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variants of this functions
PolyLog




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > PolyLog[2,z] > Limit representations





http://functions.wolfram.com/10.07.09.0001.01









  


  








Input Form





PolyLog[2, z] == Limit[(Hypergeometric2F1[\[CurlyEpsilon], \[CurlyEpsilon], 1, z] - 1)/\[CurlyEpsilon]^2, \[CurlyEpsilon] -> 0]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z"]], "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[CurlyEpsilon]", ",", "\[CurlyEpsilon]", ",", "1", ",", "z"]], "]"]], "-", "1"]], SuperscriptBox["\[CurlyEpsilon]", "2"]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", "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> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#949; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mfrac> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#949; </mi> <mo> , </mo> <mi> &#949; </mi> </mrow> <mo> ; </mo> <mn> 1 </mn> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;\[CurlyEpsilon]&quot;, Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[CurlyEpsilon]&quot;, Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;1&quot;, Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, Hypergeometric2F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> <mo> - </mo> <mn> 1 </mn> </mrow> <msup> <mi> &#949; </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> PolyLog </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <limit /> <bvar> <ci> &#949; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#949; </ci> <cn type='integer'> 0 </cn> </apply> </condition> <apply> <times /> <apply> <plus /> <apply> <ci> Hypergeometric2F1 </ci> <ci> &#949; </ci> <ci> &#949; </ci> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <power /> <ci> &#949; </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[CurlyEpsilon]", ",", "\[CurlyEpsilon]", ",", "1", ",", "z"]], "]"]], "-", "1"]], SuperscriptBox["\[CurlyEpsilon]", "2"]], ",", RowBox[List["\[CurlyEpsilon]", "\[Rule]", "0"]]]], "]"]]]]]]


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





2001-10-29