html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 BesselY

 http://functions.wolfram.com/03.03.21.0016.01

 Input Form

 Integrate[z^(\[Alpha] - 1) BesselY[\[Nu], a z^r], z] == (1/r) ((2^(-1 - \[Nu]) z^\[Alpha] Csc[Pi \[Nu]] ((-4^\[Nu]) Gamma[(\[Alpha] - r \[Nu])/(2 r)] HypergeometricPFQRegularized[{(\[Alpha] - r \[Nu])/(2 r)}, {1 - \[Nu], (1/2) (2 + \[Alpha]/r - \[Nu])}, (-(1/4)) a^2 z^(2 r)] + (a z^r)^(2 \[Nu]) Cos[Pi \[Nu]] Gamma[(\[Alpha] + r \[Nu])/(2 r)] HypergeometricPFQRegularized[{(\[Alpha] + r \[Nu])/(2 r)}, {1 + \[Nu], (\[Alpha] + r (2 + \[Nu]))/(2 r)}, (-(1/4)) a^2 z^(2 r)]))/ (a z^r)^\[Nu])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "r"], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["4", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "-", RowBox[List["r", " ", "\[Nu]"]]]], RowBox[List["2", " ", "r"]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["\[Alpha]", "-", RowBox[List["r", " ", "\[Nu]"]]]], RowBox[List["2", " ", "r"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox["\[Alpha]", "r"], "-", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], RowBox[List["2", " ", "r"]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], RowBox[List["2", " ", "r"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", "r"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 Y ν ( a z r ) z 1 r ( 2 - ν - 1 z α ( a z r ) - ν csc ( π ν ) ( ( a z r ) 2 ν cos ( π ν ) Γ ( α + r ν 2 r ) 1 F ~ 2 ( α + r ν 2 r ; ν + 1 , α + r ( ν + 2 ) 2 r ; - 1 4 a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], RowBox[List["2", " ", "r"]]], HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "2"]], ")"]]]]]], RowBox[List["2", " ", "r"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] - 4 ν Γ ( α - r ν 2 r ) 1 F ~ 2 ( α - r ν 2 r ; 1 - ν , 1 2 ( α r - ν + 2 ) ; - 1 4 a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[FractionBox[RowBox[List["\[Alpha]", "-", RowBox[List["r", " ", "\[Nu]"]]]], RowBox[List["2", " ", "r"]]], HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["\[Alpha]", "r"], "-", "\[Nu]", "+", "2"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) ) z z α -1 BesselY ν a z r 1 r -1 2 -1 ν -1 z α a z r -1 ν ν a z r 2 ν ν Gamma α r ν 2 r -1 HypergeometricPFQRegularized α r ν 2 r -1 ν 1 α r ν 2 2 r -1 -1 1 4 a 2 z 2 r -1 4 ν Gamma α -1 r ν 2 r -1 HypergeometricPFQRegularized α -1 r ν 2 r -1 1 -1 ν 1 2 α r -1 -1 ν 2 -1 1 4 a 2 z 2 r [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["4", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "-", RowBox[List["r", " ", "\[Nu]"]]]], RowBox[List["2", " ", "r"]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["\[Alpha]", "-", RowBox[List["r", " ", "\[Nu]"]]]], RowBox[List["2", " ", "r"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", FractionBox["\[Alpha]", "r"], "-", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], RowBox[List["2", " ", "r"]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", "\[Nu]"]]]], RowBox[List["2", " ", "r"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox[RowBox[List["\[Alpha]", "+", RowBox[List["r", " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]]]], RowBox[List["2", " ", "r"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]]]], ")"]]]], "r"]]]]]

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

 2001-10-29