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http://functions.wolfram.com/01.02.20.0018.01
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D[(c z^2 + b)^a, {z, \[Alpha]}] == ((b^a/(z^\[Alpha] Gamma[1 - \[Alpha]]))
AppellF1[-\[Alpha], -a, -a, 1 - \[Alpha], z/(Sqrt[-(b/c)] + z),
-(z/(Sqrt[-(b/c)] - z))])/((Sqrt[-(b/c)]/(Sqrt[-(b/c)] - z))^a
(Sqrt[-(b/c)]/(Sqrt[-(b/c)] + z))^a) /;
!(Element[\[Alpha], Integers] && \[Alpha] > 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", "b"]], ")"]], "a"]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["b", "a"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], RowBox[List[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], "-", "z"]]], ")"]], RowBox[List["-", "a"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], RowBox[List[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], "+", "z"]]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", RowBox[List["-", "a"]], ",", RowBox[List["-", "a"]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", FractionBox["z", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], "+", "z"]]], ",", RowBox[List["-", FractionBox["z", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], "-", "z"]]]]]]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[Alpha]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Alpha]", ">", "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> <msup> <mo> ∂ </mo> <mi> α </mi> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> b </mi> </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> <mfrac> <mrow> <msup> <mi> b </mi> <mi> a </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mtext> </mtext> <msup> <mrow> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> </mrow> </msqrt> <mrow> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> </mrow> </msqrt> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> </mrow> </msqrt> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> </mrow> <mo> ; </mo> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mrow> <msqrt> <mrow> <mo> - </mo> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> </mrow> </msqrt> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> α </mi> <mo> ∉ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> α </ci> </degree> </bvar> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> b </ci> <ci> a </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </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> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <ci> AppellF1 </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> α </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </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[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", "b_"]], ")"]], "a_"]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["b", "a"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], RowBox[List[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], "-", "z"]]], ")"]], RowBox[List["-", "a"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], RowBox[List[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], "+", "z"]]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], ",", RowBox[List["-", "a"]], ",", RowBox[List["-", "a"]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", FractionBox["z", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], "+", "z"]]], ",", RowBox[List["-", FractionBox["z", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["b", "c"]]]], "-", "z"]]]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["\[Alpha]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Alpha]", ">", "0"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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