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ArcSinh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.25.16.0207.01









  


  










Input Form





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










Standard Form





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MathML Form







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










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





2007-05-02