ArcCosh[z] == (Sqrt[z - 1]/Sqrt[1 - z]) (Pi/2 - (z/(2 Sqrt[Pi])) (1/(2 Pi I)) ContourIntegrate[ (Gamma[s] Gamma[1/2 - s]^2)/Gamma[3/2 - s]/(-z^2)^s, {s, \[ScriptCapitalL]}]) /; Abs[Arg[-z^2]] < Pi

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCosh", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["z", "-", "1"]]], " "]], SqrtBox[RowBox[List["1", "-", "z"]]]], RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List[FractionBox["z", RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], FractionBox["1", RowBox[List[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], " "]]], RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], "2"]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "s"]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "s"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]]]], ")"]]]]]], "/;", " ", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["-", SuperscriptBox["z", "2"]]], "]"]], "]"]], "<", "\[Pi]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <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> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mtext> </mtext> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mfrac> <mi> z </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mo> &#8747; </mo> <mi> &#8466; </mi> </msub> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mtext> </mtext> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> s </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mi> &#960; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> cosh </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> z </ms> </list> </apply> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> &#960; </ms> <ms> 2 </ms> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> z </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> SqrtBox </ci> <ms> &#960; </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> <ms> &#960; </ms> <ms> &#8520; </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8747; </ms> </apply> <ms> &#8466; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <ms> s </ms> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> s </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8518; </ms> <ms> s </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#62979; </ms> <apply> <ci> RowBox </ci> <list> <ms> arg </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#62980; </ms> </list> </apply> <ms> &lt; </ms> <ms> &#960; </ms> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCosh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["z", "-", "1"]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", FractionBox[RowBox[List["z", " ", RowBox[List["ContourIntegrate", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "s"]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "s"]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox["\[Pi]"]]], ")"]], " ", RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]]]], ")"]]]], SqrtBox[RowBox[List["1", "-", "z"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["-", SuperscriptBox["z", "2"]]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]

2001-10-29