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/03.03.21.0020.01

 Input Form

 Integrate[E^(I a z^r) BesselY[\[Nu], a z^r], z] == (2^\[Nu] z Csc[Pi \[Nu]] HypergeometricPFQ[{1/2 - \[Nu], 1/r - \[Nu]}, {1 - 2 \[Nu], 1 + 1/r - \[Nu]}, 2 I a z^r])/(a z^r)^\[Nu]/ ((-1 + r \[Nu]) Gamma[1 - \[Nu]]) + (z (a z^r)^\[Nu] Cot[Pi \[Nu]] HypergeometricPFQ[ {1/2 + \[Nu], 1/r + \[Nu]}, {1 + 1/r + \[Nu], 1 + 2 \[Nu]}, 2 I a z^r])/ 2^\[Nu]/((1 + r \[Nu]) Gamma[1 + \[Nu]])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "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[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "r"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["1", "+", FractionBox["1", "r"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], "\[Nu]"], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "r"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["1", "r"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["r", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]], ")"]]]]]]]]]]

 MathML Form

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

 Rule Form

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

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

 2001-10-29