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ArcSec






Mathematica Notation

Traditional Notation









Elementary Functions > ArcSec[z] > Representations through equivalent functions > With related functions > Involving cosh-1 > Involving sec-1(z) > Involving sec-1(z) and cosh-1(2 (z2-1)1/2/z2)





http://functions.wolfram.com/01.18.27.1228.01









  


  










Input Form





ArcSec[z] == Pi/2 + ((z Sqrt[-2 + z^2])/(2 Sqrt[z^2 (2 - z^2)])) ArcCosh[(2 Sqrt[z^2 - 1])/z^2] - (Pi/4) Sqrt[(z^2 - 2)/z^2] Sqrt[z^2/(-2 + z^2)] (Sqrt[z^2]/z + Sqrt[(z - Sqrt[2])/z] Sqrt[z/(-Sqrt[2] + z)] - Sqrt[z/(Sqrt[2] + z)] Sqrt[(Sqrt[2] + z)/z])










Standard Form





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





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










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





2003-08-21