Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
HermiteH






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HermiteH[nu,z] > Series representations > Asymptotic series expansions





http://functions.wolfram.com/07.01.06.0006.01









  


  










Input Form





HermiteH[\[Nu], z] \[Proportional] (-((Sin[\[Nu] Pi] (z^2)^(-\[Nu]/2 - 1))/(2 Sqrt[Pi]))) (E^z^2 (Sqrt[z^2] - z) Gamma[\[Nu] + 1] HypergeometricPFQ[ {(1 + \[Nu])/2, (2 + \[Nu])/2}, {}, 1/z^2] + 2^\[Nu] Sqrt[Pi] Sqrt[-z^2] (-z^4)^(\[Nu]/2) (Sqrt[-z^2] Csc[(\[Nu] Pi)/2] + z Sec[(\[Nu] Pi)/2]) HypergeometricPFQ[{-(\[Nu]/2), (1 - \[Nu])/2}, {}, -(1/z^2)]) /; (Abs[z] -> Infinity)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "2"]], "-", "1"]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]], " ", RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], "-", "z"]], ")"]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SqrtBox["\[Pi]"], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "4"]]], ")"]], RowBox[List["\[Nu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Csc", "[", FractionBox[RowBox[List["\[Nu]", " ", "\[Pi]"]], "2"], "]"]]]], "+", RowBox[List["z", " ", RowBox[List["Sec", "[", FractionBox[RowBox[List["\[Nu]", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[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> <mrow> <mrow> <msub> <mi> H </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 0 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mo> &#8202; </mo> <mo> ; </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;0&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;2&quot;]], &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[&quot;1&quot;, SuperscriptBox[&quot;z&quot;, &quot;2&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <msup> <mn> 2 </mn> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> sec </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 0 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mo> &#8202; </mo> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;0&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Nu]&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;1&quot;, SuperscriptBox[&quot;z&quot;, &quot;2&quot;]]]], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> HermiteH </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <sin /> <apply> <times /> <ci> &#957; </ci> <pi /> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <csc /> <apply> <times /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <sec /> <apply> <times /> <ci> &#957; </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]], " ", RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], "-", "z"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "4"]]], ")"]], RowBox[List["\[Nu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Csc", "[", FractionBox[RowBox[List["\[Nu]", " ", "\[Pi]"]], "2"], "]"]]]], "+", RowBox[List["z", " ", RowBox[List["Sec", "[", FractionBox[RowBox[List["\[Nu]", " ", "\[Pi]"]], "2"], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2001-10-29