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

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

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> </mrow> <mo> - </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mrow> <mo> - </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mo> + </mo> <mn> 2 </mn> <mo> - </mo> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <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> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccosh /> <ci> z </ci> </apply> <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> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <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> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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> <ci> z </ci> <apply> <power /> <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> <cn type='integer'> -1 </cn> </apply> </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 /> <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> <power /> <ci> z </ci> <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> <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> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <arccot /> <apply> <times /> <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> <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> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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

2003-08-21