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

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Csc






Mathematica Notation

Traditional Notation









Elementary Functions > Csc[z] > Integration > Indefinite integration > Involving functions of the direct function and a power function > Involving powers of the direct function and a power function > Involving powers of csch and power > Involving znand linear arguments





http://functions.wolfram.com/01.10.21.0115.01









  


  










Input Form





Integrate[z Csc[c z]^5, z] == (-(1/(24 c^2))) (9 Csc[c z] + 9 c z Cot[c z] Csc[c z] + 2 Csc[c z]^3 + 6 c z Cot[c z] Csc[c z]^3 - 9 c z Log[1 - E^(I c z)] + 9 c z Log[1 + E^(I c z)] - 9 I PolyLog[2, -E^(I c z)] + 9 I PolyLog[2, E^(I c z)])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List["z", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "5"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["24", " ", SuperscriptBox["c", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["9", " ", RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["9", " ", "c", " ", "z", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "3"]]], "+", RowBox[List["6", " ", "c", " ", "z", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "3"]]], "-", RowBox[List["9", " ", "c", " ", "z", " ", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], "+", RowBox[List["9", " ", "c", " ", "z", " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], "-", RowBox[List["9", " ", "\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], "+", RowBox[List["9", " ", "\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "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> <mo> &#8747; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mn> 5 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> c </ci> <ci> z </ci> <apply> <cot /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <ci> c </ci> <ci> z </ci> <apply> <cot /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <csc /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <ci> c </ci> <ci> z </ci> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <ci> c </ci> <ci> z </ci> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <imaginaryi /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <imaginaryi /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["z_", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c_", " ", "z_"]], "]"]], "5"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["9", " ", RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["9", " ", "c", " ", "z", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "3"]]], "+", RowBox[List["6", " ", "c", " ", "z", " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["c", " ", "z"]], "]"]], "3"]]], "-", RowBox[List["9", " ", "c", " ", "z", " ", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], "+", RowBox[List["9", " ", "c", " ", "z", " ", RowBox[List["Log", "[", RowBox[List["1", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]], "-", RowBox[List["9", " ", "\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]]]]], "]"]]]], "+", RowBox[List["9", " ", "\[ImaginaryI]", " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]]], "]"]]]]]], RowBox[List["24", " ", SuperscriptBox["c", "2"]]]]]]]]]]










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





2002-12-18