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/03.06.21.0069.01

 Input Form

 Integrate[z^(\[Alpha] - 1) BesselK[1, (2/3) (a z^r)^(3/2)] AiryBi[a z^r], z] == (1/(2^(1/3) 3^(1/6))) (Sqrt[Pi] (-((1/(6 Pi^3 r)) (z^\[Alpha] MeijerG[{{1/6, 2/3, 1 - \[Alpha]/(3 r)}, {}}, {{-(1/2), -(1/6), 1/2, 5/6}, {-(\[Alpha]/(3 r))}}, (2/3)^(2/3) a z^r, 1/3])) + (1/(6 Pi r)) (z^\[Alpha] MeijerG[{{1/6, 2/3, 1 - \[Alpha]/(3 r)}, {-(1/12), 5/12}}, {{-(1/2), -(1/6), 1/2, 5/6}, {-(1/12), 5/12, -(\[Alpha]/(3 r))}}, (2/3)^(2/3) a z^r, 1/3]) - (1/(3 r)) (2 z^\[Alpha] ((3/2) Log[a z^r] - Log[(a z^r)^(3/2)]) MeijerG[{{1/6, 2/3, 1 - \[Alpha]/(3 r)}, {2/3, 7/6}}, {{1/2, 5/6}, {-(1/2), -(1/6), 2/3, 7/6, -(\[Alpha]/(3 r))}}, (2/3)^(2/3) a z^r, 1/3])))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["BesselK", "[", RowBox[List["1", ",", 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", RowBox[List[SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["1", "/", "6"]]]]]], RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", RowBox[List["6", " ", SuperscriptBox["\[Pi]", "3"], " ", "r"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox["1", "2"], ",", FractionBox["5", "6"]]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], ")"]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List["6", " ", "\[Pi]", " ", "r"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "12"]]], ",", FractionBox["5", "12"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox["1", "2"], ",", FractionBox["5", "6"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "12"]]], ",", FractionBox["5", "12"], ",", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["3", " ", "r"]]], RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["3", "2"], " ", RowBox[List["Log", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "-", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["2", "3"], ",", FractionBox["7", "6"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["5", "6"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox["2", "3"], ",", FractionBox["7", "6"], ",", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]]]], "}"]], ",", 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 1 ( 2 3 ( a z r ) 3 / 2 ) Bi ( a z r ) z 1 2 3 3 6 ( π ( - z α 6 π 3 r G 3 , 5 4 , 3 ( ( 2 3 ) 2 / 3 a z r , 1 3 1 6 , 2 3 , 1 - α 3 r - 1 2 , - 1 6 , 1 2 , 5 6 , - α 3 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "5"]], RowBox[List["4", ",", "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[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List["-", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "6"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] - 2 z α ( 3 2 log ( a z r ) - log ( ( a z r ) 3 / 2 ) ) 3 r G 5 , 7 2 , 3 ( ( 2 3 ) 2 / 3 a z r , 1 3 1 6 , 2 3 , 1 - α 3 r , 2 3 , 7 6 1 2 , 5 6 , - 1 2 , - 1 6 , 2 3 , 7 6 , - α 3 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[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["7", "6"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "6"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["7", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + z α 6 π r G 5 , 7 4 , 3 ( ( 2 3 ) 2 / 3 a z r , 1 3 1 6 , 2 3 , 1 - α 3 r , - 1 12 , 5 12 - 1 2 , - 1 6 , 1 2 , 5 6 , - 1 12 , 5 12 , - α 3 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["5", ",", "7"]], RowBox[List["4", ",", "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[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "12"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "12"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List["-", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "6"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "12"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "12"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) ) z z α -1 BesselK 1 2 3 a z r 3 2 AiryBi a z r 1 2 1 3 3 1 6 -1 1 2 -1 z α 6 3 r -1 MeijerG 1 6 2 3 1 -1 α 3 r -1 -1 1 2 -1 1 6 1 2 5 6 -1 α 3 r -1 2 3 2 3 a z r 1 3 -1 2 z α 3 2 a z r -1 a z r 3 2 3 r -1 MeijerG 1 6 2 3 1 -1 α 3 r -1 2 3 7 6 1 2 5 6 -1 1 2 -1 1 6 2 3 7 6 -1 α 3 r -1 2 3 2 3 a z r 1 3 z α 6 r -1 MeijerG 1 6 2 3 1 -1 α 3 r -1 -1 1 12 5 12 -1 1 2 -1 1 6 1 2 5 6 -1 1 12 5 12 -1 α 3 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["1", ",", 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]", FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox["1", "2"], ",", FractionBox["5", "6"]]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], RowBox[List["6", " ", SuperscriptBox["\[Pi]", "3"], " ", "r"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "12"]]], ",", FractionBox["5", "12"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox["1", "2"], ",", FractionBox["5", "6"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "12"]]], ",", FractionBox["5", "12"], ",", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], RowBox[List["6", " ", "\[Pi]", " ", "r"]]], "-", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["3", "2"], " ", RowBox[List["Log", "[", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], "]"]]]], "-", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", SuperscriptBox["z", "r"]]], ")"]], RowBox[List["3", "/", "2"]]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["2", "3"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["2", "3"], ",", FractionBox["7", "6"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["5", "6"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox["2", "3"], ",", FractionBox["7", "6"], ",", RowBox[List["-", FractionBox["\[Alpha]", RowBox[List["3", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], RowBox[List["3", " ", "r"]]]]], ")"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["1", "/", "6"]]]]]]]]]]

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

 2001-10-29