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/03.03.21.0070.01

 Input Form

 Integrate[z^(\[Alpha] - 1) BesselY[\[Nu] - 1, a z^r] BesselY[\[Nu], a z^r], z] == (-(1/2)) z^\[Alpha] Csc[Pi \[Nu]]^2 ((4^\[Nu] (a z^r)^(1 - 2 \[Nu]) HypergeometricPFQ[ {3/2 - \[Nu], 1/2 + \[Alpha]/(2 r) - \[Nu]}, {2 - 2 \[Nu], 2 - \[Nu], 3/2 + \[Alpha]/(2 r) - \[Nu]}, (-a^2) z^(2 r)])/ ((r + \[Alpha] - 2 r \[Nu]) Gamma[1 - \[Nu]] Gamma[2 - \[Nu]]) - (1/a) ((2 Cos[Pi \[Nu]] ((2^(1 - 2 \[Nu]) (a z^r)^(2 \[Nu]) Cos[Pi \[Nu]] HypergeometricPFQ[{1/2 + \[Nu], -(1/2) + \[Alpha]/(2 r) + \[Nu]}, {2 \[Nu], 1 + \[Nu], 1/2 + \[Alpha]/(2 r) + \[Nu]}, (-a^2) z^(2 r)])/((\[Alpha] + r (-1 + 2 \[Nu])) Gamma[\[Nu]] Gamma[1 + \[Nu]]) + (1/(Pi (r - \[Alpha]))) (HypergeometricPFQ[{1/2, -(1/2) + \[Alpha]/(2 r)}, {1/2 + \[Alpha]/(2 r), 1 - \[Nu], \[Nu]}, (-a^2) z^(2 r)] Sin[Pi \[Nu]])))/z^r) + (2 Cot[Pi \[Nu]] (Pi (r + \[Alpha]) (-1 + \[Nu]) \[Nu] BesselJ[1 - \[Nu], a z^r] BesselJ[\[Nu], a z^r] + a r z^r HypergeometricPFQ[{3/2, 1/2 + \[Alpha]/(2 r)}, {3/2 + \[Alpha]/(2 r), 2 - \[Nu], 1 + \[Nu]}, (-a^2) z^(2 r)] Sin[Pi \[Nu]]))/((-r + \[Alpha]) (r + \[Alpha]) Gamma[2 - \[Nu]] Gamma[1 + \[Nu]]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], 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 MathML Form

 z α - 1 Y ν - 1 ( a z r ) Y ν ( a z r ) z - 1 2 z α csc 2 ( π ν ) ( - 1 a ( 2 z - r cos ( π ν ) ( 2 1 - 2 ν ( a z r ) 2 ν cos ( π ν ) ( α + r ( 2 ν - 1 ) ) Γ ( ν ) Γ ( ν + 1 ) 2 F 3 ( ν + 1 2 , α 2 r + ν - 1 2 ; 2 ν , ν + 1 , α 2 r + ν + 1 2 ; - a 2 z 2 r ) 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["\[Nu]", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]", "-", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", " ", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", "\[Nu]", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + sin ( π ν ) π ( r - α ) 2 F 3 ( 1 2 , α 2 r - 1 2 ; α 2 r + 1 2 , 1 - ν , ν ; - a 2 z 2 r ) 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[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Nu]", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) ) + 4 ν ( a z r ) 1 - 2 ν ( - 2 ν r + r + α ) Γ ( 1 - ν ) Γ ( 2 - ν ) 2 F 3 ( 3 2 - ν , α 2 r - ν + 1 2 ; 2 - 2 ν , 2 - ν , α 2 r - ν + 3 2 ; - a 2 z 2 r ) 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[FractionBox["3", "2"], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", RowBox[List["2", " ", "\[Nu]"]]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", "\[Nu]", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 2 cot ( π ν ) ( α - r ) ( r + α ) Γ ( 2 - ν ) Γ ( ν + 1 ) ( a r 2 F 3 ( 3 2 , α 2 r + 1 2 ; α 2 r + 3 2 , 2 - ν , ν + 1 ; - a 2 z 2 r ) 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[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] sin ( π ν ) z r + π ( r + α ) ( ν - 1 ) ν J 1 - ν ( a z r ) J ν ( a z r ) ) ) z z α -1 BesselY ν -1 a z r BesselY ν a z r -1 1 2 z α ν 2 -1 1 a -1 2 z -1 r ν 2 1 -1 2 ν a z r 2 ν ν α r 2 ν -1 Gamma ν Gamma ν 1 -1 HypergeometricPFQ ν 1 2 α 2 r -1 ν -1 1 2 2 ν ν 1 α 2 r -1 ν 1 2 -1 a 2 z 2 r ν r -1 α -1 HypergeometricPFQ 1 2 α 2 r -1 -1 1 2 α 2 r -1 1 2 1 -1 ν ν -1 a 2 z 2 r 4 ν a z r 1 -1 2 ν -2 ν r r α Gamma 1 -1 ν Gamma 2 -1 ν -1 HypergeometricPFQ 3 2 -1 ν α 2 r -1 -1 ν 1 2 2 -1 2 ν 2 -1 ν α 2 r -1 -1 ν 3 2 -1 a 2 z 2 r 2 ν α -1 r r α Gamma 2 -1 ν Gamma ν 1 -1 a r HypergeometricPFQ 3 2 α 2 r -1 1 2 α 2 r -1 3 2 2 -1 ν ν 1 -1 a 2 z 2 r ν z r r α ν -1 ν BesselJ 1 -1 ν a z r BesselJ ν a z 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[RowBox[List["\[Nu]_", "-", "1"]], ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", 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 Date Added to functions.wolfram.com (modification date)

 2001-10-29

© 1998-2013 Wolfram Research, Inc.