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 HypergeometricU

 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

 U ( 0 , 1 , 0 ) TagBox[SuperscriptBox["U", RowBox[List["(", RowBox[List["0", ",", "1", ",", "0"]], ")"]]], HypergeometricU] ( a , b , z ) Γ ( 1 - b ) ( ψ TagBox["\[Psi]", PolyGamma] ( a - b + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( 1 - b ) ) Γ ( a - b + 1 ) 1 F 1 ( a ; b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["a", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] - Γ ( b - 1 ) ( log ( z ) - ψ TagBox["\[Psi]", PolyGamma] ( b - 1 ) ) Γ ( a ) z 1 - b 1 F 1 ( a - b + 1 ; 2 - b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["a", "-", "b", "+", "1"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", "b"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] + a z Γ ( - b ) b Γ ( a - b + 1 ) F 2 0 1 1 1 2 ( a + 1 ; 1 ; 1 , b ; 2 , b + 1 ; ; b + 1 ; z , z ) + z 2 - b Γ ( b - 1 ) ( b - 2 ) 2 Γ ( a ) ( ( a - b + 1 ) F 2 0 1 1 1 2 ( a - b + 2 ; 1 ; 1 , 2 - b ; 2 , 3 - b ; ; 3 - b ; z , z ) + ( b - 2 ) F 2 0 1 1 1 2 ( a - b + 2 ; 1 ; 1 , a - b + 1 ; 2 , 3 - b ; ; a - b + 2 ; z , z ) ) /; b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] U ( 0 , 1 , 0 ) TagBox[SuperscriptBox["U", RowBox[List["(", RowBox[List["0", ",", "1", ",", "0"]], ")"]]], HypergeometricU] ( a , b , z ) Γ ( 1 - b ) ( ψ TagBox["\[Psi]", PolyGamma] ( a - b + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( 1 - b ) ) Γ ( a - b + 1 ) 1 F 1 ( a ; b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["a", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] - Γ ( b - 1 ) ( log ( z ) - ψ TagBox["\[Psi]", PolyGamma] ( b - 1 ) ) Γ ( a ) z 1 - b 1 F 1 ( a - b + 1 ; 2 - b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["a", "-", "b", "+", "1"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", "b"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] + a z Γ ( - b ) b Γ ( a - b + 1 ) F 2 0 1 1 1 2 ( a + 1 ; 1 ; 1 , b ; 2 , b + 1 ; ; b + 1 ; z , z ) + z 2 - b Γ ( b - 1 ) ( b - 2 ) 2 Γ ( a ) ( ( a - b + 1 ) F 2 0 1 1 1 2 ( a - b + 2 ; 1 ; 1 , 2 - b ; 2 , 3 - b ; ; 3 - b ; z , z ) + ( b - 2 ) F 2 0 1 1 1 2 ( a - b + 2 ; 1 ; 1 , a - b + 1 ; 2 , 3 - b ; ; a - b + 2 ; z , z ) ) /; b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] [/itex]

 Rule Form

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

 2001-10-29