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ArcCos






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.13.27.0386.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcCos", "[", FractionBox[SqrtBox[RowBox[List["z", "-", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]], SqrtBox[RowBox[List["2", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox["z"], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", RowBox[List["ArcSin", "[", FractionBox["1", "z"], "]"]]]], "+", RowBox[List[FractionBox["\[Pi]", "4"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox[RowBox[List[SqrtBox[FractionBox["1", "z"]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", "z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], "+", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]]]], SqrtBox[FractionBox["z", RowBox[List["1", "+", "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> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </msqrt> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; 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Rule Form





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










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





2003-08-21