Logarithm: Differentiation (formula 01.05.20.0009)

Log

$Mathematica Notation$

Elementary Functions

Log[a,z]

Differentiation

Fractional integro-differentiation

With respect to a

http://functions.wolfram.com/01.05.20.0009.01

Input Form

D[Log[a, z], {a, \[Alpha]}] == ((UnitStep[Re[-\[Alpha]]] a^(-\[Alpha] - 1))/Gamma[-\[Alpha]]) Log[z] Sum[(Pochhammer[1 + \[Alpha], k] ExpIntegralEi[(1 + k) Log[a]])/k!/a^k, {k, 0, Infinity}] + ((UnitStep[-Re[-\[Alpha]]] a^(-\[Alpha] - 1))/ Gamma[Floor[\[Alpha]] - \[Alpha] + 1]) Log[a, z] Sum[(Pochhammer[\[Alpha] - Floor[\[Alpha]], k]/(a^k k!)) Sum[Binomial[Floor[\[Alpha]] + 1, Floor[\[Alpha]] + 1 - m] Pochhammer[m - \[Alpha] - k, Floor[\[Alpha]] + 1 - m] Sum[(1/p!) ((-1)^p p! + (1 + k) Log[a] HypergeometricPFQRegularized[ {1, 1}, {2, 1 - p}, (1 + k) Log[a]]) (Sum[Log[a]^(1 - h) StirlingS1[m, h] p! Sum[(-1)^j/(j! (p - j - h + 1)!), {j, 0, p - 1}], {h, 0, m}] + (-1)^(m + p + 1) (m - 1)!), {p, 0, m - 1}], {m, 0, Floor[\[Alpha]] + 1}], {k, 0, Infinity}]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]]], RowBox[List["Log", "[", RowBox[List["a", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Re", "[", RowBox[List["-", "\[Alpha]"]], "]"]], "]"]], SuperscriptBox["a", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "1"]]], " "]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Alpha]"]], "]"]]], RowBox[List["Log", "[", "z", "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Alpha]"]], ",", "k"]], "]"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], " ", RowBox[List["Log", "[", "a", "]"]]]], "]"]]]], RowBox[List["k", "!"]]], " ", SuperscriptBox["a", RowBox[List["-", "k"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", RowBox[List["-", "\[Alpha]"]], "]"]]]], "]"]], SuperscriptBox["a", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "1"]]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]], RowBox[List["Log", "[", RowBox[List["a", ",", "z"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Alpha]", "-", RowBox[List["Floor", "[", "\[Alpha]", "]"]]]], ",", "k"]], "]"]], SuperscriptBox["a", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "+", "1"]]], " ", RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "+", "1"]], ",", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "+", "1", "-", "m"]]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "\[Alpha]", "-", "k"]], ",", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "+", "1", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["m", "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List["p", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["p", "!"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], " ", RowBox[List["Log", "[", "a", "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", RowBox[List["1", "-", "p"]]]], "}"]], ",", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], " ", RowBox[List["Log", "[", "a", "]"]]]]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "m"], RowBox[List[SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], RowBox[List["1", "-", "h"]]], RowBox[List["StirlingS1", "[", RowBox[List["m", ",", "h"]], "]"]], RowBox[List["p", "!"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["p", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " "]], RowBox[List[RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["p", "-", "j", "-", "h", "+", "1"]], ")"]], "!"]]]]]]]]]]], " ", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "p", "+", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]]]]], ")"]]]]]]]]]]]]]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> α </mi> </msup> <mrow> <msub> <mi> log </mi> <mi> a </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> a </mi> <mi> α </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <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> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Alpha]", "+", "1"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> Ei </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> log </mi> <mi> a </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mi> α </mi> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mi> α </mi> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Alpha]", "-", RowBox[List["\[LeftFloor]", "\[Alpha]", "\[RightFloor]"]]]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <mo> ⌊ </mo> <mi> α </mi> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> ⌊ </mo> <mi> α </mi> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mi> α </mi> <mo> ⌋ </mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["\[LeftFloor]", "\[Alpha]", "\[RightFloor]"]], "+", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["1", "-", "m", "+", RowBox[List["\[LeftFloor]", "\[Alpha]", "\[RightFloor]"]]]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mi> α </mi> <mo> ⌋ </mo> </mrow> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "k", "-", "\[Alpha]"]], ")"]], RowBox[List["1", "-", "m", "+", RowBox[List["\[LeftFloor]", "\[Alpha]", "\[RightFloor]"]]]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> p </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> p </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> p </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> p </mi> <mo> ! </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> p </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox["2", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "p"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List["log", "(", "a", ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> h </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mrow> <msup> <mi> log </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> h </mi> </mrow> </msup> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mi> m </mi> <mrow> <mo> ( </mo> <mi> h </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ⁢ </mo> <mrow> <mi> p </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> j </mi> <mo> + </mo> <mi> p </mi> <mo> - </mo> <mi> h </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> a </ci> <degree> <ci> α </ci> </degree> </bvar> <apply> <log /> <logbase> <ci> a </ci> </logbase> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <ci> UnitStep </ci> <apply> <real /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </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> <ci> Pochhammer </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <ci> ExpIntegralEi </ci> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ln /> <ci> a </ci> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <log /> <logbase> <ci> a </ci> </logbase> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <apply> <floor /> <ci> α </ci> </apply> </apply> </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> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <ci> α </ci> </apply> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <floor /> <ci> α </ci> </apply> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <apply> <plus /> <apply> <floor /> <ci> α </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <floor /> <ci> α </ci> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <floor /> <ci> α </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> p </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <apply> <factorial /> <ci> p </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ln /> <ci> a </ci> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> </list> <list> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> </list> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ln /> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> m </ci> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> h </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <power /> <apply> <ln /> <ci> a </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> h </ci> </apply> </apply> </apply> <apply> <ci> StirlingS1 </ci> <ci> m </ci> <ci> h </ci> </apply> <apply> <factorial /> <ci> p </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> h </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </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["a_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["Log", "[", RowBox[List["a_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Re", "[", RowBox[List["-", "\[Alpha]"]], "]"]], "]"]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "1"]]]]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Alpha]"]], ",", "k"]], "]"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], " ", RowBox[List["Log", "[", "a", "]"]]]], "]"]]]], ")"]], " ", SuperscriptBox["a", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Alpha]"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", RowBox[List["-", "\[Alpha]"]], "]"]]]], "]"]], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "1"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["a", ",", "z"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Alpha]", "-", RowBox[List["Floor", "[", "\[Alpha]", "]"]]]], ",", "k"]], "]"]], " ", SuperscriptBox["a", RowBox[List["-", "k"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "+", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "+", "1"]], ",", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "+", "1", "-", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "\[Alpha]", "-", "k"]], ",", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "+", "1", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["p", "!"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], " ", RowBox[List["Log", "[", "a", "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["2", ",", RowBox[List["1", "-", "p"]]]], "}"]], ",", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], " ", RowBox[List["Log", "[", "a", "]"]]]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "m"], RowBox[List[SuperscriptBox[RowBox[List["Log", "[", "a", "]"]], RowBox[List["1", "-", "h"]]], " ", RowBox[List["StirlingS1", "[", RowBox[List["m", ",", "h"]], "]"]], " ", RowBox[List["p", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["p", "-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["p", "-", "j", "-", "h", "+", "1"]], ")"]], "!"]]]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "p", "+", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]]]]], ")"]]]], RowBox[List["p", "!"]]]]]]]]]]], RowBox[List["k", "!"]]]]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]]

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

2001-10-29

Log[z]