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ArcCosh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.26.27.1532.01









  


  










Input Form





ArcCosh[z Sqrt[(1 - Sqrt[1 + z^2])/(2 z^2)]] == (1 - I z Sqrt[-z^(-2)] - Sqrt[(z + I)/z] Sqrt[z/(z + I)]) ((Pi I)/2) + ((Sqrt[-1 - I z] Sqrt[I z] Sqrt[z] Sqrt[-1 - z^2])/ (2 Sqrt[-z] Sqrt[z (-I + z)] Sqrt[1 + z^2])) ArcSinh[z]










Standard Form





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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> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </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> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </msqrt> </mrow> <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> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <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> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> </mfrac> </msqrt> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; 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Rule Form





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










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





2003-08-21