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http://functions.wolfram.com/01.27.27.1096.01
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ArcTanh[(1 - z)/(2 Sqrt[-z])] == ((2 Sqrt[-z])/Sqrt[z]) ArcCot[1/Sqrt[z]] -
(Pi Sqrt[-1 - z])/(2 Sqrt[1 + z])
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Cell[BoxData[RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], SqrtBox["z"]], " ", RowBox[List["ArcCot", "[", FractionBox["1", SqrtBox["z"]], "]"]]]], "-", FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", "z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]]]]]]]
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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> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <msqrt> <mi> z </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> z </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arctanh /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </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> <plus /> <apply> <times /> <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> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccot /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <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 /> <apply> <plus /> <ci> z </ci> <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> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["1", "-", "z_"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z_"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], ")"]], " ", RowBox[List["ArcCot", "[", FractionBox["1", SqrtBox["z"]], "]"]]]], SqrtBox["z"]], "-", FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", "z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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