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HypergeometricU




Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricU[a,b,z] > Differentiation > Low-order differentiation > With respect to b





http://functions.wolfram.com/07.33.20.0004.01









  


  








Input Form





Derivative[0, 1, 0][HypergeometricU][a, b, z] == ((Gamma[1 - b] (PolyGamma[1 + a - b] - PolyGamma[1 - b]))/Gamma[1 + a - b]) Hypergeometric1F1[a, b, z] - ((z^(1 - b) Gamma[-1 + b])/Gamma[a]) (Log[z] - PolyGamma[b - 1]) Hypergeometric1F1[1 + a - b, 2 - b, z] + ((a z Gamma[-b])/(b Gamma[1 + a - b])) HypergeometricPFQ[ {{1 + a}, {1}, {1, b}}, {{2, 1 + b}, {}, {1 + b}}, z, z] + ((z^(2 - b) Gamma[b - 1])/((b - 2)^2 Gamma[a])) ((1 + a - b) HypergeometricPFQ[{{2 + a - b}, {1}, {1, 2 - b}}, {{2, 3 - b}, {}, {3 - b}}, z, z] + (b - 2) HypergeometricPFQ[{{2 + a - b}, {1}, {1, 1 + a - b}}, {{2, 3 - b}, {}, {2 + a - b}}, z, z]) /; !Element[b, Integers]








Standard Form





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MathML Form







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</mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <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;1&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;b&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, &quot;b&quot;]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, Hypergeometric1F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; 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</mo> <mn> 0 </mn> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> &#8290; </mo> <mn> 1 </mn> <mo> &#8290; </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mn> 1 </mn> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 2 </mn> <mo> , </mo> <mrow> <mrow> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ; </mo> </mrow> <mo> ; </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> F </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mn> 0 </mn> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> &#8290; </mo> <mn> 1 </mn> <mo> &#8290; </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mn> 1 </mn> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 2 </mn> <mo> , </mo> <mrow> <mrow> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ; </mo> </mrow> <mo> ; </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> b </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29