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variants of this functions
ArcTan




Mathematica Notation

Traditional Notation









Elementary Functions > ArcTan[z] > Representations through equivalent functions > With related functions > Involving csch-1 > Involving tan-1(z1/2/(1-z)1/2) > Involving tan-1(z1/2/(1-z)1/2) and csch-1(1/(-z)1/2)





http://functions.wolfram.com/01.14.27.2409.01









  


  








Input Form





ArcTan[Sqrt[z]/Sqrt[1 - z]] == (-(Sqrt[-z]/Sqrt[z])) ArcCsch[1/Sqrt[-z]] + (Pi/2) (Sqrt[-z + 1] Sqrt[1/(-z + 1)] - 1)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcTan", "[", FractionBox[SqrtBox["z"], SqrtBox[RowBox[List["1", "-", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SqrtBox[RowBox[List["-", "z"]]], SqrtBox["z"]]]], RowBox[List["ArcCsch", "[", FractionBox["1", SqrtBox[RowBox[List["-", "z"]]]], "]"]]]], "+", RowBox[List[FractionBox["\[Pi]", "2"], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "z"]], "+", "1"]]], SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "z"]], "+", "1"]]]]]], "-", "1"]], ")"]]]]]]]]]]








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> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mi> z </mi> </msqrt> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <msqrt> <mi> z </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arctan /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <pi /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccsch /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </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["ArcTan", "[", FractionBox[SqrtBox["z_"], SqrtBox[RowBox[List["1", "-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", "z"]]], " ", RowBox[List["ArcCsch", "[", FractionBox["1", SqrtBox[RowBox[List["-", "z"]]]], "]"]]]], SqrtBox["z"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "z"]], "+", "1"]]], " ", SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "z"]], "+", "1"]]]]]], "-", "1"]], ")"]]]]]]]]]]


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





2003-08-21