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http://functions.wolfram.com/02.03.09.0009.01
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Pi == Limit[(\[Alpha] n)/Sum[KroneckerDelta[Sign[Cos[k \[Alpha]]],
-Sign[Cos[(k + 1) \[Alpha]]]], {k, 0, n}], n -> Infinity] /;
0 <= \[Alpha] <= Pi
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Cell[BoxData[RowBox[List[RowBox[List["\[Pi]", "\[Equal]", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List["\[Alpha]", " ", "n"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["Sign", "[", RowBox[List["Cos", "[", RowBox[List["k", " ", "\[Alpha]"]], "]"]], "]"]], ",", RowBox[List["-", RowBox[List["Sign", "[", RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", "\[Alpha]"]], "]"]], "]"]]]]]], "]"]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List["0", "\[LessEqual]", "\[Alpha]", "\[LessEqual]", "\[Pi]"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> π </mi> <mo> ⩵ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> ≤ </mo> <mi> α </mi> <mo> ≤ </mo> <mi> π </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <pi /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <ci> α </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> KroneckerDelta </ci> <apply> <ci> Sign </ci> <apply> <cos /> <apply> <times /> <ci> k </ci> <ci> α </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <apply> <cos /> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <ci> α </ci> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> α </ci> <pi /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "\[Pi]", "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List["\[Alpha]", " ", "n"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["Sign", "[", RowBox[List["Cos", "[", RowBox[List["k", " ", "\[Alpha]"]], "]"]], "]"]], ",", RowBox[List["-", RowBox[List["Sign", "[", RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", "\[Alpha]"]], "]"]], "]"]]]]]], "]"]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List["0", "\[LessEqual]", "\[Alpha]", "\[LessEqual]", "\[Pi]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
