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; }

 AiryBi

 http://functions.wolfram.com/03.06.21.0067.01

 Input Form

 Integrate[z^(\[Alpha] - 1) BesselK[\[Nu], (2/3) (a z^r)^(3/2)] AiryBi[a z^r], z] == (1/2) Pi Csc[Pi \[Nu]] (-((1/r) (2^(2/3 + \[Nu]) 3^(-(7/6) - \[Nu]) Sqrt[Pi] z^\[Alpha] ((a z^r)^(3/2))^\[Nu] MeijerG[{{(1/6) (1 - 3 \[Nu]), (1/6) (4 - 3 \[Nu]), 1 - \[Alpha]/(3 r) - \[Nu]/2}, {1/6, 2/3}}, {{0, 1/3}, {1/6, 2/3, 1/3 - \[Nu], -\[Nu], -((2 \[Alpha] + 3 r \[Nu])/(6 r))}}, (2/3)^(2/3) a z^r, 1/3])) + (1/r) ((2^(2/3 - \[Nu]) 3^(-(7/6) + \[Nu]) Sqrt[Pi] z^\[Alpha] MeijerG[{{1 - \[Alpha]/(3 r) + \[Nu]/2, (1/6) (1 + 3 \[Nu]), (1/6) (4 + 3 \[Nu])}, {1/6, 2/3}}, {{0, 1/3}, {1/6, 2/3, -(\[Alpha]/(3 r)) + \[Nu]/2, \[Nu], 1/3 + \[Nu]}}, (2/3)^(2/3) a z^r, 1/3])/((a z^r)^(3/2))^\[Nu]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], RowBox[List["AiryBi", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", "r"], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["2", "3"], "+", "\[Nu]"]]], " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", FractionBox["7", "6"]]], "-", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], ")"]], "\[Nu]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"], ",", RowBox[List[FractionBox["1", "3"], "-", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", "\[Alpha]"]], "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], ")"]]]]]], "+", RowBox[List[FractionBox["1", "r"], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["2", "3"], "-", "\[Nu]"]]], " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", FractionBox["7", "6"]]], "+", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"], ",", RowBox[List[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], "+", FractionBox["\[Nu]", "2"]]], ",", "\[Nu]", ",", RowBox[List[FractionBox["1", "3"], "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 K ν ( 2 3 ( a z r ) 3 / 2 ) Bi ( a z r ) z 1 2 π csc ( π ν ) ( 2 2 3 - ν 3 ν - 7 6 π z α ( ( a z r ) 3 / 2 ) - ν r G 5 , 7 2 , 3 ( ( 2 3 ) 2 / 3 a z r , 1 3 - α 3 r + ν 2 + 1 , 1 6 ( 3 ν + 1 ) , 1 6 ( 3 ν + 4 ) , 1 6 , 2 3 0 , 1 3 , 1 6 , 2 3 , ν 2 - α 3 r , ν , ν + 1 3 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["5", ",", "7"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], "+", FractionBox["\[Nu]", "2"], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "+", "4"]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Nu]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["1", "3"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] - 2 ν + 2 3 3 - ν - 7 6 π z α ( ( a z r ) 3 / 2 ) ν r G 5 , 7 2 , 3 ( ( 2 3 ) 2 / 3 a z r , 1 3 1 6 ( 1 - 3 ν ) , 1 6 ( 4 - 3 ν ) , - α 3 r - ν 2 + 1 , 1 6 , 2 3 0 , 1 3 , 1 6 , 2 3 , 1 3 - ν , - ν , - 2 α + 3 r ν 6 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["5", ",", "7"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], "-", FractionBox["\[Nu]", "2"], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "3"], "-", "\[Nu]"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "\[Nu]"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", "\[Alpha]"]], "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) z z α -1 BesselK ν 2 3 a z r 3 2 AiryBi a z r 1 2 ν 2 2 3 -1 ν 3 ν -1 7 6 1 2 z α a z r 3 2 -1 ν r -1 MeijerG -1 α 3 r -1 ν 2 -1 1 1 6 3 ν 1 1 6 3 ν 4 1 6 2 3 0 1 3 1 6 2 3 ν 2 -1 -1 α 3 r -1 ν ν 1 3 2 3 2 3 a z r 1 3 -1 2 ν 2 3 3 -1 ν -1 7 6 1 2 z α a z r 3 2 ν r -1 MeijerG 1 6 1 -1 3 ν 1 6 4 -1 3 ν -1 α 3 r -1 -1 ν 2 -1 1 1 6 2 3 0 1 3 1 6 2 3 1 3 -1 ν -1 ν -1 2 α 3 r ν 6 r -1 2 3 2 3 a z r 1 3 [/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["BesselK", "[", RowBox[List["\[Nu]_", ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], " ", RowBox[List["AiryBi", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["2", "3"], "+", "\[Nu]"]]], " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", FractionBox["7", "6"]]], "-", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], ")"]], "\[Nu]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]], "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"], ",", RowBox[List[FractionBox["1", "3"], "-", "\[Nu]"]], ",", RowBox[List["-", "\[Nu]"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", "\[Alpha]"]], "+", RowBox[List["3", " ", "r", " ", "\[Nu]"]]]], RowBox[List["6", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], "r"]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["2", "3"], "-", "\[Nu]"]]], " ", SuperscriptBox["3", RowBox[List[RowBox[List["-", FractionBox["7", "6"]]], "+", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List["4", "+", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"], ",", RowBox[List[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], "+", FractionBox["\[Nu]", "2"]]], ",", "\[Nu]", ",", RowBox[List[FractionBox["1", "3"], "+", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], "r"]]], ")"]]]]]]]]

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

 2001-10-29