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variants of this functions
PolyLog






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[nu,z] > Series representations > Generalized power series > Expansions at z==1 > For the function itself > General case





http://functions.wolfram.com/10.08.06.0032.01









  


  










Input Form





PolyLog[\[Nu], z] == Gamma[1 - \[Nu]] (-(z - 1))^(\[Nu] - 1) Sum[((-1)^k (z - 1)^k)/(k + 1), {k, 0, Infinity}]^(\[Nu] - 1) + Zeta[\[Nu]] + Sum[Sum[(StirlingS1[k, j] Zeta[\[Nu] - j])/k!, {j, 1, k}] (z - 1)^k, {k, 1, Infinity}] /; !(Element[\[Nu], Integers] && \[Nu] > 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]], RowBox[List["k", "+", "1"]]]]], ")"]], RowBox[List["\[Nu]", "-", "1"]]]]], "+", RowBox[List["Zeta", "[", "\[Nu]", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], FractionBox[RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["\[Nu]", "-", "j"]], "]"]]]], RowBox[List["k", "!"]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", ">", "0"]]]], "]"]]]]]]










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> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mtext> </mtext> <mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> + </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;\[Nu]&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mfrac> <mrow> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, StirlingS1] </annotation> </semantics> <mi> k </mi> <mrow> <mo> ( </mo> <mi> j </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;-&quot;, &quot;j&quot;]], Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> &#172; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyLog </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <ci> &#957; </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <ci> StirlingS1 </ci> <ci> k </ci> <ci> j </ci> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <not /> <apply> <in /> <ci> &#957; </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]], RowBox[List["k", "+", "1"]]]]], ")"]], RowBox[List["\[Nu]", "-", "1"]]]]], "+", RowBox[List["Zeta", "[", "\[Nu]", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], FractionBox[RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["\[Nu]", "-", "j"]], "]"]]]], RowBox[List["k", "!"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "k"]]]]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", ">", "0"]]]], ")"]]]]]]]]]]










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





2007-05-02





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