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variants of this functions
ArcTan






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.14.16.0243.01









  


  










Input Form





ArcTan[x] - ArcCosh[y] == Pi Floor[(-Arg[1 - I x] + Arg[I x + 1] + Pi)/(2 Pi)] - 2 I Pi (Floor[(-Arg[((1 - I x)/(I x + 1))^(I/2)] - Arg[1/(y + Sqrt[y - 1] Sqrt[y + 1])] + Pi)/(2 Pi)] + Floor[(Im[Log[y + Sqrt[y - 1] Sqrt[y + 1]]] + Pi)/(2 Pi)] + Floor[(Pi - (1/2) Re[Log[(1 - I x)/(I x + 1)]])/(2 Pi)]) + Log[((1 - I x)/(I x + 1))^(I/2)/(y + Sqrt[y - 1] Sqrt[y + 1])]










Standard Form





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MathML Form







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










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcTan", "[", "x_", "]"]], "-", RowBox[List["ArcCosh", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], "]"]]]], "+", RowBox[List["Arg", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], "]"]]]], "-", RowBox[List["Arg", "[", FractionBox["1", RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]], "]"]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Re", "[", RowBox[List["Log", "[", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], "]"]], "]"]]]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List["Log", "[", FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "1"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], RowBox[List["y", "+", RowBox[List[SqrtBox[RowBox[List["y", "-", "1"]]], " ", SqrtBox[RowBox[List["y", "+", "1"]]]]]]]], "]"]]]]]]]]










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





2007-05-02