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 HypergeometricU

 http://functions.wolfram.com/07.33.06.0020.01

 Input Form

 HypergeometricU[a, b, z] \[Proportional] (1/Gamma[1 + a - b]) (2 I E^(I b Pi) Pi Floor[Arg[-x + z]/(2 Pi)] Hypergeometric1F1Regularized[a, b, x]) + HypergeometricU[a, b, x]/ E^(2 I b Pi Floor[Arg[-x + z]/(2 Pi)]) + a ((1/Gamma[1 + a - b]) (2 I E^(I b Pi) Pi Floor[Arg[-x + z]/(2 Pi)] Hypergeometric1F1Regularized[1 + a, 1 + b, x]) - HypergeometricU[1 + a, 1 + b, x]/E^(2 I b Pi Floor[Arg[-x + z]/(2 Pi)])) (z - x) + ((a (1 + a))/2) ((1/Gamma[1 + a - b]) (2 I E^(I b Pi) Pi Floor[Arg[-x + z]/(2 Pi)] Hypergeometric1F1Regularized[2 + a, 2 + b, x]) + HypergeometricU[2 + a, 2 + b, x]/ E^(2 I b Pi Floor[Arg[-x + z]/(2 Pi)])) (z - x)^2 + O[(z - x)^3] /; x < 0

 Standard Form

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 MathML Form

 U TagBox["U", HypergeometricU] ( a , b , z ) - 2 b π arg ( z - x ) 2 π U TagBox["U", HypergeometricU] ( a , b , x ) + 2 π Γ ( a - b + 1 ) b π arg ( z - x ) 2 π 1 F ~ 1 ( a ; b ; x ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["a", Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox["x", Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] + a ( 2 b π π arg ( z - x ) 2 π 1 F ~ 1 ( a + 1 ; b + 1 ; x ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["a", "+", "1"]], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["b", "+", "1"]], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox["x", Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] Γ ( a - b + 1 ) - - 2 b π arg ( z - x ) 2 π U TagBox["U", HypergeometricU] ( a + 1 , b + 1 , x ) ) ( z - x ) + 1 2 ( a ( a + 1 ) ) ( 2 b π π arg ( z - x ) 2 π 1 F ~ 1 ( a + 2 ; b + 2 ; x ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["a", "+", "2"]], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["b", "+", "2"]], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox["x", Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] Γ ( a - b + 1 ) + - 2 b π arg ( z - x ) 2 π U TagBox["U", HypergeometricU] ( a + 2 , b + 2 , x ) ) ( z - x ) 2 + O ( ( z - x ) 3 ) /; x < 0 Condition Proportional HypergeometricU a b z -2 b z -1 x 2 -1 HypergeometricU a b x 2 Gamma a -1 b 1 -1 b z -1 x 2 -1 Hypergeometric1F1Regularized a b x a 2 b z -1 x 2 -1 Hypergeometric1F1Regularized a 1 b 1 x Gamma a -1 b 1 -1 -1 -2 b z -1 x 2 -1 HypergeometricU a 1 b 1 x z -1 x 1 2 a a 1 2 b z -1 x 2 -1 Hypergeometric1F1Regularized a 2 b 2 x Gamma a -1 b 1 -1 -2 b z -1 x 2 -1 HypergeometricU a 2 b 2 x z -1 x 2 O z -1 x 3 x 0 [/itex]

 Rule Form

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

 2007-05-02