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ArcCsch






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.29.27.1438.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcCsch", "[", FractionBox[SqrtBox[RowBox[List["2", "z"]]], SqrtBox[RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]], "-", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " "]], RowBox[List["2", " ", SqrtBox["z"]]]], SqrtBox[FractionBox["1", "z"]], " ", SqrtBox[FractionBox[SuperscriptBox["z", "2"], RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], RowBox[List["ArcSinh", "[", FractionBox["1", "z"], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " "]], RowBox[List["4", " ", SqrtBox["z"]]]], RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["-", "z"]]], "+", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], " ", SqrtBox[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> csch </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> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msqrt> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </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> <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 /> <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='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <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='integer'> -1 </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> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <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 /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 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='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arcsinh /> <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", "[", FractionBox[SqrtBox[RowBox[List["2", " ", "z_"]]], SqrtBox[RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]], "-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]], ")"]], " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox[FractionBox[SuperscriptBox["z", "2"], RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], " ", RowBox[List["ArcSinh", "[", FractionBox["1", "z"], "]"]]]], RowBox[List["2", " ", SqrtBox["z"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["-", "z"]]], "+", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]]], ")"]]]]]], ")"]]]], RowBox[List["4", " ", SqrtBox["z"]]]]]]]]]]










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





2003-08-21