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http://functions.wolfram.com/01.28.27.3278.01
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ArcCoth[Sqrt[z]] ==
(Pi/8) (2 I (-1 + Sqrt[(-1 + z)/z] Sqrt[z/(-1 + z)] +
(3 I z)/(2 Sqrt[-z^2])) (1 - ((z + 1)/(z - 1))
Sqrt[((z - 1)/(z + 1))^2]) + (z/Sqrt[-z^2])
(1 + ((z + 1)/(z - 1)) Sqrt[((z - 1)/(z + 1))^2])) +
(1/4) (1 - ((z + 1)/(z - 1)) Sqrt[((z - 1)/(z + 1))^2] +
Sqrt[1/(1 + z)] Sqrt[1 + z] (1 + ((z + 1)/(z - 1))
Sqrt[((z - 1)/(z + 1))^2])) ArcSech[(z - 1)/(2 Sqrt[-z])] /;
Abs[z] != 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCoth", "[", SqrtBox["z"], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "z"]]]]]], "+", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "z"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]], "+", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "+", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List["z", "+", "1"]], RowBox[List["z", "-", "1"]]], SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcSech", "[", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[NotEqual]", "1"]]]]]]
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<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> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mi> π </mi> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <mi> z </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> z </mi> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> z </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ≠ </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccoth /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <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> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <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> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <arcsech /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <neq /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCoth", "[", SqrtBox["z_"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "8"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "z"]]]]]], "+", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "z"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]], RowBox[List["z", "-", "1"]]]]], ")"]]]], "+", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]], RowBox[List["z", "-", "1"]]]]], ")"]]]], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]], RowBox[List["z", "-", "1"]]], "+", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "+", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["z", "+", "1"]]], ")"]], "2"]]]], RowBox[List["z", "-", "1"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["ArcSech", "[", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[NotEqual]", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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