Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











ArcCos






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCos[z] > Representations through equivalent functions > With related functions > Involving cosh-1 > Involving cos-1(a (b zc)m) > Involving cos-1(a (b zc)m) and cosh-1(a bm zm c)





http://functions.wolfram.com/01.13.27.1361.01









  


  










Input Form





ArcCos[a (b z^c)^m] == Pi/2 - ((b z^c)^m/(b^m z^(m c))) (Pi/2 - (Sqrt[1 - a b^m z^(m c)]/Sqrt[a b^m z^(m c) - 1]) ArcCosh[a b^m z^(m c)]) /; Element[2 m, Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCos", "[", RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "m"]]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "m"], RowBox[List[SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]]], RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List[FractionBox[SqrtBox[RowBox[List["1", "-", RowBox[List["a", " ", SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]]]]], SqrtBox[RowBox[List[RowBox[List["a", " ", SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]], "-", "1"]]]], " ", RowBox[List["ArcCosh", "[", RowBox[List["a", " ", SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]], "]"]]]]]], ")"]]]]]]]], "/;", RowBox[List["Element", "[", RowBox[List[RowBox[List["2", " ", "m"]], ",", "Integers"]], "]"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mrow> <msup> <mi> b </mi> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> b </mi> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> </mrow> </msqrt> <mtext> </mtext> </mrow> <msqrt> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> b </mi> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> b </mi> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccos /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> b </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> b </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> b </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccosh /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> b </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCos", "[", RowBox[List["a_", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b_", " ", SuperscriptBox["z_", "c_"]]], ")"]], "m_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "m"], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["a", " ", SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]]]]], " ", RowBox[List["ArcCosh", "[", RowBox[List["a", " ", SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]], "]"]]]], SqrtBox[RowBox[List[RowBox[List["a", " ", SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]], "-", "1"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]]]]], "/;", RowBox[List[RowBox[List["2", " ", "m"]], "\[Element]", "Integers"]]]]]]]]










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





2003-08-21