ArcSinh[x] + I ArcSin[y] == ((I (I x y + Sqrt[1 + x^2] Sqrt[1 - y^2]))/ Sqrt[(I x y + Sqrt[1 + x^2] Sqrt[1 - y^2])^2]) ArcSin[Sqrt[1 + x^2] y - I x Sqrt[1 - y^2]] + (1/2) Pi I (1 - (I x y + Sqrt[1 + x^2] Sqrt[1 - y^2])/ Sqrt[(I x y + Sqrt[1 + x^2] Sqrt[1 - y^2])^2]) - Pi I (1 + (I x y + Sqrt[1 + x^2] Sqrt[1 - y^2])/ Sqrt[(I x y + Sqrt[1 + x^2] Sqrt[1 - y^2])^2]) Floor[(Pi - Arg[x + Sqrt[1 + x^2]] - Arg[I y + Sqrt[1 - y^2]])/(2 Pi)] + Pi I (1 - (I x y + Sqrt[1 + x^2] Sqrt[1 - y^2])/ Sqrt[(I x y + Sqrt[1 + x^2] Sqrt[1 - y^2])^2]) Floor[(Arg[x + Sqrt[1 + x^2]] + Arg[I y + Sqrt[1 - y^2]])/(2 Pi)]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSinh", "[", "x", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", "y", "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"]]], " ", RowBox[List["ArcSin", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"]]]]], ")"]]]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", RowBox[List["x", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]], "]"]], "-", RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List["x", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]

<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> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <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> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <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> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <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> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <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> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <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> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <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> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <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> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <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> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mi> &#960; </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <arcsinh /> <ci> x </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <arcsin /> <ci> y </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <ci> y </ci> </apply> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <ci> y </ci> </apply> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arcsin /> <apply> <plus /> <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> <ci> y </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> x </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <arg /> <apply> <plus /> <ci> x </ci> <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> <arg /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <ci> y </ci> </apply> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <ci> y </ci> </apply> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <pi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <ci> y </ci> </apply> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <ci> y </ci> </apply> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <ci> y </ci> </apply> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <ci> y </ci> </apply> <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 /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <plus /> <ci> x </ci> <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> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcSinh", "[", "x_", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", "y_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]]]], ")"]], " ", RowBox[List["ArcSin", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", "y"]], "-", RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], "]"]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"]]]]], ")"]]]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", RowBox[List["x", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]], "]"]], "-", RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x", " ", "y"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]]]], ")"]], "2"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List["x", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]]]]]

2007-05-02