ArcSinh[x] - ArcCosh[y] == -ArcSinh[x y - Sqrt[1 + x^2] Sqrt[y^2 - 1]] - Pi I /; -1 < y < 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> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> - </mo> <mrow> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> y </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <arcsinh /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccosh /> <ci> y </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <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> x </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> y </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> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> </apply> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> y </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29