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http://functions.wolfram.com/07.20.06.0006.01
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Hypergeometric1F1[a, b, z] \[Proportional]
Gamma[b] (AsymptoticHypergeometricPFQRegularizedSeries[Power][{a}, {b},
{z, ComplexInfinity, Infinity}] +
AsymptoticHypergeometricPFQRegularizedSeries[Exp][{a}, {b},
{z, ComplexInfinity, Infinity}]) /; (Abs[z] -> Infinity)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric1F1", "[", RowBox[List["a", ",", "b", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["AsymptoticHypergeometricPFQRegularizedSeries", "[", "Power", "]"]], "[", RowBox[List[RowBox[List["{", "a", "}"]], ",", RowBox[List["{", "b", "}"]], ",", RowBox[List["{", RowBox[List["z", ",", "ComplexInfinity", ",", "\[Infinity]"]], "}"]]]], "]"]], "+", RowBox[List[RowBox[List["AsymptoticHypergeometricPFQRegularizedSeries", "[", "Exp", "]"]], "[", RowBox[List[RowBox[List["{", "a", "}"]], ",", RowBox[List["{", "b", "}"]], ",", RowBox[List["{", RowBox[List["z", ",", "ComplexInfinity", ",", "\[Infinity]"]], "}"]]]], "]"]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[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> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <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["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["a", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] </annotation> </semantics> <mo> ∝ </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msubsup> <mi> 𝒜 </mi> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mo> ( </mo> <mi> power </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mi> a </mi> <mo> ; </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> b </mi> <mo> ; </mo> </mrow> </mtd> </mtr> </mtable> <mo> ⁢ </mo> <mrow> <mo> { </mo> <mrow> <mi> z </mi> <mo> , </mo> <mover> <mi> ∞ </mi> <mo> ~ </mo> </mover> <mo> , </mo> <mi> ∞ </mi> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msubsup> <mi> 𝒜 </mi> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mo> ( </mo> <mi> exp </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mi> a </mi> <mo> ; </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> b </mi> <mo> ; </mo> </mrow> </mtd> </mtr> </mtable> <mo> ⁢ </mo> <mrow> <mo> { </mo> <mrow> <mi> z </mi> <mo> , </mo> <mover> <mi> ∞ </mi> <mo> ~ </mo> </mover> <mo> , </mo> <mi> ∞ </mi> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Hypergeometric1F1 </ci> <ci> a </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <plus /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> 𝒜 </ci> <apply> <ci> OverTilde </ci> <ci> F </ci> </apply> </apply> <ci> power </ci> </apply> <apply> <times /> <list> <list> <apply> <ci> CompoundExpression </ci> <ci> a </ci> <ci> Null </ci> </apply> </list> <list> <apply> <ci> CompoundExpression </ci> <ci> b </ci> <ci> Null </ci> </apply> </list> </list> <list> <ci> z </ci> <apply> <ci> OverTilde </ci> <infinity /> </apply> <infinity /> </list> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> 𝒜 </ci> <apply> <ci> OverTilde </ci> <ci> F </ci> </apply> </apply> <ci> exp </ci> </apply> <apply> <times /> <list> <list> <apply> <ci> CompoundExpression </ci> <ci> a </ci> <ci> Null </ci> </apply> </list> <list> <apply> <ci> CompoundExpression </ci> <ci> b </ci> <ci> Null </ci> </apply> </list> </list> <list> <ci> z </ci> <apply> <ci> OverTilde </ci> <infinity /> </apply> <infinity /> </list> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric1F1", "[", RowBox[List["a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["AsymptoticHypergeometricPFQRegularizedSeries", "[", "Power", "]"]], "[", RowBox[List[RowBox[List["{", "a", "}"]], ",", RowBox[List["{", "b", "}"]], ",", RowBox[List["{", RowBox[List["z", ",", "ComplexInfinity", ",", "\[Infinity]"]], "}"]]]], "]"]], "+", RowBox[List[RowBox[List["AsymptoticHypergeometricPFQRegularizedSeries", "[", "Exp", "]"]], "[", RowBox[List[RowBox[List["{", "a", "}"]], ",", RowBox[List["{", "b", "}"]], ",", RowBox[List["{", RowBox[List["z", ",", "ComplexInfinity", ",", "\[Infinity]"]], "}"]]]], "]"]]]], ")"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]] |
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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[{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] | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] | |
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