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http://functions.wolfram.com/03.01.26.0102.01
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BesselJ[\[Nu], z] Hypergeometric0F1Regularized[1 - \[Nu], z^2/4] ==
2^(\[Nu]/2) Sqrt[Pi] MeijerG[{{}, {(2 - \[Nu])/4}},
{{\[Nu]/4, (\[Nu] + 2)/4}, {(3 \[Nu])/4, -(\[Nu]/4), (2 - \[Nu])/4}},
z/(2 Sqrt[2]), 1/4]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "/", "2"]]], " ", SqrtBox["\[Pi]"], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["\[Nu]", "4"], ",", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", " ", "\[Nu]"]], "4"], ",", RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "}"]]]], "}"]], ",", FractionBox["z", RowBox[List["2", SqrtBox["2"]]]], ",", FractionBox["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> <mrow> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 0 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mo>   </mo> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ; </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["0", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Nu]"]], Hypergeometric0F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], Hypergeometric0F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1Regularized] </annotation> </semantics> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> ν </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 5 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mfrac> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mi> ν </mi> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "5"]], RowBox[List["2", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[FractionBox["z", RowBox[List["2", " ", SqrtBox["2"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[TagBox[FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox[FractionBox["\[Nu]", "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["3", " ", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "4"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> J </ci> <ci> ν </ci> </apply> <ci> z </ci> </apply> <apply> <ci> Hypergeometric0F1Regularized </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> MeijerG </ci> <list> <list /> <list> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> </list> <list> <list> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <apply> <times /> <cn type='integer'> 3 </cn> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> </list> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], " ", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]_"]], ",", FractionBox[SuperscriptBox["z_", "2"], "4"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "/", "2"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["\[Nu]", "4"], ",", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", " ", "\[Nu]"]], "4"], ",", RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "}"]]]], "}"]], ",", FractionBox["z", RowBox[List["2", " ", SqrtBox["2"]]]], ",", FractionBox["1", "4"]]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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