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 http://functions.wolfram.com/06.36.20.0005.01

 Input Form

 D[LogIntegral[z], {z, \[Alpha]}] == (UnitStep[Re[1 - \[Alpha]]]/(z^\[Alpha] Gamma[1 - \[Alpha]])) Sum[(Pochhammer[\[Alpha], k] ExpIntegralEi[(1 + k) Log[z]])/k!/z^k, {k, 0, Infinity}] + (UnitStep[-Re[1 - \[Alpha]]]/ (z^\[Alpha] (Gamma[Floor[\[Alpha]] - \[Alpha] + 1] Log[z]))) Sum[(Pochhammer[\[Alpha] - Floor[\[Alpha]], k]/(z^k k!)) Sum[Binomial[Floor[\[Alpha]], Floor[\[Alpha]] - m] Pochhammer[1 + m - \[Alpha] - k, Floor[\[Alpha]] - m] Sum[(1/p!) ((-1)^p p! + (1 + k) Log[z] HypergeometricPFQRegularized[ {1, 1}, {2, 1 - p}, (1 + k) Log[z]]) (Sum[Log[z]^(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["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["LogIntegral", "[", "z", "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Re", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]], "]"]], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " "]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Alpha]", ",", "k"]], "]"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "]"]]]], RowBox[List["k", "!"]]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]], "]"]], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]], 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["\[Alpha]", "-", RowBox[List["Floor", "[", "\[Alpha]", "]"]]]], ",", "k"]], "]"]], SuperscriptBox["z", 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["Floor", "[", "\[Alpha]", "]"]], ",", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "-", "m"]]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m", "-", "\[Alpha]", "-", "k"]], ",", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "-", "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", "[", "z", "]"]], " ", 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", "[", "z", "]"]]]]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "m"], RowBox[List[SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], 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

 α li ( z ) z α θ UnitStep ( Re ( 1 - α ) ) z - α Γ ( 1 - α ) k = 0 ( α ) k TagBox[SubscriptBox[RowBox[List["(", "\[Alpha]", ")"]], "k"], Pochhammer] Ei ( ( k + 1 ) log ( z ) ) z - k k ! + θ UnitStep ( - Re ( 1 - α ) ) z - α Γ ( α - α + 1 ) log ( z ) k = 0 ( α - α ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Alpha]", "-", RowBox[List["\[LeftFloor]", "\[Alpha]", "\[RightFloor]"]]]], ")"]], "k"], Pochhammer] z - k k ! m = 0 α + 1 ( α α - m ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["\[LeftFloor]", "\[Alpha]", "\[RightFloor]"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List[RowBox[List["\[LeftFloor]", "\[Alpha]", "\[RightFloor]"]], "-", "m"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( m - k - α + 1 ) α - m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["m", "-", "k", "-", "\[Alpha]", "+", "1"]], ")"]], RowBox[List[RowBox[List["\[LeftFloor]", "\[Alpha]", "\[RightFloor]"]], "-", "m"]]], Pochhammer] p = 0 m - 1 1 p ! ( ( ( - 1 ) p p ! + ( k + 1 ) log ( z ) 2 F ~ 2 ( 1 , 1 ; 2 , 1 - p ; ( k + 1 ) log ( z ) ) 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", "(", "z", ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) ( ( - 1 ) m + p + 1 ( m - 1 ) ! + h = 0 m log 1 - h ( z ) S TagBox["S", StirlingS1] m ( h ) p ! j = 0 p - 1 ( - 1 ) j j ! ( p - h - j + 1 ) ! ) ) z α LogIntegral z UnitStep 1 -1 α z -1 α Gamma 1 -1 α -1 k 0 Pochhammer α k ExpIntegralEi k 1 z z -1 k k -1 UnitStep -1 1 -1 α z -1 α Gamma α -1 α 1 z -1 k 0 Pochhammer α -1 α k z -1 k k -1 m 0 α 1 Binomial α α -1 m Pochhammer m -1 k -1 α 1 α -1 m p 0 m -1 1 p -1 -1 p p k 1 z HypergeometricPFQRegularized 1 1 2 1 -1 p k 1 z -1 m p 1 m -1 h 0 m z 1 -1 h StirlingS1 m h p j 0 p -1 -1 j j p -1 h -1 j 1 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["LogIntegral", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Re", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Alpha]", ",", "k"]], "]"]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", 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["z", 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["Floor", "[", "\[Alpha]", "]"]], ",", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "-", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "m", "-", "\[Alpha]", "-", "k"]], ",", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "-", "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", "[", "z", "]"]], " ", 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", "[", "z", "]"]]]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "m"], RowBox[List[SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], 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[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["Floor", "[", "\[Alpha]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]]]]]]]]]

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

 2001-10-29