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
 











HypergeometricU






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricU[a,b,z] > Specific values > For fixed z > For fixed z and a=-5/2





http://functions.wolfram.com/07.33.03.0239.01









  


  










Input Form





HypergeometricU[-(5/2), 6, z] == (1/(64 Sqrt[Pi] z^4)) (E^(z/2) (z (-360 + z (-495 + 2 z (-225 + 2 z (-1155 + 4 z (210 + z (-39 + 2 z)))))) BesselK[0, z/2] + (-1440 + z (-1980 + z (-1845 + 2 z (-825 + 2 z (-525 + 4 z (174 + z (-37 + 2 z))))))) BesselK[1, z/2]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["-", FractionBox["5", "2"]]], ",", "6", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["64", " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "4"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "360"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "495"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "225"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1155"]], "+", RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List["210", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "39"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["BesselK", "[", RowBox[List["0", ",", FractionBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1440"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1980"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1845"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "825"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "525"]], "+", RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List["174", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "37"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["BesselK", "[", RowBox[List["1", ",", FractionBox["z", "2"]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox[&quot;U&quot;, HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 5 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mn> 6 </mn> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 64 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> z </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 39 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 210 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1155 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 225 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 495 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 360 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> K </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 37 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 174 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 525 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 825 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1845 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1980 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1440 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> K </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricU </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 5 <sep /> 2 </cn> </apply> <cn type='integer'> 6 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -39 </cn> </apply> </apply> <cn type='integer'> 210 </cn> </apply> </apply> <cn type='integer'> -1155 </cn> </apply> </apply> <cn type='integer'> -225 </cn> </apply> </apply> <cn type='integer'> -495 </cn> </apply> </apply> <cn type='integer'> -360 </cn> </apply> <apply> <ci> BesselK </ci> <cn type='integer'> 0 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -37 </cn> </apply> </apply> <cn type='integer'> 174 </cn> </apply> </apply> <cn type='integer'> -525 </cn> </apply> </apply> <cn type='integer'> -825 </cn> </apply> </apply> <cn type='integer'> -1845 </cn> </apply> </apply> <cn type='integer'> -1980 </cn> </apply> </apply> <cn type='integer'> -1440 </cn> </apply> <apply> <ci> BesselK </ci> <cn type='integer'> 1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <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["HypergeometricU", "[", RowBox[List[RowBox[List["-", FractionBox["5", "2"]]], ",", "6", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "360"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "495"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "225"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1155"]], "+", RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List["210", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "39"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["BesselK", "[", RowBox[List["0", ",", FractionBox["z", "2"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1440"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1980"]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1845"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "825"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "525"]], "+", RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List["174", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "37"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["BesselK", "[", RowBox[List["1", ",", FractionBox["z", "2"]]], "]"]]]]]], ")"]]]], RowBox[List["64", " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "4"]]]]]]]]










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





2007-05-02