Generalized hypergeometric function 4F3: Specific values (formula 07.28.03.0083)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]

Specific values

For fixed z

http://functions.wolfram.com/07.28.03.0083.01

Input Form

HypergeometricPFQ[{4/3, 5/3, 2, 2}, {7/3, 8/3, 3}, z] == (10/(3 z^(8/3))) (3 z (z^(1/3) - 4) Log[1 - z^(1/3)] + z^(2/3) (6 + 4 z^(1/3) - z^(2/3)) Log[1 - z] - 2 Sqrt[3] z (z^(1/3) + 4) ArcTan[(Sqrt[3] z^(1/3))/(2 + z^(1/3))])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["4", "3"], ",", FractionBox["5", "3"], ",", "2", ",", "2"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "3"], ",", FractionBox["8", "3"], ",", "3"]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["10", RowBox[List["3", SuperscriptBox["z", RowBox[List["8", "/", "3"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["3", "z", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["1", "/", "3"]]], "-", "4"]], ")"]], RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["2", "/", "3"]]], RowBox[List["(", RowBox[List["6", "+", RowBox[List["4", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "-", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], ")"]], RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], "-", RowBox[List["2", SqrtBox["3"], "z", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["1", "/", "3"]]], "+", "4"]], ")"]], RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox["3"], SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], RowBox[List["2", "+", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]]], "]"]]]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 7 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 8 </mn> <mn> 3 </mn> </mfrac> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["4", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["7", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["8", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["3", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 10 </mn> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 8 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 3 </mn> </mroot> </mrow> <mo> - </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 3 </mn> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mroot> <mi> z </mi> <mn> 3 </mn> </mroot> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> ⁢ </mo> <mroot> <mi> z </mi> <mn> 3 </mn> </mroot> </mrow> <mrow> <mroot> <mi> z </mi> <mn> 3 </mn> </mroot> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mroot> <mi> z </mi> <mn> 3 </mn> </mroot> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mroot> <mi> z </mi> <mn> 3 </mn> </mroot> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='rational'> 4 <sep /> 3 </cn> <cn type='rational'> 5 <sep /> 3 </cn> <cn type='integer'> 2 </cn> <cn type='integer'> 2 </cn> </list> <list> <cn type='rational'> 7 <sep /> 3 </cn> <cn type='rational'> 8 <sep /> 3 </cn> <cn type='integer'> 3 </cn> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 8 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <ci> z </ci> <apply> <arctan /> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -4 </cn> </apply> <ci> z </ci> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["4", "3"], ",", FractionBox["5", "3"], ",", "2", ",", "2"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "3"], ",", FractionBox["8", "3"], ",", "3"]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["10", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "z", " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["1", "/", "3"]]], "-", "4"]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List["4", " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "-", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox["3"], " ", "z", " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["1", "/", "3"]]], "+", "4"]], ")"]], " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List[SqrtBox["3"], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], RowBox[List["2", "+", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]]], "]"]]]]]], ")"]]]], RowBox[List["3", " ", SuperscriptBox["z", RowBox[List["8", "/", "3"]]]]]]]]]]

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

2001-10-29

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]