Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











ArcCot






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCot[z] > Transformations > Products, sums, and powers of the direct function > Differences of the direct function





http://functions.wolfram.com/01.16.16.0188.01









  


  










Input Form





ArcCot[x] - ArcCot[y] == ArcCot[(x y + 1)/(y - x)] + (Pi/2) (1 - Sqrt[x^2]/x + Sqrt[y^2]/y + ((y - x)/(1 + x y)) Sqrt[((1 + x y)/(y - x))^2]) + Pi Floor[(Pi - 2 Arg[-1 - I x] - 2 Arg[-1 + I y] + 2 Arg[x - y])/(4 Pi)] - Pi Floor[(1/(4 Pi)) (3 Pi - 2 Arg[-1 + I x] - 2 Arg[-1 - I y] + 2 Arg[x - y])]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCot", "[", "x", "]"]], "-", RowBox[List["ArcCot", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["ArcCot", "[", FractionBox[RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]], RowBox[List["y", "-", "x"]]], "]"]], "+", RowBox[List[FractionBox["\[Pi]", "2"], RowBox[List["(", RowBox[List["1", "-", FractionBox[SqrtBox[SuperscriptBox["x", "2"]], "x"], "+", FractionBox[SqrtBox[SuperscriptBox["y", "2"]], "y"], "+", RowBox[List[FractionBox[RowBox[List["y", "-", "x"]], RowBox[List["1", "+", RowBox[List["x", " ", "y"]]]]], SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", RowBox[List["x", " ", "y"]]]], RowBox[List["y", "-", "x"]]], ")"]], "2"]]]]]], ")"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List["x", "-", "y"]], "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox["1", RowBox[List["4", " ", "\[Pi]"]]], RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Pi]"]], "-", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List["x", "-", "y"]], "]"]]]]]], ")"]], "]"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> y </mi> <mo> - </mo> <mi> x </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> y </mi> <mo> - </mo> <mi> x </mi> </mrow> </mrow> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> y </mi> <mo> - </mo> <mi> x </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </msqrt> </mrow> <mo> + </mo> <mfrac> <msqrt> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </msqrt> <mi> y </mi> </mfrac> <mo> - </mo> <mfrac> <msqrt> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </msqrt> <mi> x </mi> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <arccot /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccot /> <ci> y </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <arccot /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> y </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> y </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> y </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arg /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> x </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arg /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arg /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arg /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arg /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arg /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </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["ArcCot", "[", "x_", "]"]], "-", RowBox[List["ArcCot", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArcCot", "[", FractionBox[RowBox[List[RowBox[List["x", " ", "y"]], "+", "1"]], RowBox[List["y", "-", "x"]]], "]"]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SqrtBox[SuperscriptBox["x", "2"]], "x"], "+", FractionBox[SqrtBox[SuperscriptBox["y", "2"]], "y"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["y", "-", "x"]], ")"]], " ", SqrtBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", RowBox[List["x", " ", "y"]]]], RowBox[List["y", "-", "x"]]], ")"]], "2"]]]], RowBox[List["1", "+", RowBox[List["x", " ", "y"]]]]]]], ")"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", "x"]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List["x", "-", "y"]], "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["3", " ", "\[Pi]"]], "-", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", "x"]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Arg", "[", RowBox[List["x", "-", "y"]], "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "]"]]]]]]]]]]










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





2007-05-02