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Csc






Mathematica Notation

Traditional Notation









Elementary Functions > Csc[z] > Series representations > Generalized power series > Expansions at z==z0 > For the function itself





http://functions.wolfram.com/01.10.06.0026.01









  


  










Input Form





Csc[z] == Csc[Subscript[z, 0]] Sum[(KroneckerDelta[k] + (k + 1) Sum[(((-1)^(k + j) 2^(1 - m) (m - 2 j)^k Csc[Subscript[z, 0]]^m)/ ((1 + m) j! (m - j)! (k - m)!)) Cos[((m - k) Pi)/2 + (m - 2 j) Subscript[z, 0]], {m, 0, k}, {j, 0, Floor[(m - 1)/2]}]) (z - Subscript[z, 0])^k, {k, 0, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Csc", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["Csc", "[", SubscriptBox["z", "0"], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", "k", "]"]], "+", " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["m", "-", "1"]], "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "j"]]], " ", SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", "j"]]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["Csc", "[", SubscriptBox["z", "0"], "]"]], "m"]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "j"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]]]]], RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]], " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", "j"]]]], ")"]], " ", SubscriptBox["z", "0"]]]]], "]"]]]]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mi> k </mi> </msub> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mi> m </mi> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mtext> </mtext> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> j </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <csc /> <ci> z </ci> </apply> <apply> <times /> <apply> <csc /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <apply> <ci> KroneckerDelta </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <csc /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Csc", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Csc", "[", SubscriptBox["zz", "0"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", "k", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["m", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "j"]]], " ", SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "j"]]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["Csc", "[", SubscriptBox["zz", "0"], "]"]], "m"]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]], " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "j"]]]], ")"]], " ", SubscriptBox["zz", "0"]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]]]]]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]]]]]]]]]]










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





2007-05-02





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