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; }

 AiryBiPrime

 http://functions.wolfram.com/03.08.21.0057.01

 Input Form

 Integrate[BesselK[0, (2/3) (a z^r)^(3/2)] AiryBiPrime[a z^r], z] == (-(1/(2 2^(2/3) 3^(5/6) Pi^(3/2) r))) (z (MeijerG[{{1/3, 5/6, 1 - 1/(3 r)}, {-(1/6)}}, {{0, 0, 2/3, 2/3}, {-(1/6), -(1/(3 r))}}, (-(2/3)^(2/3)) a z^r, 1/3] + Pi ((-3 Log[(-a) z^r] + 2 Log[(a z^r)^(3/2)]) MeijerG[{{1/3, 5/6, 1 - 1/(3 r)}, {1/3}}, {{0, 2/3}, {0, 1/3, 2/3, -(1/(3 r))}}, (-(2/3)^(2/3)) a z^r, 1/3] + Pi MeijerG[{{1/3, 5/6, 1 - 1/(3 r)}, {-(1/6), 1/6, 1/2}}, {{0, 0, 2/3, 2/3}, {-(1/6), 1/6, 1/2, -(1/(3 r))}}, (-(2/3)^(2/3)) a z^r, 1/3])))

 Standard Form

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 MathML Form

 K 0 ( 2 3 ( a z r ) 3 / 2 ) Bi ( a z r ) z - 1 2 2 2 / 3 3 5 / 6 π 3 / 2 r ( z ( G 4 , 6 4 , 3 ( - ( 2 3 ) 2 / 3 a z r , 1 3 1 3 , 5 6 , 1 - 1 3 r , - 1 6 0 , 0 , 2 3 , 2 3 , - 1 6 , - 1 3 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "6"]], RowBox[List["4", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[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", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "6"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "6"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + π ( ( 2 log ( ( a z r ) 3 / 2 ) - 3 log ( - a z r ) ) G 4 , 6 2 , 3 ( - ( 2 3 ) 2 / 3 a z r , 1 3 1 3 , 5 6 , 1 - 1 3 r , 1 3 0 , 2 3 , 0 , 1 3 , 2 3 , - 1 3 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "6"]], RowBox[List["2", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[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", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + π G 6 , 8 4 , 3 ( - ( 2 3 ) 2 / 3 a z r , 1 3 1 3 , 5 6 , 1 - 1 3 r , - 1 6 , 1 6 , 1 2 0 , 0 , 2 3 , 2 3 , - 1 6 , 1 6 , 1 2 , - 1 3 r ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["6", ",", "8"]], RowBox[List["4", ",", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[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", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["5", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "6"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["2", "3"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", "6"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "6"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["1", RowBox[List["3", " ", "r"]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) ) ) z BesselK 0 2 3 a z r 3 2 AiryBiPrime a z r -1 1 2 2 2 3 3 5 6 3 2 r -1 z MeijerG 1 3 5 6 1 -1 1 3 r -1 -1 1 6 0 0 2 3 2 3 -1 1 6 -1 1 3 r -1 -1 2 3 2 3 a z r 1 3 2 a z r 3 2 -1 3 -1 a z r MeijerG 1 3 5 6 1 -1 1 3 r -1 1 3 0 2 3 0 1 3 2 3 -1 1 3 r -1 -1 2 3 2 3 a z r 1 3 MeijerG 1 3 5 6 1 -1 1 3 r -1 -1 1 6 1 6 1 2 0 0 2 3 2 3 -1 1 6 1 6 1 2 -1 1 3 r -1 -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[RowBox[List["BesselK", "[", RowBox[List["0", ",", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], " ", RowBox[List["AiryBiPrime", "[", RowBox[List["a_", " ", SuperscriptBox["z_", "r_"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", FractionBox["5", "6"], ",", RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["1", "6"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", FractionBox["2", "3"], ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", RowBox[List["-", FractionBox["1", RowBox[List["3", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "r"]]], "]"]]]], "+", RowBox[List["2", " ", 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", "3"], ",", FractionBox["5", "6"], ",", RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "3"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "3"], ",", FractionBox["2", "3"], ",", RowBox[List["-", FractionBox["1", RowBox[List["3", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "3"], ",", FractionBox["5", "6"], ",", RowBox[List["1", "-", FractionBox["1", RowBox[List["3", " ", "r"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox["1", "6"], ",", FractionBox["1", "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", FractionBox["2", "3"], ",", FractionBox["2", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox["1", "6"], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["1", RowBox[List["3", " ", "r"]]]]]]], "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["2", "/", "3"]]]]], " ", "a", " ", SuperscriptBox["z", "r"]]], ",", FractionBox["1", "3"]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", "r"]]]]]]]]]

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

 2001-10-29