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ArcCos




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.13.27.1574.01









  


  








Input Form





ArcCos[z] == (Sqrt[z^2 - 1]/(2 Sqrt[1 - z^2])) ArcTanh[(1 - 2 z^2)/(2 z Sqrt[z^2 - 1])] + (2 - Sqrt[z^2]/z - Sqrt[I z] Sqrt[-(I/z)] + Sqrt[(-I) z] Sqrt[I/z] - Sqrt[z + 1] Sqrt[1/(z + 1)] + Sqrt[-z + 1] Sqrt[1/(-z + 1)]) (Pi/4)








Standard Form





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MathML Form







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</mo> <msqrt> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </msqrt> </mrow> <mo> - </mo> <mrow> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <mo> - </mo> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCos", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "-", FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], "-", RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]]]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]]]], "-", RowBox[List[SqrtBox[RowBox[List["z", "+", "1"]]], " ", SqrtBox[FractionBox["1", RowBox[List["z", "+", "1"]]]]]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "z"]], "+", "1"]]], " ", SqrtBox[FractionBox["1", RowBox[List[RowBox[List["-", "z"]], "+", "1"]]]]]]]], ")"]], " ", "\[Pi]"]]]]]]]]


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





2003-08-21