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ArcCsch






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.29.16.0184.01









  


  










Input Form





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










Standard Form





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MathML Form







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<plus /> <apply> <times /> <ci> y </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </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> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arcsinh /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn 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<apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </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> <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 /> <ci> y </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </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> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> y </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </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> <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 /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <plus /> <ci> y </ci> <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> <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>










Rule Form





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










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





2007-05-02