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http://functions.wolfram.com/07.18.09.0001.01
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Hypergeometric0F1Regularized[b, z] ==
z^((1 - b)/2) Limit[\[Lambda]^(b - 1) LegendreP[\[Lambda], 1 - b, 3,
Cosh[(2 Sqrt[z])/\[Lambda]]], \[Lambda] -> Infinity]
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Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["z", FractionBox[RowBox[List["1", "-", "b"]], "2"]], RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["\[Lambda]", RowBox[List["b", "-", "1"]]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Lambda]", ",", RowBox[List["1", "-", "b"]], ",", "3", ",", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", SqrtBox["z"]]], "\[Lambda]"], "]"]]]], "]"]]]], ",", RowBox[List["\[Lambda]", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]]]
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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> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mo>   </mo> <mo> ; </mo> <mi> b </mi> <mo> ; </mo> <mi> z </mi> </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["b", Hypergeometric0F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric0F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1Regularized] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <msup> <mi> z </mi> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> λ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <msup> <mi> λ </mi> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <semantics> <mi> 𝔓 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalP]", LegendreQ] </annotation> </semantics> <mi> λ </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mi> λ </mi> </mfrac> <mo> ) </mo> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Hypergeometric0F1Regularized </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <limit /> <bvar> <ci> λ </ci> </bvar> <condition> <apply> <tendsto /> <ci> λ </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <power /> <ci> λ </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔓 </ci> </apply> <ci> λ </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> λ </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["z", FractionBox[RowBox[List["1", "-", "b"]], "2"]], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["\[Lambda]", RowBox[List["b", "-", "1"]]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Lambda]", ",", RowBox[List["1", "-", "b"]], ",", "3", ",", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", SqrtBox["z"]]], "\[Lambda]"], "]"]]]], "]"]]]], ",", RowBox[List["\[Lambda]", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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HypergeometricPFQRegularized[{a},{b},z] | HypergeometricPFQRegularized[{a1,a2},{b1},z] | HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] | |
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