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ArcCsch






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.29.27.1656.01









  


  










Input Form





ArcCsch[Sqrt[(2 z)/(Sqrt[z^2 - 1] - z)]] == (Sqrt[(-I) z]/2) Sqrt[I/z] ArcCosh[1/z] + (Pi/4) (3 I Sqrt[1/z] Sqrt[z] + (2 z)/Sqrt[-z^2] + Sqrt[-z^4]/z^2 - I (1 + Sqrt[I/z] Sqrt[(-I) z] + Sqrt[(-1 + z)/z] Sqrt[z/(-1 + z)]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcCsch", "[", SqrtBox[FractionBox[RowBox[List["2", "z"]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], "-", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], "2"], " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], RowBox[List["ArcCosh", "[", FractionBox["1", "z"], "]"]]]], "+", RowBox[List[FractionBox["\[Pi]", "4"], RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"]]], "+", FractionBox[RowBox[List["2", " ", "z"]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]], "+", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "4"]]]], SuperscriptBox["z", "2"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]], "+", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List["-", "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> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> + </mo> <mfrac> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> <mo> + </mo> <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> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsch /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <apply> <plus /> <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> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <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> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <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> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <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 /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </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> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsch", "[", SqrtBox[FractionBox[RowBox[List["2", " ", "z_"]], RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]], "-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", RowBox[List["ArcCosh", "[", FractionBox["1", "z"], "]"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"]]], "+", FractionBox[RowBox[List["2", " ", "z"]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]], "+", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "4"]]]], SuperscriptBox["z", "2"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]], "+", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "z"]]]]]]]], ")"]]]]]], ")"]]]]]]]]]]










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





2003-08-21