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
 











ArcCsc




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCsc[z] > Representations through equivalent functions > With related functions > Involving tan-1 > Involving csc-1((a2 z2 c+1)1/2/a zc) > Involving csc-1((a2 z2 c+1)1/2/a zc) and tan-1(a zc)





http://functions.wolfram.com/01.17.27.0019.02









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcCsc", "[", FractionBox[SqrtBox[RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "c"]]]]], "+", "1"]]], RowBox[List["a", " ", SuperscriptBox["z", "c"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["z", "c"]]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["z", "c"]]]]]]]], "-", RowBox[List[SqrtBox[FractionBox["1", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["z", "c"]]], "+", "1"]]]], " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["z", "c"]]], "+", "1"]]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["ArcTan", "[", RowBox[List["a", " ", SuperscriptBox["z", "c"]]], "]"]], " ", SqrtBox[FractionBox["1", RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "c"]]]]], "+", "1"]]]], " ", SqrtBox[RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "c"]]]]], "+", "1"]]]]]]]]]]]








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> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> </mrow> </msqrt> </mrow> <mo> - </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsc /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <pi /> <apply> <plus /> <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> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </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> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> a </ci> <imaginaryi /> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </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 /> <ci> a </ci> <imaginaryi /> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </apply> </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 /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arctan /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2001-10-29