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ArcSech




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.30.27.0473.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSech", "[", FractionBox["z", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", SqrtBox[FractionBox["z", RowBox[List["z", "+", "1"]]]], SqrtBox[FractionBox[RowBox[List["z", "+", "1"]], "z"]]]], "-", "2", "-", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox["z"]]], SqrtBox[RowBox[List["-", "z"]]]], SqrtBox[FractionBox["\[ImaginaryI]", "z"]]]], "-", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["-", "z"]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]]]], SqrtBox["z"]], " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]]]]]], ")"]], FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], "+", RowBox[List["z", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]], SqrtBox[FractionBox["1", RowBox[List["1", "+", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", 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> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> <mo> - </mo> <mn> 2 </mn> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mtext> </mtext> </mrow> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </msqrt> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <msqrt> <mi> z </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; 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Rule Form





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


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





2003-08-21