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 Hypergeometric0F1

 http://functions.wolfram.com/07.17.06.0024.01

 Input Form

 Hypergeometric0F1[b, z] \[Proportional] (Gamma[b]/Sqrt[Pi]) (-z)^((1 - 2 b)/4) (Cos[((2 b - 1)/4) Pi - 2 Sqrt[-z]] (Sum[Pochhammer[(3 - 2 b)/4, k] Pochhammer[(5 - 2 b)/4, k] Pochhammer[(2 b - 1)/4, k] (Pochhammer[(2 b + 1)/4, k]/ Pochhammer[1/2, k]) (1/(4 z))^k, {k, 0, n}] + O[1/z^(n + 1)]) + (((2 b - 1) (2 b - 3))/(16 Sqrt[-z])) Sin[((2 b - 1)/4) Pi - 2 Sqrt[-z]] (Sum[Pochhammer[(5 - 2 b)/4, k] Pochhammer[(7 - 2 b)/4, k] Pochhammer[(2 b + 1)/4, k] (Pochhammer[(2 b + 3)/4, k]/ Pochhammer[3/2, k]) (1/(4 z))^k, {k, 0, n}] + O[1/z^(n + 1)])) /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric0F1", "[", RowBox[List["b", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", "b", "]"]]]], SqrtBox["\[Pi]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], FractionBox[RowBox[List["1", "-", RowBox[List["2", "b"]]]], "4"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "b"]], "-", "1"]], "4"], " ", "\[Pi]"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Sum", "[", RowBox[List[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "-", RowBox[List["2", "b"]]]], "4"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["5", "-", RowBox[List["2", "b"]]]], "4"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "b"]], "-", "1"]], "4"], ",", "k"]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "b"]], "+", "1"]], "4"], ",", "k"]], "]"]], "/", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["4", " ", "z"]]], ")"]], "k"]]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "n"]], "}"]]]], "]"]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "b"]], "-", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["2", "b"]], "-", "3"]], ")"]]]], RowBox[List["16", " ", SqrtBox[RowBox[List["-", "z"]]]]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "b"]], "-", "1"]], "4"], " ", "\[Pi]"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["Sum", "[", RowBox[List[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["5", "-", RowBox[List["2", "b"]]]], "4"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["7", "-", RowBox[List["2", "b"]]]], "4"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "b"]], "+", "1"]], "4"], ",", "k"]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", "b"]], "+", "3"]], "4"], ",", "k"]], "]"]], "/", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]], SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["4", " ", "z"]]], ")"]], "k"]]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "n"]], "}"]]]], "]"]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 0 F 1 ( ; b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric0F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1] Γ ( b ) π ( - z ) 1 4 ( 1 - 2 b ) ( cos ( 1 4 ( 2 b - 1 ) π - 2 - z ) ( k = 0 n ( 1 4 ( 3 - 2 b ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 5 - 2 b ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 2 b - 1 ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "1"]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 2 b + 1 ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", "1"]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( 1 4 z ) k + O ( 1 z n + 1 ) ) + ( 2 b - 1 ) ( 2 b - 3 ) 16 - z sin ( 1 4 ( 2 b - 1 ) π - 2 - z ) ( k = 0 n ( 1 4 ( 5 - 2 b ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 7 - 2 b ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["7", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 2 b + 1 ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", "1"]], ")"]]]], ")"]], "k"], Pochhammer] ( 1 4 ( 2 b + 3 ) ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", "3"]], ")"]]]], ")"]], "k"], Pochhammer] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ( 1 4 z ) k + O ( 1 z n + 1 ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional Hypergeometric0F1 b z Gamma b 1 2 -1 -1 z 1 4 1 -1 2 b 1 4 2 b -1 -1 2 -1 z 1 2 k 0 n Pochhammer 1 4 3 -1 2 b k Pochhammer 1 4 5 -1 2 b k Pochhammer 1 4 2 b -1 k Pochhammer 1 4 2 b 1 k Pochhammer 1 2 k -1 1 4 z -1 k O 1 z n 1 -1 2 b -1 2 b -3 16 -1 z 1 2 -1 1 4 2 b -1 -1 2 -1 z 1 2 k 0 n Pochhammer 1 4 5 -1 2 b k Pochhammer 1 4 7 -1 2 b k Pochhammer 1 4 2 b 1 k Pochhammer 1 4 2 b 3 k Pochhammer 3 2 k -1 1 4 z -1 k O 1 z n 1 -1 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric0F1", "[", RowBox[List["b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "1"]], ")"]], " ", "\[Pi]"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "1"]], ")"]]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", "1"]], ")"]]]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["4", " ", "z"]]], ")"]], "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "3"]], ")"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "1"]], ")"]], " ", "\[Pi]"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["7", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", "1"]], ")"]]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", "3"]], ")"]]]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["4", " ", "z"]]], ")"]], "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]], RowBox[List["16", " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], ")"]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2007-05-02