Riemann-Siegel function Z: Series representations (formula 10.04.06.0010)

RiemannSiegelZ

$Mathematica Notation$

Zeta Functions and Polylogarithms

RiemannSiegelZ[z]

Series representations

Asymptotic series expansions

http://functions.wolfram.com/10.04.06.0010.01

Input Form

RiemannSiegelZ[x] \[Proportional] 2 Sum[Cos[RiemannSiegelTheta[x] - x Log[k]]/Sqrt[k], {k, 1, \[Nu]}] + (-1)^(\[Nu] - 1) (2 Pi)^(1/4) (\[CapitalOmega][p]/x^4^(-1) - (1/(x^(3/4) (48 Sqrt[2] Pi^(3/2)))) Derivative[3][\[CapitalOmega]][p] + (2 Pi (Derivative[2][\[CapitalOmega]][p]/(64 Pi^2) + Derivative[6][\[CapitalOmega]][p]/(18432 Pi^4)))/x^(5/4) - ((2 Pi)^(3/2) (Derivative[1][\[CapitalOmega]][p]/(64 Pi^2) + Derivative[5][\[CapitalOmega]][p]/(3840 Pi^4) + Derivative[9][\[CapitalOmega]][p]/(5308416 Pi^6)))/x^(7/4)) /; \[Nu] == Floor[Sqrt[x/(2 Pi)]] && p == Sqrt[x/(2 Pi)] - \[Nu] && \[CapitalOmega][p] == Cos[2 Pi (p^2 - p - 1/16)]/Cos[2 Pi p] && Element[x, Reals] && (x -> Infinity)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RiemannSiegelZ", "[", "x", "]"]], "\[Proportional]", RowBox[List[RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Nu]"], FractionBox[RowBox[List["Cos", "[", RowBox[List[RowBox[List["RiemannSiegelTheta", "[", "x", "]"]], "-", RowBox[List["x", " ", RowBox[List["Log", "[", "k", "]"]]]]]], "]"]], SqrtBox["k"]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["x", RowBox[List[RowBox[List["-", "1"]], "/", "4"]]], RowBox[List["\[CapitalOmega]", "[", "p", "]"]]]], "-", RowBox[List[FractionBox[SuperscriptBox["x", RowBox[List[RowBox[List["-", "3"]], "/", "4"]]], RowBox[List["48", " ", SqrtBox["2"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]]]], "+", RowBox[List["2", "\[Pi]", " ", SuperscriptBox["x", RowBox[List[RowBox[List["-", "5"]], "/", "4"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["64", " ", SuperscriptBox["\[Pi]", "2"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "6", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["18432", " ", SuperscriptBox["\[Pi]", "4"]]]]]], ")"]]]], "-", " ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", "\[Pi]"]], ")"]], RowBox[List["3", "/", "2"]]], SuperscriptBox["x", RowBox[List[RowBox[List["-", "7"]], "/", "4"]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", "\[Prime]", Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["64", " ", SuperscriptBox["\[Pi]", "2"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "5", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["3840", " ", SuperscriptBox["\[Pi]", "4"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "9", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["5308416", " ", SuperscriptBox["\[Pi]", "6"]]]]]], ")"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "\[Equal]", RowBox[List["Floor", "[", SqrtBox[FractionBox["x", RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]], "\[And]", RowBox[List["p", "\[Equal]", RowBox[List[SqrtBox[FractionBox["x", RowBox[List["2", " ", "\[Pi]"]]]], "-", "\[Nu]"]]]], "\[And]", RowBox[List[RowBox[List["\[CapitalOmega]", "[", "p", "]"]], "\[Equal]", FractionBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["p", "2"], "-", "p", "-", RowBox[List["1", "/", "16"]]]], ")"]]]], "]"]], RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "p"]], "]"]]]]], " ", "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["(", RowBox[List["x", "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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> <semantics> <mi> Z </mi> <annotation encoding='Mathematica'> TagBox["Z", RiemannSiegelZ] </annotation> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ν </mi> </munderover> <mfrac> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ϑ </mi> <annotation encoding='Mathematica'> TagBox["\[CurlyTheta]", RiemannSiegelTheta] </annotation> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <msqrt> <mi> k </mi> </msqrt> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mroot> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> Ω </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mroot> <mi> x </mi> <mn> 4 </mn> </mroot> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msup> <mi> Ω </mi> <semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "3", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 48 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> x </mi> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> Ω </mi> <mi> ′′ </mi> </msup> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mrow> <mn> 64 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mi> Ω </mi> <semantics> <mrow> <mo> ( </mo> <mn> 6 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "6", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mrow> <mn> 18432 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> x </mi> <mrow> <mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mtext> </mtext> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> Ω </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mrow> <mn> 64 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mi> Ω </mi> <semantics> <mrow> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "5", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mrow> <mn> 3840 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mi> Ω </mi> <semantics> <mrow> <mo> ( </mo> <mn> 9 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "9", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mrow> <mn> 5308416 