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ArcSinh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSinh[z] > Transformations > Related transformations > Differences involving the direct function > Involving tanh-1(z)





http://functions.wolfram.com/01.25.16.0203.01









  


  










Input Form





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










Standard Form





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MathML Form







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</cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> </apply> </apply> <apply> <power /> <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> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> 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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> <cn type='integer'> -1 </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> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










Date Added to functions.wolfram.com (modification date)





2007-05-02