|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.27.03.0896.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
HypergeometricPFQ[{1, (2 + I)/4, (2 - I)/4}, {(6 + I)/4, (6 - I)/4}, -1] ==
(5/16) I (PolyGamma[1/4 - I/8] - PolyGamma[1/4 + I/8] -
PolyGamma[3/4 - I/8] + PolyGamma[3/4 + I/8])
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["2", "+", "\[ImaginaryI]"]], "4"], ",", FractionBox[RowBox[List["2", "-", "\[ImaginaryI]"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["6", "+", "\[ImaginaryI]"]], "4"], ",", FractionBox[RowBox[List["6", "-", "\[ImaginaryI]"]], "4"]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["5", "16"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[ImaginaryI]", "8"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[ImaginaryI]", "8"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[ImaginaryI]", "8"]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[ImaginaryI]", "8"]]], "]"]]]], ")"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mfrac> <mrow> <mn> 6 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 6 </mn> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["2", "+", "\[ImaginaryI]"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "\[ImaginaryI]"]], "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["6", "+", "\[ImaginaryI]"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["6", "-", "\[ImaginaryI]"]], "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 16 </mn> </mfrac> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> ⅈ </mi> <mn> 8 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> ⅈ </mi> <mn> 8 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mn> 8 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mi> ⅈ </mi> <mn> 8 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='complex-cartesian'> 2 <sep /> -1 </cn> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <apply> <times /> <apply> <plus /> <cn type='integer'> 6 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='complex-cartesian'> 6 <sep /> -1 </cn> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='rational'> 5 <sep /> 16 </cn> <imaginaryi /> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["2", "+", "\[ImaginaryI]"]], "4"], ",", FractionBox[RowBox[List["2", "-", "\[ImaginaryI]"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["6", "+", "\[ImaginaryI]"]], "4"], ",", FractionBox[RowBox[List["6", "-", "\[ImaginaryI]"]], "4"]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["5", "16"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[ImaginaryI]", "8"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[ImaginaryI]", "8"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[ImaginaryI]", "8"]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[ImaginaryI]", "8"]]], "]"]]]], ")"]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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,a4},{b1,b2,b3},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,...,ap},{b1,...,bq},z] | |
|
|
|