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
 











LerchPhi






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > LerchPhi[z,s,a] > Differentiation > Symbolic differentiation > With respect to s > For Phi(z,s,a)





http://functions.wolfram.com/10.06.20.0036.01









  


  










Input Form





D[LerchPhi[z, s, a], {s, n}] == ((-1)^n/2^n) Sum[(Log[(a + k)^2]^n z^k)/((a + k)^2)^(s/2), {k, 0, Infinity}] /; !(Element[-a, Integers] && -a >= 0) && Abs[z] < 1 && Re[s] > 1 && Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["s", ",", "n"]], "}"]]], RowBox[List["LerchPhi", "[", RowBox[List["z", ",", "s", ",", "a"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], SuperscriptBox["2", "n"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox[RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], "]"]], "n"], SuperscriptBox["z", "k"]]]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], ")"]], RowBox[List["s", "/", "2"]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["-", "a"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], "]"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "1"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <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> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> s </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mtext> </mtext> </mrow> <msup> <mn> 2 </mn> <mi> n </mi> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <mrow> <msup> <mi> log </mi> <mi> n </mi> </msup> <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> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <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> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8713; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> s </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> LerchPhi </ci> <ci> z </ci> <ci> s </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <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> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <apply> <ln /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> n </ci> </apply> <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> <and /> <apply> <notin /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <apply> <real /> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["s_", ",", "n_"]], "}"]]]]], RowBox[List["LerchPhi", "[", RowBox[List["z_", ",", "s_", ",", "a_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], "]"]], "n"], " ", SuperscriptBox["z", "k"]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], ")"]], RowBox[List["s", "/", "2"]]]]]]]], SuperscriptBox["2", "n"]], "/;", RowBox[List[RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "&&", RowBox[List[RowBox[List["Re", "[", "s", "]"]], ">", "1"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.