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/03.08.21.0032.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Cosh[b + (2/3) (a z)^(3/2)] AiryBiPrime[a z], z] == (z^\[Alpha] (9 (2 + \[Alpha]) Gamma[5/3] HypergeometricPFQ[ {-(1/6), (2 \[Alpha])/3}, {-(1/3), 1 + (2 \[Alpha])/3}, (-(4/3)) (a z)^(3/2)] + 9 E^(2 b) (2 + \[Alpha]) Gamma[5/3] HypergeometricPFQ[{-(1/6), (2 \[Alpha])/3}, {-(1/3), 1 + (2 \[Alpha])/3}, (4/3) (a z)^(3/2)] + 3^(2/3) a^2 z^2 \[Alpha] Gamma[1/3] (HypergeometricPFQ[{7/6, 4/3 + (2 \[Alpha])/3}, {7/3, 7/3 + (2 \[Alpha])/3}, (-(4/3)) (a z)^(3/2)] + E^(2 b) HypergeometricPFQ[{7/6, 4/3 + (2 \[Alpha])/3}, {7/3, 7/3 + (2 \[Alpha])/3}, (4/3) (a z)^(3/2)])))/E^b/ (6 3^(5/6) \[Alpha] (2 + \[Alpha]) Gamma[1/3] Gamma[5/3])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Cosh", "[", RowBox[List["b", "+", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], RowBox[List["AiryBiPrime", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["9", " ", RowBox[List["(", RowBox[List["2", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], "+", RowBox[List["9", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["7", "6"], ",", RowBox[List[FractionBox["4", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "3"], ",", RowBox[List[FractionBox["7", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["7", "6"], ",", RowBox[List[FractionBox["4", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "3"], ",", RowBox[List[FractionBox["7", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["6", " ", SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List["2", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 cosh ( 2 3 ( a z ) 3 / 2 + b ) Bi ( a z ) z 1 6 3 5 / 6 α ( α + 2 ) Γ ( 1 3 ) Γ ( 5 3 ) ( - b z α ( 3 2 / 3 a 2 α Γ ( 1 3 ) ( 2 b 2 F 2 ( 7 6 , 2 α 3 + 4 3 ; 7 3 , 2 α 3 + 7 3 ; 4 3 ( a z ) 3 / 2 ) 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[FractionBox["7", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], "+", FractionBox["4", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["7", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], "+", FractionBox["7", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 2 F 2 ( 7 6 , 2 α 3 + 4 3 ; 7 3 , 2 α 3 + 7 3 ; 1 3 ( - 4 ) ( a z ) 3 / 2 ) 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[FractionBox["7", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], "+", FractionBox["4", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["7", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], "+", FractionBox["7", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) z 2 + 9 2 b ( α + 2 ) Γ ( 5 3 ) 2 F 2 ( - 1 6 , 2 α 3 ; - 1 3 , 2 α 3 + 1 ; 4 3 ( a z ) 3 / 2 ) 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", "6"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["1", "3"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 9 ( α + 2 ) Γ ( 5 3 ) 2 F 2 ( - 1 6 , 2 α 3 ; - 1 3 , 2 α 3 + 1 ; 1 3 ( - 4 ) ( a z ) 3 / 2 ) 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", "6"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["1", "3"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) ) z z α -1 2 3 a z 3 2 b AiryBiPrime a z 1 6 3 5 6 α α 2 Gamma 1 3 Gamma 5 3 -1 -1 b z α 3 2 3 a 2 α Gamma 1 3 2 b HypergeometricPFQ 7 6 2 α 3 -1 4 3 7 3 2 α 3 -1 7 3 4 3 a z 3 2 HypergeometricPFQ 7 6 2 α 3 -1 4 3 7 3 2 α 3 -1 7 3 1 3 -4 a z 3 2 z 2 9 2 b α 2 Gamma 5 3 HypergeometricPFQ -1 1 6 2 α 3 -1 -1 1 3 2 α 3 -1 1 4 3 a z 3 2 9 α 2 Gamma 5 3 HypergeometricPFQ -1 1 6 2 α 3 -1 -1 1 3 2 α 3 -1 1 1 3 -4 a z 3 2 [/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["Cosh", "[", RowBox[List["b_", "+", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", " ", "z_"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], " ", RowBox[List["AiryBiPrime", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "b"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["9", " ", RowBox[List["(", RowBox[List["2", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], "+", RowBox[List["9", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "6"]]], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["7", "6"], ",", RowBox[List[FractionBox["4", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "3"], ",", RowBox[List[FractionBox["7", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["(", RowBox[List["-", "4"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "b"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["7", "6"], ",", RowBox[List[FractionBox["4", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "3"], ",", RowBox[List[FractionBox["7", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List[FractionBox["4", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["6", " ", SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List["2", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["5", "3"], "]"]]]]]]]]]

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

 2001-10-29