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ArcCosh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCosh[z] > Transformations > Transformations and argument simplifications > Argument involving basic arithmetic operations > Involving cosh-1((z-(z2-1)1/2)1/2/(2z)1/2) > Involving cosh-1((z-(z2-1)1/2)1/2/(2z)1/2) and cosh-1(1/z)





http://functions.wolfram.com/01.26.16.0154.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcCosh", "[", FractionBox[SqrtBox[RowBox[List["z", "-", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]]], SqrtBox[RowBox[List["2", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", "z"], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List["-", "z"]]], " ", SqrtBox[SuperscriptBox["z", "2"]]]], "+", RowBox[List[SqrtBox[FractionBox["z", RowBox[List["z", "-", "1"]]]], SqrtBox[FractionBox[RowBox[List["z", "-", "1"]], "z"]]]], "+", RowBox[List["2", SqrtBox[FractionBox["z", RowBox[List["1", "+", "z"]]]], SqrtBox[FractionBox[RowBox[List["1", "+", "z"]], "z"]]]], "-", "3"]], ")"]], FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "4"]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcCosh", "[", 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> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </msqrt> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <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> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccosh /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </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> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </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> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </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'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </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> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -3 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <arccosh /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










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





2003-08-21





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