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http://functions.wolfram.com/01.02.20.0027.01
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D[(b z^c)^a, {z, \[Alpha]}] ==
Piecewise[{{((-1)^\[Alpha] Pochhammer[(-a) c, \[Alpha]] (b z^c)^a)/
z^\[Alpha], Element[\[Alpha], Integers] && Element[a c, Integers] &&
a c < 0 && a c < \[Alpha]},
{((((-1)^(a c - 1) (Log[b z^c] + c PolyGamma[(-a) c] -
c PolyGamma[1 + a c - \[Alpha]]))/(c (-1 - a c)!
Gamma[1 + a c - \[Alpha]])) (b z^c)^a)/z^\[Alpha],
Element[a c, Integers] && a c < 0}},
((Gamma[1 + a c]/Gamma[1 - \[Alpha] + a c]) (b z^c)^a)/z^\[Alpha]]
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "a"]]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Alpha]"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "c"]], ",", "\[Alpha]"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "a"], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], " ", ",", RowBox[List[RowBox[List["\[Alpha]", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", " ", "c"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", " ", "c"]], "<", "0"]], "&&", RowBox[List[RowBox[List["a", " ", "c"]], "<", "\[Alpha]"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["a", " ", "c"]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], "]"]], "+", RowBox[List["c", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "a"]], " ", "c"]], "]"]]]], "-", RowBox[List["c", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", RowBox[List["a", " ", "c"]], "-", "\[Alpha]"]], "]"]]]]]], ")"]]]], RowBox[List["c", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["a", " ", "c"]]]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List["a", " ", "c"]], "-", "\[Alpha]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "a"], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], " ", ",", RowBox[List[RowBox[List[RowBox[List["a", " ", "c"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", " ", "c"]], "<", "0"]]]]]], "}"]]]], "}"]], ",", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List["a", " ", "c"]]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]", "+", RowBox[List["a", " ", "c"]]]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "a"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]]]], " ", "]"]]]]]]
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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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> α </mi> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> α </mi> </msup> </mrow> </mfrac> <mo>  </mo> <mrow> <mo>  </mo> <mtable> <mtr> <mtd> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "c"]], ")"]], "\[Alpha]"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mrow> <mi> α </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> < </mo> <mi> α </mi> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> </mrow> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> α </ci> </degree> </bvar> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> a </ci> </apply> </apply> <piecewise> <piece> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> c </ci> </apply> <ci> α </ci> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> α </ci> <integers /> </apply> <apply> <in /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <integers /> </apply> <apply> <lt /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <ci> α </ci> </apply> </apply> </piece> <piece> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <ci> PolyGamma </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> c </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <apply> <factorial /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <integers /> </apply> <apply> <lt /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </piece> <otherwise> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> a </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> </otherwise> </piecewise> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b_", " ", SuperscriptBox["z_", "c_"]]], ")"]], "a_"]]], "]"]], "\[RuleDelayed]", RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Alpha]"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], " ", "c"]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "a"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List[RowBox[List["\[Alpha]", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", " ", "c"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", " ", "c"]], "<", "0"]], "&&", RowBox[List[RowBox[List["a", " ", "c"]], "<", "\[Alpha]"]]]]], List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["a", " ", "c"]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], "]"]], "+", RowBox[List["c", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "a"]], " ", "c"]], "]"]]]], "-", RowBox[List["c", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", RowBox[List["a", " ", "c"]], "-", "\[Alpha]"]], "]"]]]]]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "a"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["c", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["a", " ", "c"]]]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List["a", " ", "c"]], "-", "\[Alpha]"]], "]"]]]]], RowBox[List[RowBox[List[RowBox[List["a", " ", "c"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["a", " ", "c"]], "<", "0"]]]]], List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List["a", " ", "c"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "a"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]", "+", RowBox[List["a", " ", "c"]]]], "]"]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
