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/03.03.21.0040.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Sin[b + a z^r] BesselY[\[Nu], a z^r], z] == (z^\[Alpha] (-((4^\[Nu] a z^r Cos[b] Csc[Pi \[Nu]] HypergeometricPFQ[ {3/4 - \[Nu]/2, 5/4 - \[Nu]/2, 1/2 + \[Alpha]/(2 r) - \[Nu]/2}, {3/2, 1 - \[Nu], 3/2 - \[Nu], 3/2 + \[Alpha]/(2 r) - \[Nu]/2}, (-a^2) z^(2 r)])/((r + \[Alpha] - r \[Nu]) Gamma[1 - \[Nu]])) + ((a z^r)^(1 + 2 \[Nu]) Cos[b] Cot[Pi \[Nu]] HypergeometricPFQ[ {3/4 + \[Nu]/2, 5/4 + \[Nu]/2, 1/2 + \[Alpha]/(2 r) + \[Nu]/2}, {3/2, 3/2 + \[Alpha]/(2 r) + \[Nu]/2, 1 + \[Nu], 3/2 + \[Nu]}, (-a^2) z^(2 r)])/((r + \[Alpha] + r \[Nu]) Gamma[1 + \[Nu]]) + (-((1/((\[Alpha] - r \[Nu]) Gamma[1 - \[Nu]])) (4^\[Nu] Csc[Pi \[Nu]] HypergeometricPFQ[{1/4 - \[Nu]/2, 3/4 - \[Nu]/2, \[Alpha]/(2 r) - \[Nu]/2}, {1/2, 1/2 - \[Nu], 1 - \[Nu], 1 + \[Alpha]/(2 r) - \[Nu]/2}, (-a^2) z^(2 r)])) + (1/((\[Alpha] + r \[Nu]) Gamma[1 + \[Nu]])) ((a z^r)^(2 \[Nu]) Cot[Pi \[Nu]] HypergeometricPFQ[{1/4 + \[Nu]/2, 3/4 + \[Nu]/2, \[Alpha]/(2 r) + \[Nu]/2}, {1/2, 1 + \[Alpha]/(2 r) + \[Nu]/2, 1/2 + \[Nu], 1 + \[Nu]}, (-a^2) z^(2 r)])) Sin[b]))/ (2^\[Nu] (a z^r)^\[Nu])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Sin", "[", RowBox[List["b", "+", 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[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["4", "\[Nu]"], " ", "a", " ", SuperscriptBox["z", "r"], " ", RowBox[List["Cos", "[", "b", "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", 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RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", RowBox[List["Cos", "[", "b", "]"]], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " 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"4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]]]], "]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sin", "[", "b", "]"]]]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 sin ( a z r + b ) Y ν ( a z r ) z 2 - ν z α ( a z r ) - ν ( ( a z r ) 2 ν + 1 cos ( b ) cot ( π ν ) ( ν r + r + α ) Γ ( ν + 1 ) 3 F 4 ( ν 2 + 3 4 , ν 2 + 5 4 , α 2 r + ν 2 + 1 2 ; 3 2 , α 2 r + ν 2 + 3 2 , ν + 1 , ν + 3 2 ; - a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"], "+", FractionBox["1", "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[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"], "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[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] + ( ( a z r ) 2 ν cot ( π ν ) ( α + r ν ) Γ ( ν + 1 ) 3 F 4 ( ν 2 + 1 4 , ν 2 + 3 4 , α 2 r + ν 2 ; 1 2 , α 2 r + ν 2 + 1 , ν + 1 2 , ν + 1 ; - a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], 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[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "+", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], 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] - 4 ν csc ( π ν ) ( α - r ν ) Γ ( 1 - ν ) 3 F 4 ( 1 4 - ν 2 , 3 4 - ν 2 , α 2 r - ν 2 ; 1 2 , 1 2 - ν , 1 - ν , α 2 r - ν 2 + 1 ; - a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]], 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[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", FractionBox["\[Nu]", "2"], "+", "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 ( b ) - 4 ν a z r cos ( b ) csc ( π ν ) ( - ν r + r + α ) Γ ( 1 - ν ) 3 F 4 ( 3 4 - ν 2 , 5 4 - ν 2 , α 2 r - ν 2 + 1 2 ; 3 2 , 1 - ν , 3 2 - ν , α 2 r - ν 2 + 3 2 ; - a 2 z 2 r ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", FractionBox["\[Nu]", "2"], "+", FractionBox["1", "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["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "2"], "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", RowBox[List["2", " ", "r"]]], "-", FractionBox["\[Nu]", "2"], "+", 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] ) z z α -1 a z r b BesselY ν a z r 2 -1 ν z α a z r -1 ν a z r 2 ν 1 b ν ν r r α Gamma ν 1 -1 HypergeometricPFQ ν 2 -1 3 4 ν 2 -1 5 4 α 2 r -1 ν 2 -1 1 2 3 2 α 2 r -1 ν 2 -1 3 2 ν 1 ν 3 2 -1 a 2 z 2 r a z r 2 ν ν α r ν Gamma ν 1 -1 HypergeometricPFQ ν 2 -1 1 4 ν 2 -1 3 4 α 2 r -1 ν 2 -1 1 2 α 2 r -1 ν 2 -1 1 ν 1 2 ν 1 -1 a 2 z 2 r -1 4 ν ν α -1 r ν Gamma 1 -1 ν -1 HypergeometricPFQ 1 4 -1 ν 2 -1 3 4 -1 ν 2 -1 α 2 r -1 -1 ν 2 -1 1 2 1 2 -1 ν 1 -1 ν α 2 r -1 -1 ν 2 -1 1 -1 a 2 z 2 r b -1 4 ν a z r b ν -1 ν r r α Gamma 1 -1 ν -1 HypergeometricPFQ 3 4 -1 ν 2 -1 5 4 -1 ν 2 -1 α 2 r -1 -1 ν 2 -1 1 2 3 2 1 -1 ν 3 2 -1 ν α 2 r -1 -1 ν 2 -1 3 2 -1 a 2 z 2 r [/itex]

 Rule Form

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

 2001-10-29