Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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] > Series representations > Generalized power series > Expansions at z==0 > For small integer powers of the function > For the second power





http://functions.wolfram.com/01.17.06.0039.01









  


  










Input Form





ArcCsc[z]^2 \[Proportional] (-(1/4)) Log[-(1/z^2)]^2 + Log[-(1/z^2)] (-Log[2] + z^2/4 + (3 z^4)/32 + O[z^6]) + (-Log[2]^2 + (Log[2]/2) z^2 + ((3 Log[2] - 1)/16) z^4 + O[z^6])










Standard Form





Cell[BoxData[RowBox[List[SuperscriptBox[RowBox[List["ArcCsc", "[", "z", "]"]], "2"], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]], "2"]]], "+", " ", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "2", "]"]]]], "+", FractionBox[SuperscriptBox["z", "2"], "4"], "+", FractionBox[RowBox[List["3", SuperscriptBox["z", "4"]]], "32"], "+", RowBox[List["O", "[", SuperscriptBox["z", "6"], "]"]]]], ")"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "2"]]], "+", RowBox[List[FractionBox[RowBox[List["Log", "[", "2", "]"]], RowBox[List["2", " "]]], SuperscriptBox["z", "2"]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["3", " ", RowBox[List["Log", "[", "2", "]"]]]], "-", "1"]], "16"], SuperscriptBox["z", "4"]]], "+", RowBox[List["O", "[", SuperscriptBox["z", "6"], "]"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msup> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8733; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mn> 32 </mn> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 16 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <power /> <apply> <arccsc /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 32 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 16 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcCsc", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]], "2"]]], "+", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "2", "]"]]]], "+", FractionBox[SuperscriptBox["z", "2"], "4"], "+", FractionBox[RowBox[List["3", " ", SuperscriptBox["z", "4"]]], "32"], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "6"]]], ")"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["Log", "[", "2", "]"]], "2"]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", "2", "]"]], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[FractionBox["1", "16"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", RowBox[List["Log", "[", "2", "]"]]]], "-", "1"]], ")"]], " ", SuperscriptBox["z", "4"]]], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "6"]]], ")"]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.