LerchPhi[z, s, a] == LerchPhiClassicalRegularized[z, s, a] - Sum[z^k/(a + k)^s - z^k/((a + k)^2)^(s/2), {k, 0, Floor[-Re[a]]}] /; !(Element[-a, Integers] && -a > 0)

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LerchPhi", "[", RowBox[List["z", ",", "s", ",", "a"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["LerchPhiClassicalRegularized", "[", RowBox[List["z", ",", "s", ",", "a"]], "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]], RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["z", "k"], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "s"]], "-", FractionBox[SuperscriptBox["z", "k"], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], ")"]], RowBox[List["s", "/", "2"]]]]]], ")"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["-", "a"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["-", "a"]], ">", "0"]]]], "]"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mi> &#934; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[CapitalPhi]&quot;, &quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;s&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;a&quot;, Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mrow> <mover> <mi> &#934; </mi> <mo> ~ </mo> </mover> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> z </mi> <mi> k </mi> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> </mfrac> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mi> k </mi> </msup> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mi> s </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8713; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LerchPhi </ci> <ci> z </ci> <ci> s </ci> <ci> a </ci> </apply> <apply> <plus /> <apply> <apply> <ci> OverTilde </ci> <ci> &#934; </ci> </apply> <ci> z </ci> <ci> s </ci> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </uplimit> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> s </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LerchPhi", "[", RowBox[List["z_", ",", "s_", ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["LerchPhiClassicalRegularized", "[", RowBox[List["z", ",", "s", ",", "a"]], "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]], RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["z", "k"], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "s"]], "-", FractionBox[SuperscriptBox["z", "k"], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], ")"]], RowBox[List["s", "/", "2"]]]]]], ")"]]]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], ">", "0"]]]], ")"]]]]]]]]]]

2001-10-29