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 BesselY

 http://functions.wolfram.com/03.03.06.0047.01

 Input Form

 BesselY[n, z] == Subscript[F, Infinity][z, n] /; (Subscript[F, m][z, n] == (2/Pi) Log[z/2] BesselJ[n, z] - (1/Pi) Sum[((n - k - 1)!/k!) (z/2)^(2 k - n), {k, 0, n - 1}] - (1/Pi) (z/2)^n Sum[(((-1)^k (PolyGamma[k + 1] + PolyGamma[k + n + 1]))/ (k! (k + n)!)) (z/2)^(2 k), {k, 0, m}] == BesselY[n, z] - (((-1)^m 2^(-2 m - n - 1) z^(2 + n + 2 m))/ (Pi (m + 1)! (m + n + 1)!)) Log[z/2] HypergeometricPFQ[{1}, {2 + m, 2 + m + n}, -(z^2/4)] - (-1)^n MeijerG[{{n/2 + m + 1, n/2 + m + 1}, {-((n + 1)/2)}}, {{n/2 + m + 1, n/2 + m + 1}, {n/2, -(n/2), -((n + 1)/2)}}, z^2/4] && Element[n, Integers] && n >= 0) && Element[m, Integers] && m >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "n"]], "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["2", "\[Pi]"], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], RowBox[List["BesselJ", "[", RowBox[List["n", ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox["1", "\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]], " "]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "-", "n"]]]]]]]]], "-", RowBox[List[FractionBox["1", "\[Pi]"], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "n", "+", "1"]], "]"]]]], ")"]], " "]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["n", ",", "z"]], "]"]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "m"]], "-", "n", "-", "1"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "n", "+", RowBox[List["2", "m"]]]]], " "]], RowBox[List["\[Pi]", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n", "+", "1"]], ")"]], "!"]]]]], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", "n"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["n", "2"], "+", "m", "+", "1"]], ",", RowBox[List[FractionBox["n", "2"], "+", "m", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox[RowBox[List["n", "+", "1"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["n", "2"], "+", "m", "+", "1"]], ",", RowBox[List[FractionBox["n", "2"], "+", "m", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["n", "2"], ",", RowBox[List["-", FractionBox["n", "2"]]], ",", RowBox[List["-", FractionBox[RowBox[List["n", "+", "1"]], "2"]]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 Y n ( z ) F ( z , n ) /; ( F m ( z , n ) 2 π log ( z 2 ) J n ( z ) - 1 π k = 0 n - 1 ( - k + n - 1 ) ! ( z 2 ) 2 k - n k ! - 1 π ( z 2 ) n k = 0 m ( - 1 ) k ( - 2 log ( z 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + n + 1 ) ) k ! ( k + n ) ! ( z 2 ) 2 k Y n ( z ) - ( - 1 ) m 2 - 2 m - n - 1 z 2 m + n + 2 π ( m + 1 ) ! ( m + n + 1 ) ! log ( z 2 ) 1 F 2 ( 1 ; m + 2 , m + n + 2 ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["m", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", "n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - ( - 1 ) n G 3 , 5 2 , 2 ( z 2 4 m + n 2 + 1 , m + n 2 + 1 , - 1 2 ( n + 1 ) m + n 2 + 1 , m + n 2 + 1 , n 2 , - n 2 , - 1 2 ( n + 1 ) ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "5"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["m", "+", FractionBox["n", "2"], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", FractionBox["n", "2"], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List["m", "+", FractionBox["n", "2"], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", FractionBox["n", "2"], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["n", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["n", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] n ) m Condition BesselY n z Subscript F z n Subscript F m z n 2 -1 z 2 -1 BesselJ n z -1 1 -1 k 0 n -1 -1 k n -1 z 2 -1 2 k -1 n k -1 -1 1 -1 z 2 -1 n k 0 m -1 k -2 z 2 -1 PolyGamma k 1 PolyGamma k n 1 k k n -1 z 2 -1 2 k BesselY n z -1 -1 m 2 -2 m -1 n -1 z 2 m n 2 m 1 m n 1 -1 z 2 -1 HypergeometricPFQ 1 m 2 m n 2 -1 z 2 4 -1 -1 -1 n MeijerG m n 2 -1 1 m n 2 -1 1 -1 1 2 n 1 m n 2 -1 1 m n 2 -1 1 n 2 -1 -1 n 2 -1 -1 1 2 n 1 z 2 4 -1 n m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselY", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "n"]], "]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "n"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List["n", ",", "z"]], "]"]]]], "\[Pi]"], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "-", "n"]]]]], RowBox[List["k", "!"]]]]], "\[Pi]"], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "n", "+", "1"]], "]"]]]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]]]]]], "\[Pi]"]]], "\[Equal]", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["n", ",", "z"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "m"]], "-", "n", "-", "1"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "n", "+", RowBox[List["2", " ", "m"]]]]]]], ")"]], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "m"]], ",", RowBox[List["2", "+", "m", "+", "n"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n", "+", "1"]], ")"]], "!"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["n", "2"], "+", "m", "+", "1"]], ",", RowBox[List[FractionBox["n", "2"], "+", "m", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["n", "2"], "+", "m", "+", "1"]], ",", RowBox[List[FractionBox["n", "2"], "+", "m", "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["n", "2"], ",", RowBox[List["-", FractionBox["n", "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02