Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
BernoulliB






Mathematica Notation

Traditional Notation









Polynomials > BernoulliB[n,z] > Specific values > Specialized values > For fixed n





http://functions.wolfram.com/05.14.03.0013.01









  


  










Input Form





BernoulliB[n, p/q] == ((n I^n)/(2 Pi q)^n) Sum[Subscript[c, n][k/q] E^((2 Pi I p k)/q), {k, 0, q - 1}] /; Subscript[c, n][z] == D[Log[Sin[Pi z]], {z, n}] && Subscript[c, n][0] == -2 KroneckerDelta[Mod[n, 2], 0] Zeta[n] (n - 1)! && Element[n, Integers] && n > 1 && Element[p, Integers] && p >= 0 && Element[q, Integers] && q > 0 && p <= q










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["n", ",", FractionBox["p", "q"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["n", " ", SuperscriptBox["\[ImaginaryI]", "n"]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "q"]], ")"]], "n"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "1"]]], RowBox[List[RowBox[List[SubscriptBox["c", "n"], "[", FractionBox["k", "q"], "]"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p", " ", "k"]], "q"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["c", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], "]"]]]]]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["c", "n"], "[", "0", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "2"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["n", ",", "2"]], "]"]], ",", "0"]], "]"]], RowBox[List["Zeta", "[", "n", "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]]], "\[And]", RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "1"]], "\[And]", RowBox[List["Element", "[", RowBox[List["p", ",", "Integers"]], "]"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Element", "[", RowBox[List["q", ",", "Integers"]], "]"]], "\[And]", RowBox[List["q", ">", "0"]], "\[And]", RowBox[List["p", "\[LessEqual]", "q"]]]]]]]]










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> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msup> <mi> &#8520; </mi> <mi> n </mi> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <msub> <mi> c </mi> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mi> k </mi> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> p </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msub> <mi> c </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mi> n </mi> </msub> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <semantics> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 2 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`n </ci> <cn type='integer'> 2 </cn> </apply> </annotation-xml> </semantics> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;n&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> q </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8804; </mo> <mi> q </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BernoulliB </ci> <ci> n </ci> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> n </ci> <apply> <power /> <imaginaryi /> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> q </ci> </apply> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> n </ci> </apply> <apply> <times /> <ci> k </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <exp /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> p </ci> <ci> k </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> n </ci> </apply> <ci> z </ci> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ln /> <apply> <sin /> <apply> <times /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> n </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <rem /> <ci> $CellContext`n </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Zeta </ci> <ci> n </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <integers /> </apply> <apply> <in /> <ci> p </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> q </ci> <integers /> </apply> <apply> <leq /> <ci> p </ci> <ci> q </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BernoulliB", "[", RowBox[List["n_", ",", FractionBox["p_", "q_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", " ", SuperscriptBox["\[ImaginaryI]", "n"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["q", "-", "1"]]], RowBox[List[RowBox[List[SubscriptBox["c", "n"], "[", FractionBox["k", "q"], "]"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p", " ", "k"]], "q"]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "q"]], ")"]], "n"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["c", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]]]], RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], "]"]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["c", "n"], "[", "0", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["n", ",", "2"]], "]"]], ",", "0"]], "]"]], " ", RowBox[List["Zeta", "[", "n", "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "1"]], "&&", RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "0"]], "&&", RowBox[List["q", "\[Element]", "Integers"]], "&&", RowBox[List["q", ">", "0"]], "&&", RowBox[List["p", "\[LessEqual]", "q"]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.