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ArcSec




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.18.27.0488.01









  


  








Input Form





ArcSec[(z^2 + 1)/(z^2 - 1)] == (-((2 Sqrt[z^2])/z)) Sqrt[(I - z)/(I + z)] Sqrt[(I + z)/(I - z)] ArcTan[z] + Pi (-1 + Sqrt[1/(1 - I z)] Sqrt[1 - I z] + Sqrt[1/(1 + I z)] Sqrt[1 + I z])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSec", "[", FractionBox[RowBox[List[SuperscriptBox["z", "2"], "+", "1"]], RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", " ", SqrtBox[SuperscriptBox["z", "2"]], " "]], "z"]]], SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], RowBox[List["\[ImaginaryI]", "+", "z"]]]], " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", "+", "z"]], RowBox[List["\[ImaginaryI]", "-", "z"]]]], RowBox[List["ArcTan", "[", "z", "]"]]]], "+", RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]], "+", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "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> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msqrt> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mrow> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> &#8520; </mi> <mo> - </mo> <mi> z </mi> </mrow> <mrow> <mi> &#8520; </mi> <mo> + </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> &#8520; </mi> <mo> + </mo> <mi> z </mi> </mrow> <mrow> <mi> &#8520; </mi> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsec /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </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='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <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> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </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 /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <imaginaryi /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <imaginaryi /> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <imaginaryi /> <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> <arctan /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2003-08-21