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 HypergeometricU

 http://functions.wolfram.com/07.33.06.0016.01

 Input Form

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

 Standard Form

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 MathML Form

 U TagBox["U", HypergeometricU] ( a , b , z ) ( 1 z 0 ) - b arg ( z - z 0 ) 2 π z 0 - b arg ( z - z 0 ) 2 π U TagBox["U", HypergeometricU] ( a , b , z 0 ) + b π ( 2 π ) Γ ( a - b + 1 ) arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π 1 F ~ 1 ( a ; b ; z 0 ) 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[SubscriptBox["z", "0"], Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] + a ( ( 2 π ) b π Γ ( a - b + 1 ) arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π 1 F ~ 1 ( a + 1 ; b + 1 ; z 0 ) 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[SubscriptBox["z", "0"], Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] - ( 1 z 0 ) - b arg ( z - z 0 ) 2 π z 0 - b arg ( z - z 0 ) 2 π U TagBox["U", HypergeometricU] ( a + 1 , b + 1 , z 0 ) ) ( z - z 0 ) + a ( a + 1 ) 2 ( ( 1 z 0 ) - b arg ( z - z 0 ) 2 π z 0 - b arg ( z - z 0 ) 2 π U TagBox["U", HypergeometricU] ( a + 2 , b + 2 , z 0 ) + b π 2 π Γ ( a - b + 1 ) arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π 1 F ~ 1 ( a + 2 ; b + 2 ; z 0 ) 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[SubscriptBox["z", "0"], Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] ) ( z - z 0 ) 2 + O ( ( z - z 0 ) 3 ) Proportional HypergeometricU a b z 1 Subscript z 0 -1 -1 b z -1 Subscript z 0 2 -1 Subscript z 0 -1 b z -1 Subscript z 0 2 -1 HypergeometricU a b Subscript z 0 b 2 Gamma a -1 b 1 -1 z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 Hypergeometric1F1Regularized a b Subscript z 0 a 2 b Gamma a -1 b 1 -1 z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 Hypergeometric1F1Regularized a 1 b 1 Subscript z 0 -1 1 Subscript z 0 -1 -1 b z -1 Subscript z 0 2 -1 Subscript z 0 -1 b z -1 Subscript z 0 2 -1 HypergeometricU a 1 b 1 Subscript z 0 z -1 Subscript z 0 a a 1 2 -1 1 Subscript z 0 -1 -1 b z -1 Subscript z 0 2 -1 Subscript z 0 -1 b z -1 Subscript z 0 2 -1 HypergeometricU a 2 b 2 Subscript z 0 b 2 Gamma a -1 b 1 -1 z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 Hypergeometric1F1Regularized a 2 b 2 Subscript z 0 z -1 Subscript z 0 2 O z -1 Subscript z 0 3 [/itex]

 Rule Form

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

 2007-05-02