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 | | http://functions.wolfram.com/07.21.06.0008.01 | 
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 | | Hypergeometric1F1Regularized[a, b, z] \[Proportional] 
  (1/Gamma[a]) E^z z^(a - b) (1 + O[1/z]) + ((1/Gamma[b - a]) (1 + O[1/z]))/
    (-z)^a /; (Abs[z] -> Infinity) | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric1F1Regularized", "[", RowBox[List["a", ",", "b", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", "a", "]"]]], SuperscriptBox["\[ExponentialE]", "z"], SuperscriptBox["z", RowBox[List["a", "-", "b"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["b", "-", "a"]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]]]], "/;", 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>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <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[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["a", Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] </annotation>  </semantics>  <mo> ∝ </mo>  <mrow>  <mrow>  <mfrac>  <mn> 1 </mn>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mo> - </mo>  <mi> a </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> + </mo>  <mrow>  <mi> O </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mn> 1 </mn>  <mi> z </mi>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mtext>   </mtext>  <mo> + </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> a </mi>  <mo> ) </mo>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mi> z </mi>  </msup>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> + </mo>  <mrow>  <mi> O </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mfrac>  <mn> 1 </mn>  <mi> z </mi>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </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> Hypergeometric1F1Regularized </ci>  <ci> a </ci>  <ci> b </ci>  <ci> z </ci>  </apply>  <apply>  <plus />  <apply>  <times />  <apply>  <times />  <cn type='integer'> 1 </cn>  <apply>  <power />  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <ci> b </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> a </ci>  </apply>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> z </ci>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> a </ci>  </apply>  </apply>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <ci> O </ci>  <apply>  <times />  <cn type='integer'> 1 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <times />  <cn type='integer'> 1 </cn>  <apply>  <power />  <apply>  <ci> Gamma </ci>  <ci> a </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <power />  <exponentiale />  <ci> z </ci>  </apply>  <apply>  <power />  <ci> z </ci>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  </apply>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <ci> O </ci>  <apply>  <times />  <cn type='integer'> 1 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </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["Hypergeometric1F1Regularized", "[", RowBox[List["a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", SuperscriptBox["z", RowBox[List["a", "-", "b"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["Gamma", "[", "a", "]"]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["b", "-", "a"]], "]"]]]]], "/;", 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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 | | HypergeometricPFQRegularized[{},{b},z] |  | HypergeometricPFQRegularized[{a1,a2},{b1},z] |  | HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] |  |  | 
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