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http://functions.wolfram.com/01.26.16.0019.01
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ArcCosh[x] - ArcSinh[y] == ArcSinh[x y - Sqrt[1 + y^2] Sqrt[x^2 - 1]] +
Pi I /; -1 < x < 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["ArcCosh", "[", "x", "]"]], "-", RowBox[List["ArcSinh", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["ArcSinh", "[", RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]], " ", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "-", "1"]]]]]]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]]]]], "/;", RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "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> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> - </mo> <mrow> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> < </mo> <mi> x </mi> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arccosh /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arcsinh /> <ci> y </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <arcsinh /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <imaginaryi /> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcCosh", "[", "x_", "]"]], "-", RowBox[List["ArcSinh", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]], " ", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "-", "1"]]]]]]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]]]], "/;", RowBox[List[RowBox[List["-", "1"]], "<", "x", "<", "0"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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