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http://functions.wolfram.com/10.06.20.0024.01
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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)
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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"]]]], "]"]]]]]]
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<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> ∂ </mo> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </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["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["s", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", LerchPhi, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], 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> ∂ </mo> <mi> s </mi> </mrow> </mfrac> <mo>  </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <mo> ∂ </mo> <semantics> <mrow> <mi> Φ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["s", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "-", RowBox[List["\[LeftFloor]", RowBox[List["Re", "(", "a", ")"]], "\[RightFloor]"]]]], LerchPhi, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], 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> ∂ </mo> <mi> s </mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mo> ⌊ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> ⁡ </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> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </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> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> ⁡ </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> ¬ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> a </mi> <mo> > </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>
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| 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"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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