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http://functions.wolfram.com/01.27.27.3054.01
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ArcTanh[Sqrt[z^2]/Sqrt[1 + z^2]] == ArcCsch[1/z] /;
Re[z] > 0 || (Element[I z, Reals] && -1 < I z < 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], "]"]], "\[Equal]", RowBox[List["ArcCsch", "[", FractionBox["1", "z"], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Element]", "Reals"]], "\[And]", RowBox[List[RowBox[List["-", "1"]], "<", RowBox[List["\[ImaginaryI]", " ", "z"]], "<", "0"]]]], ")"]]]]]]]]
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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> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ∈ </mo> <mi> ℝ </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> < </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arctanh /> <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 /> <apply> <power /> <apply> <plus /> <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> <apply> <arccsch /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <ci> ℝ </ci> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox[SuperscriptBox["z_", "2"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcCsch", "[", FractionBox["1", "z"], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Element]", "Reals"]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", RowBox[List["\[ImaginaryI]", " ", "z"]], "<", "0"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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