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ArcSinh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.25.16.0171.01









  


  










Input Form





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










Standard Form





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MathML Form







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type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <apply> <times /> <imaginaryi /> <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 /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn 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</cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </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["ArcCoth", "[", "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"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]]]], "y"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List[FractionBox["x", "y"], "+", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]]]], ")"]]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[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"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]]]], "y"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]], "-", "x"]], "]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", FractionBox["\[ImaginaryI]", "y"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", 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"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]]]], "y"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "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"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]]]], "y"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["x", "2"], "+", "1"]]], "-", "x"]], "]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", FractionBox["\[ImaginaryI]", "y"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]], ")"]]]]]]]]]]










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





2007-05-02