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http://functions.wolfram.com/07.33.06.0006.01
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HypergeometricU[a, 0, z] \[Proportional] (1/Gamma[a + 1])
(1 + a (Log[z] - 1 + 2 EulerGamma + PolyGamma[1 + a]) z (1 + O[z])) /;
(z -> 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricU", "[", RowBox[List["a", ",", "0", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", "1", "+", RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a"]], "]"]]]], ")"]], "z", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "z", "]"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]
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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> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox["U", HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> HypergeometricU </ci> <ci> a </ci> <cn type='integer'> 0 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> <apply> <plus /> <apply> <ln /> <ci> z </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <eulergamma /> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricU", "[", RowBox[List["a_", ",", "0", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["1", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", "1", "+", RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a"]], "]"]]]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "z", "]"]]]], ")"]]]]]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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