowBox[List["z_", "-", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[Pi]", " ", SqrtBox[RowBox[List["-", FractionBox["1", RowBox[List["1", "+", "z"]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", "z"]], SuperscriptBox["z", "2"]]]]], " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List[SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", "z"]], "-", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox["z"], " ", SqrtBox[RowBox[List["-", FractionBox["1", RowBox[List["1", "+", "z"]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", "z"]], SuperscriptBox["z", "2"]]]]]]], ")"]], " ", RowBox[List["ArcCosh", "[", FractionBox["1", "z"], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "z"]]]]]]]]]]]]]










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





2003-08-21