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 HypergeometricU

 http://functions.wolfram.com/07.33.16.0002.01

 Input Form

 HypergeometricU[a, b, -z] HypergeometricU[a, b, z] == ((Pi (b - 2 a) z^(2 - b/2))/((-z)^(b/2) (b (b - 2) Gamma[a] Gamma[1 + a - b]))) Csc[(b Pi)/2] HypergeometricPFQ[ {1 + a - b/2, 1 - a + b/2}, {3/2, 2 - b/2, 1 + b/2}, z^2/4] - ((Pi z^((3 - b)/2))/((-z)^((b + 1)/2) ((b - 1) Gamma[a] Gamma[1 + a - b]))) Sec[(b Pi)/2] HypergeometricPFQ[{(1 - b)/2 + a, (1 + b)/2 - a}, {1/2, (3 - b)/2, (1 + b)/2}, z^2/4] - ((z^(2 - b) Gamma[b - 1]^2)/((-z)^b Gamma[a]^2)) HypergeometricPFQ[{1 - a, 1 + a - b}, {2 - b, 1 - b/2, (3 - b)/2}, z^2/4] + (Gamma[1 - b]^2/Gamma[1 + a - b]^2) HypergeometricPFQ[{a, -a + b}, {(1 + b)/2, b/2, b}, z^2/4]

 Standard Form

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 MathML Form

 U TagBox["U", HypergeometricU] ( a , b , - z ) U TagBox["U", HypergeometricU] ( a , b , z ) π ( b - 2 a ) ( - z ) - b 2 z 2 - b 2 b ( b - 2 ) Γ ( a ) Γ ( a - b + 1 ) csc ( b π 2 ) 2 F 3 ( a - b 2 + 1 , 1 - a + b 2 ; 3 2 , 2 - b 2 , b 2 + 1 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "-", FractionBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "a", "+", FractionBox["b", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", FractionBox["b", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - Γ ( b - 1 ) 2 ( - z ) - b z 2 - b Γ ( a ) 2 2 F 3 ( 1 - a , a - b + 1 ; 2 - b , 1 - b 2 , 3 - b 2 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "a"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", "b", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", "b"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["b", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["3", "-", "b"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + Γ ( 1 - b ) 2 Γ ( a - b + 1 ) 2 2 F 3 ( a , b - a ; b + 1 2 , b 2 , b ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "a"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["b", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - π ( - z ) - b + 1 2 z 3 - b 2 ( b - 1 ) Γ ( a ) Γ ( a - b + 1 ) sec ( b π 2 ) 2 F 3 ( a + 1 - b 2 , b + 1 2 - a ; 1 2 , 3 - b 2 , b + 1 2 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", FractionBox[RowBox[List["1", "-", "b"]], "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["b", "+", "1"]], "2"], "-", "a"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["3", "-", "b"]], "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "+", "1"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] HypergeometricU a b -1 z HypergeometricU a b z b -1 2 a -1 z -1 b 2 -1 z 2 -1 b 2 -1 b b -2 Gamma a Gamma a -1 b 1 -1 b 2 -1 HypergeometricPFQ a -1 b 2 -1 1 1 -1 a b 2 -1 3 2 2 -1 b 2 -1 b 2 -1 1 z 2 4 -1 -1 Gamma b -1 2 -1 z -1 b z 2 -1 b Gamma a 2 -1 HypergeometricPFQ 1 -1 a a -1 b 1 2 -1 b 1 -1 b 2 -1 3 -1 b 2 -1 z 2 4 -1 Gamma 1 -1 b 2 Gamma a -1 b 1 2 -1 HypergeometricPFQ a b -1 a b 1 2 -1 b 2 -1 b z 2 4 -1 -1 -1 z -1 b 1 2 -1 z 3 -1 b 2 -1 b -1 Gamma a Gamma a -1 b 1 -1 b 2 -1 HypergeometricPFQ a 1 -1 b 2 -1 b 1 2 -1 -1 a 1 2 3 -1 b 2 -1 b 1 2 -1 z 2 4 -1 [/itex]

 Rule Form

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

 2001-10-29