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Cot




Mathematica Notation

Traditional Notation









Elementary Functions > Cot[z] > Summation > Finite summation





http://functions.wolfram.com/01.09.23.0013.01









  


  








Input Form





(1/p) Sum[Cot[k (q/p) Pi] Cot[z + k (1/p) Pi], {k, 1, p - 1}] + (1/q) Sum[Cot[k (p/q) Pi] Cot[z + k (1/q) Pi], {k, 1, q - 1}] == (-Cot[p z]) Cot[q z] + (1/(p q)) Csc[z]^2 - 1 /; Element[p, Integers] && p > 0 && Element[q, Integers] && q > 0 && GCD[p, q] == 1








Standard Form





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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> <mrow> <mfrac> <mn> 1 </mn> <mi> p </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> q </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mi> p </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mi> p </mi> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> q </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mi> q </mi> </mfrac> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> q </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> q </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> GCD </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <cot /> <apply> <times /> <ci> k </ci> <ci> q </ci> <pi /> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <cot /> <apply> <plus /> <apply> <times /> <pi /> <ci> k </ci> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <cot /> <apply> <times /> <ci> k </ci> <ci> p </ci> <pi /> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <cot /> <apply> <plus /> <apply> <times /> <pi /> <ci> k </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <csc /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> p </ci> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cot /> <apply> <times /> <ci> p </ci> <ci> z </ci> </apply> </apply> <apply> <cot /> <apply> <times /> <ci> q </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> q </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <eq /> <apply> <gcd /> <ci> p </ci> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], RowBox[List["p_", "-", "1"]]], RowBox[List[RowBox[List["Cot", "[", FractionBox[RowBox[List["k_", " ", "q_", " ", "\[Pi]"]], "p_"], "]"]], " ", RowBox[List["Cot", "[", RowBox[List["z_", "+", FractionBox[RowBox[List["k_", " ", "\[Pi]"]], "p_"]]], "]"]]]]]], "p_"], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], RowBox[List["q_", "-", "1"]]], RowBox[List[RowBox[List["Cot", "[", FractionBox[RowBox[List["k_", " ", "p_", " ", "\[Pi]"]], "q_"], "]"]], " ", RowBox[List["Cot", "[", RowBox[List["z_", "+", FractionBox[RowBox[List["k_", " ", "\[Pi]"]], "q_"]]], "]"]]]]]], "q_"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cot", "[", RowBox[List["p", " ", "z"]], "]"]]]], " ", RowBox[List["Cot", "[", RowBox[List["q", " ", "z"]], "]"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["Csc", "[", "z", "]"]], "2"], RowBox[List["p", " ", "q"]]], "-", "1"]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", ">", "0"]], "&&", RowBox[List["q", "\[Element]", "Integers"]], "&&", RowBox[List["q", ">", "0"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["p", ",", "q"]], "]"]], "\[Equal]", "1"]]]]]]]]]]










References





S. Fukuhara, "New Trigonometric Identities and Generalized Dedekind Sums", Tokyo Journal of Mathematics, v. 26, issue 1, pp. 1-14 (2003)


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





2003-08-21