Digits counts: Summation (formula 13.10.23.0001)

DigitCount

$Mathematica Notation$

Number Theory Functions

DigitCount[n,b]

Summation

Finite summation

http://functions.wolfram.com/13.10.23.0001.01

Input Form

Sum[DigitCount[k, 2, 1], {k, 0, n - 1}] == (n/2) Log[2, n] + n F[Log[2, n]] /; (F[n] == F[n + 1] && F[n] == 2^(Floor[Log[2, n]] - 1) (2 f[n/2^Floor[Log[2, n]] - 1] + (n/2^Floor[Log[2, n]]) Log[2, n/2^Floor[Log[2, n]]] - 2 (n/2^Floor[Log[2, n]] - 1)) && (f[x] == Sum[g[2^k x]/2^k, {k, 0, Infinity}] /; g[x] == (1/2) Mod[x, 1] UnitStep[1/2 - Mod[x, 1]] + (1/2) (1 - Mod[x, 1]) UnitStep[Mod[x, 1] - 1/2])) || (F[x] == Sum[Subscript[c, k] E^(2 Pi I k x), {k, 0, Infinity}] /; Subscript[c, 0] == Log[2, Pi]/2 - 1/(2 Log[2]) - 1/4 && Subscript[c, k] == (-(Log[2]/(2 I k Pi Log[2] - 4 k^2 Pi^2))) Zeta[(2 I k Pi)/Log[2]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List["DigitCount", "[", RowBox[List["k", ",", "2", ",", "1"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["n", "2"], " ", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]]]], "+", RowBox[List["n", " ", RowBox[List["F", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["F", "[", "n", "]"]], "\[Equal]", RowBox[List["F", "[", RowBox[List["n", "+", "1"]], "]"]]]], "\[And]", RowBox[List[RowBox[List["F", "[", "n", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["f", "[", RowBox[List[FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]], "-", "1"]], "]"]]]], "+", RowBox[List[FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]], RowBox[List["Log", "[", RowBox[List["2", ",", FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[FractionBox["n", SuperscriptBox["2", RowBox[List["Floor", "[", RowBox[List["Log", "[", RowBox[List["2", ",", "n"]], "]"]], "]"]]]], "-", "1"]], ")"]]]]]], ")"]]]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["f", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", RowBox[List["g", "[", RowBox[List[SuperscriptBox["2", "k"], " ", "x"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["g", "[", "x", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]], " ", RowBox[List["UnitStep", "[", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]], "-", FractionBox["1", "2"]]], "]"]]]]]]]]]], ")"]]]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["F", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["c", "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k", " ", "x"]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "0"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Log", "[", RowBox[List["2", ",", "\[Pi]"]], "]"]], "2"], "-", FractionBox["1", RowBox[List["2", " ", RowBox[List["Log", "[", "2", "]"]]]]], "-", FractionBox["1", "4"]]]]], "\[And]", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Log", "[", "2", "]"]], RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]", " ", RowBox[List["Log", "[", "2", "]"]]]], "-", RowBox[List["4", " ", SuperscriptBox["k", "2"], " ", SuperscriptBox["\[Pi]", "2"]]]]]]]], RowBox[List["Zeta", "[", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]"]], RowBox[List["Log", "[", "2", "]"]]], "]"]]]]]]]]]], ")"]]]]]]]]

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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msubsup> <mi> s </mi> <mn> 2 </mn> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <msup> <mn> 2 </mn> <mrow> <mo> ⌊ </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> </msup> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <msup> <mn> 2 </mn> <mrow> <mo> ⌊ </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> </msup> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> n </mi> <msup> <mn> 2 </mn> <mrow> <mo> ⌊ </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mfrac> <mi> n </mi> <msup> <mn> 2 </mn> <mrow> <mo> ⌊ </mo> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mi> k </mi> </msup> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <semantics> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <semantics> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <semantics> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <semantics> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 0 </mn> </msub> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msub> <mi> log </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> π </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> k </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]"]], RowBox[List["log", "(", "2", ")"]]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> s </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <ci> EllipticF </ci> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> n </ci> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <eq /> <apply> <ci> EllipticF </ci> <ci> n </ci> </apply> <apply> <ci> EllipticF </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> EllipticF </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <floor /> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> f </ci> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> f </ci> <ci> x </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> g </ci> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> g </ci> <ci> x </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <plus /> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> UnitStep </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticF </ci> <ci> x </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> k </ci> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <log /> <logbase> <cn type='integer'> 2 </cn> </logbase> <pi /> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> k </ci> <pi /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> k </ci> <pi /> <apply> <power /> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29