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http://functions.wolfram.com/01.25.27.1323.01
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ArcSinh[(2 z)/(1 - z^2)] ==
2 I^(1 - ((I + z)/(I - z)) Sqrt[((I - z)/(I + z))^2])
ArcTanh[z^(((I + z)/(I - z)) Sqrt[((I - z)/(I + z))^2])]
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Cell[BoxData[RowBox[List[RowBox[List["ArcSinh", "[", FractionBox[RowBox[List["2", " ", "z"]], RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], "]"]], "\[Equal]", RowBox[List["2", SuperscriptBox["\[ImaginaryI]", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", "+", "z"]], RowBox[List["\[ImaginaryI]", "-", "z"]]], SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], RowBox[List["\[ImaginaryI]", "+", "z"]]], ")"]], "2"]]]]]]], RowBox[List["ArcTanh", "[", SuperscriptBox["z", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", "+", "z"]], RowBox[List["\[ImaginaryI]", "-", "z"]]], SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], RowBox[List["\[ImaginaryI]", "+", "z"]]], ")"]], "2"]]]]], "]"]]]]]]]]
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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> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> ⅈ </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> ⅈ </mi> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> <mo> ⁢ </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> - </mo> <mi> z </mi> </mrow> <mrow> <mi> ⅈ </mi> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msup> <mi> z </mi> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> ⅈ </mi> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> <mo> ⁢ </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> - </mo> <mi> z </mi> </mrow> <mrow> <mi> ⅈ </mi> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsinh /> <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='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <imaginaryi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </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> <arctanh /> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <times /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </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> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSinh", "[", FractionBox[RowBox[List["2", " ", "z_"]], RowBox[List["1", "-", SuperscriptBox["z_", "2"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["2", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", "z"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], RowBox[List["\[ImaginaryI]", "+", "z"]]], ")"]], "2"]]]], RowBox[List["\[ImaginaryI]", "-", "z"]]]]]], " ", RowBox[List["ArcTanh", "[", SuperscriptBox["z", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", "z"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], RowBox[List["\[ImaginaryI]", "+", "z"]]], ")"]], "2"]]]], RowBox[List["\[ImaginaryI]", "-", "z"]]]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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