Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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 > Low-order differentiation > With respect to s > For Phi(z,s,a)





http://functions.wolfram.com/10.06.20.0024.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "s"], RowBox[List["LerchPhi", "[", RowBox[List["z", ",", "s", ",", "a"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["-", RowBox[List["Floor", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]], RowBox[List[SubscriptBox["\[PartialD]", "s"], RowBox[List["LerchPhi", "[", RowBox[List["z", ",", "s", ",", RowBox[List["a", "-", RowBox[List["Floor", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]]], "]"]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]]], FractionBox[RowBox[List["Sign", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", RowBox[List["Floor", "[", RowBox[List["Re", "[", "a", "]"]], "]"]], "]"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List["2", "a"]]]], "2"], RowBox[List["Sign", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], "+", "k"]], ")"]], "2"], "]"]], SuperscriptBox["z", RowBox[List[RowBox[List["-", RowBox[List["Sign", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], " ", "k"]]]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List["2", "a"]]]], "2"], RowBox[List["Sign", "[", RowBox[List["Re", "[", "a", "]"]], "]"]]]], "+", "k"]], ")"]], "2"], ")"]], RowBox[List["s", "/", "2"]]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["a", "\[Element]", "Integers"]], "\[And]", RowBox[List["a", ">", "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> <mo> &#8706; </mo> <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;, LerchPhi, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;s&quot;, LerchPhi, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;a&quot;, LerchPhi, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mrow> <mo> &#8706; </mo> <mi> s </mi> </mrow> </mfrac> <mo> &#63449; </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mfrac> <mrow> <mo> &#8706; </mo> <semantics> <mrow> <mi> &#934; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[CapitalPhi]&quot;, &quot;(&quot;, RowBox[List[TagBox[&quot;z&quot;, LerchPhi, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[&quot;s&quot;, LerchPhi, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, RowBox[List[&quot;\[LeftFloor]&quot;, RowBox[List[&quot;Re&quot;, &quot;(&quot;, &quot;a&quot;, &quot;)&quot;]], &quot;\[RightFloor]&quot;]]]], LerchPhi, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$], LerchPhi[ZetaDump`e1$, ZetaDump`e2$, ZetaDump`e3$]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> <mrow> <mo> &#8706; </mo> <mi> s </mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mo> &#8970; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8971; </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </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> </mrow> <mo> /; </mo> <mrow> <mo> &#172; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> s </ci> </bvar> <apply> <ci> LerchPhi </ci> <ci> z </ci> <ci> s </ci> <ci> a </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> s </ci> </bvar> <apply> <ci> LerchPhi </ci> <ci> z </ci> <ci> s </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Sign </ci> <apply> <real /> <ci> a </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> UnitStep </ci> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <abs /> <apply> <floor /> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ln /> <apply> <power /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Sign </ci> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Sign </ci> <apply> <real /> <ci> a </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </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> <not /> <apply> <and /> <apply> <in /> <ci> a </ci> <integers /> </apply> <apply> <gt /> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










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





2007-05-02