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 | | http://functions.wolfram.com/07.31.03.0159.01 | 
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 | | HypergeometricPFQ[{}, {1/2, b, 2 - b}, z] == (-((2 Pi (b - 1))/Sin[2 Pi b])) 
  (KelvinBer[2 - 2 b, 2 Sqrt[2] z^(1/4)] 
    (Cos[3 b Pi] KelvinBer[-2 + 2 b, 2 Sqrt[2] z^(1/4)] + 
     Sin[3 b Pi] KelvinBei[-2 + 2 b, 2 Sqrt[2] z^(1/4)]) + 
   KelvinBei[2 - 2 b, 2 Sqrt[2] z^(1/4)] 
    (Cos[3 b Pi] KelvinBei[-2 + 2 b, 2 Sqrt[2] z^(1/4)] - 
     Sin[3 b Pi] KelvinBer[-2 + 2 b, 2 Sqrt[2] z^(1/4)])) | 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "b", ",", RowBox[List["2", "-", "b"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", "\[Pi]", RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]]]], RowBox[List["Sin", "[", RowBox[List["2", "\[Pi]", " ", "b"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List["2", "-", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["3", " ", "b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List["3", " ", "b", " ", "\[Pi]"]], "]"]], RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]]]]]], " ", ")"]]]], "+", RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List["2", "-", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["3", " ", "b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Sin", "[", RowBox[List["3", " ", "b", " ", "\[Pi]"]], "]"]], RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]] | 
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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>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mn> 0 </mn>  </msub>  <msub>  <mi> F </mi>  <mn> 3 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mo>   </mo>  <mo> ; </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  <mo> , </mo>  <mi> b </mi>  <mo> , </mo>  <mrow>  <mn> 2 </mn>  <mo> - </mo>  <mi> b </mi>  </mrow>  </mrow>  <mo> ; </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox["F", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", 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<msqrt>  <mn> 2 </mn>  </msqrt>  <mo> ⁢ </mo>  <mroot>  <mi> z </mi>  <mn> 4 </mn>  </mroot>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mrow>  <mrow>  <msub>  <mi> bei </mi>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mn> 2 </mn>  </mrow>  </msub>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msqrt>  <mn> 2 </mn>  </msqrt>  <mo> ⁢ </mo>  <mroot>  <mi> z </mi>  <mn> 4 </mn>  </mroot>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> ber </mi>  <mrow>  <mn> 2 </mn>  <mo> - </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  </mrow>  </msub>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msqrt>  <mn> 2 </mn>  </msqrt>  <mo> ⁢ </mo>  <mroot>  <mi> z </mi>  <mn> 4 </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 />  <list>  <cn type='rational'> 1 <sep /> 2 </cn>  <ci> b </ci>  <apply>  <plus />  <cn type='integer'> 2 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  </list>  <ci> z </ci>  </apply>  <apply>  <times />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 2 </cn>  <pi />  <apply>  <plus />  <ci> b </ci>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <power />  <apply>  <sin />  <apply>  <times />  <cn type='integer'> 2 </cn>  <pi />  <ci> b </ci>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <apply>  <ci> KelvinBei </ci>  <apply>  <plus />  <cn type='integer'> 2 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> b </ci>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <apply>  <cos />  <apply>  <times />  <cn type='integer'> 3 </cn>  <ci> b </ci>  <pi />  </apply>  </apply>  <apply>  <ci> KelvinBei </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> b </ci>  </apply>  <cn type='integer'> -2 </cn>  </apply>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <apply>  <sin />  <apply>  <times />  <cn type='integer'> 3 </cn>  <ci> b </ci>  <pi />  </apply>  </apply>  <apply>  <ci> KelvinBer </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> b </ci>  </apply>  <cn type='integer'> -2 </cn>  </apply>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <plus />  <apply>  <times />  <apply>  <cos />  <apply>  <times />  <cn type='integer'> 3 </cn>  <ci> b </ci>  <pi />  </apply>  </apply>  <apply>  <ci> KelvinBer </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> b </ci>  </apply>  <cn type='integer'> -2 </cn>  </apply>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <ci> KelvinBei </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> b </ci>  </apply>  <cn type='integer'> -2 </cn>  </apply>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  </apply>  </apply>  <apply>  <sin />  <apply>  <times />  <cn type='integer'> 3 </cn>  <ci> b </ci>  <pi />  </apply>  </apply>  </apply>  </apply>  <apply>  <ci> KelvinBer </ci>  <apply>  <plus />  <cn type='integer'> 2 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> b </ci>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "b_", ",", RowBox[List["2", "-", "b_"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List["2", "-", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["3", " ", "b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List["3", " ", "b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List["2", "-", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["3", " ", "b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Sin", "[", RowBox[List["3", " ", "b", " ", "\[Pi]"]], "]"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "b"]]]], ",", RowBox[List["2", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["Sin", "[", RowBox[List["2", " ", "\[Pi]", " ", "b"]], "]"]]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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 | | 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},{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] |  |  | 
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