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variants of this functions
Hypergeometric0F1




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric0F1[b,z] > Limit representations





http://functions.wolfram.com/07.17.09.0002.01









  


  








Input Form





Hypergeometric0F1[b, z] == Gamma[b] Limit[(1/n^(b - 1)) LaguerreL[n, b - 1, -(z/n)], n -> Infinity]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric0F1", "[", RowBox[List["b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Limit", "[", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox["n", RowBox[List["b", "-", "1"]]]], RowBox[List["LaguerreL", "[", RowBox[List["n", ",", RowBox[List["b", "-", "1"]], ",", RowBox[List["-", FractionBox["z", "n"]]]]], "]"]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]]]








MathML Form







<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> &#8202; </mo> <mn> 0 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mo> &#8202; </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[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;0&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;b&quot;, Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, Hypergeometric0F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> n </mi> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mi> n </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Hypergeometric0F1 </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> n </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric0F1", "[", RowBox[List["b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List["LaguerreL", "[", RowBox[List["n", ",", RowBox[List["b", "-", "1"]], ",", RowBox[List["-", FractionBox["z", "n"]]]]], "]"]], SuperscriptBox["n", RowBox[List["b", "-", "1"]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29