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ArcCot




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCot[z] > Representations through equivalent functions > With related functions > Involving tanh-1 > Involving cot-1((z2)1/2) > Involving cot-1((z2)1/2) and tanh-1(i z)





http://functions.wolfram.com/01.16.27.1978.01









  


  








Input Form





ArcCot[Sqrt[z^2]] == (Pi/2) Sqrt[(1 - I z)/(1 + I z)] Sqrt[(1 + I z)/(1 - I z)] Sqrt[z^2] Sqrt[1/z^2] + ((I z)/Sqrt[z^2]) ArcTanh[I z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcCot", "[", SqrtBox[SuperscriptBox["z", "2"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]], SqrtBox[SuperscriptBox["z", "2"]], SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]]]], "+", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox[SuperscriptBox["z", "2"]]], RowBox[List["ArcTanh", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mfrac> </msqrt> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccot /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctanh /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCot", "[", SqrtBox[SuperscriptBox["z_", "2"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", SqrtBox[FractionBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]], " ", SqrtBox[SuperscriptBox["z", "2"]], " ", SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], " ", RowBox[List["ArcTanh", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], SqrtBox[SuperscriptBox["z", "2"]]]]]]]]]


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





2003-08-21