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ArcSec




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.18.27.0357.01









  


  








Input Form





ArcSec[z^2/(2 Sqrt[z^2 - 1])] == Pi/2 + ((z^3 Sqrt[(1 - z^2)/z^4] Sqrt[-2 + z^2] Sqrt[-1 + z^2] Sqrt[1/z] Sqrt[-((1 + z)/z)])/(2 Sqrt[1 - z] (z + 1) Sqrt[-2 + 3 z^2 - z^4])) (Pi ((Sqrt[(1 - z^2)/z^2] Sqrt[(1 - z^2)/z^4] z^3)/(1 - z^2) + Sqrt[1/z^2] z + Sqrt[1 - Sqrt[2]/z] Sqrt[-(1/z)] Sqrt[-z] Sqrt[z/(-Sqrt[2] + z)] - Sqrt[1/z] Sqrt[z] Sqrt[z/(Sqrt[2] + z)] Sqrt[(Sqrt[2] + z)/z] - 2) + 4 ArcCos[1/z])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSec", "[", FractionBox[SuperscriptBox["z", "2"], RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "3"], SqrtBox[FractionBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "4"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], SqrtBox[FractionBox["1", "z"]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", "z"]], "z"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", SuperscriptBox["z", "2"]]], "-", SuperscriptBox["z", "4"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "2"]]], SqrtBox[FractionBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "4"]]], " ", SuperscriptBox["z", "3"]]], RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]]], "+", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox[SqrtBox["2"], "z"]]]], " ", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["-", "z"]]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List["-", SqrtBox["2"]]], "+", "z"]]]]]], "-", RowBox[List[SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"], " ", SqrtBox[FractionBox["z", RowBox[List[SqrtBox["2"], "+", "z"]]]], " ", SqrtBox[FractionBox[RowBox[List[SqrtBox["2"], "+", "z"]], "z"]]]], "-", "2"]], ")"]]]], "+", RowBox[List["4", RowBox[List["ArcCos", "[", FractionBox["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> sec </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mtext> </mtext> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 2 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mtext> </mtext> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mfrac> </msqrt> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <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> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msqrt> <mn> 2 </mn> </msqrt> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mfrac> </msqrt> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsec /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <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> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <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> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <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> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <arccos /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSec", "[", FractionBox[SuperscriptBox["z_", "2"], RowBox[List["2", " ", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["\[Pi]", "2"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "3"], " ", SqrtBox[FractionBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "4"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", "z"]], "z"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "2"]]], " ", SqrtBox[FractionBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "4"]]], " ", SuperscriptBox["z", "3"]]], RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], "+", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z"]], "+", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox[SqrtBox["2"], "z"]]]], " ", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["-", "z"]]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List["-", SqrtBox["2"]]], "+", "z"]]]]]], "-", RowBox[List[SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"], " ", SqrtBox[FractionBox["z", RowBox[List[SqrtBox["2"], "+", "z"]]]], " ", SqrtBox[FractionBox[RowBox[List[SqrtBox["2"], "+", "z"]], "z"]]]], "-", "2"]], ")"]]]], "+", RowBox[List["4", " ", RowBox[List["ArcCos", "[", FractionBox["1", "z"], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", SuperscriptBox["z", "2"]]], "-", SuperscriptBox["z", "4"]]]]]]]]]]]]]


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





2003-08-21