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 6 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> x </mi> <mrow> <mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> ν </mi> <mo> ⩵ </mo> <mrow> <mo> ⌊ </mo> <msqrt> <mfrac> <mi> x </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </msqrt> <mo> ⌋ </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> ⩵ </mo> <mrow> <msqrt> <mfrac> <mi> x </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </msqrt> <mo> - </mo> <mi> ν </mi> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Ω </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> p </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 16 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> RiemannSiegelZ </ci> <ci> x </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> ν </ci> </uplimit> <apply> <times /> <apply> <cos /> <apply> <plus /> <apply> <ci> RiemannSiegelTheta </ci> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <ln /> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Ω </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> p </ci> <degree> <cn type='integer'> 3 </cn> </degree> </bvar> <apply> <ci> Ω </ci> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 48 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> x </ci> <cn type='rational'> -3 <sep /> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <plus /> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> p </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> Ω </ci> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> p </ci> <degree> <cn type='integer'> 6 </cn> </degree> </bvar> <apply> <ci> Ω </ci> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 18432 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> x </ci> <cn type='rational'> -5 <sep /> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> p </ci> </bvar> <apply> <ci> Ω </ci> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> p </ci> <degree> <cn type='integer'> 5 </cn> </degree> </bvar> <apply> <ci> Ω </ci> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3840 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <partialdiff /> <bvar> <ci> p </ci> <degree> <cn type='integer'> 9 </cn> </degree> </bvar> <apply> <ci> Ω </ci> <ci> p </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 5308416 </cn> <apply> <power /> <pi /> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> x </ci> <cn type='rational'> -7 <sep /> 4 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> ν </ci> <apply> <floor /> <apply> <power /> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <eq /> <ci> p </ci> <apply> <plus /> <apply> <power /> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Ω </ci> <ci> p </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> p </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <plus /> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 16 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <ci> Rule </ci> <ci> x </ci> <infinity /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RiemannSiegelZ", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Nu]"], FractionBox[RowBox[List["Cos", "[", RowBox[List[RowBox[List["RiemannSiegelTheta", "[", "x", "]"]], "-", RowBox[List["x", " ", RowBox[List["Log", "[", "k", "]"]]]]]], "]"]], SqrtBox["k"]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["x", RowBox[List[RowBox[List["-", "1"]], "/", "4"]]], " ", RowBox[List["\[CapitalOmega]", "[", "p", "]"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["x", RowBox[List[RowBox[List["-", "3"]], "/", "4"]]], " ", RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]]]], RowBox[List["48", " ", SqrtBox["2"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], "+", RowBox[List["2", " ", "\[Pi]", " ", SuperscriptBox["x", RowBox[List[RowBox[List["-", "5"]], "/", "4"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["64", " ", SuperscriptBox["\[Pi]", "2"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "6", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["18432", " ", SuperscriptBox["\[Pi]", "4"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["x", RowBox[List[RowBox[List["-", "7"]], "/", "4"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", "\[Prime]", Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["64", " ", SuperscriptBox["\[Pi]", "2"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "5", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["3840", " ", SuperscriptBox["\[Pi]", "4"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[CapitalOmega]", TagBox[RowBox[List["(", "9", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "p", "]"]], RowBox[List["5308416", " ", SuperscriptBox["\[Pi]", "6"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "\[Equal]", RowBox[List["Floor", "[", SqrtBox[FractionBox["x", RowBox[List["2", " ", "\[Pi]"]]]], "]"]]]], "&&", RowBox[List["p", "\[Equal]", RowBox[List[SqrtBox[FractionBox["x", RowBox[List["2", " ", "\[Pi]"]]]], "-", "\[Nu]"]]]], "&&", RowBox[List[RowBox[List["\[CapitalOmega]", "[", "p", "]"]], "\[Equal]", FractionBox[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["p", "2"], "-", "p", "-", FractionBox["1", "16"]]], ")"]]]], "]"]], RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Pi]", " ", "p"]], "]"]]]]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["(", RowBox[List["x", "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]

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

2001-10-29