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 AiryAi

 http://functions.wolfram.com/03.05.21.0028.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Sinh[(2/3) (a z)^(3/2)] AiryAi[a z], z] == (z^\[Alpha] (-3 (1 + \[Alpha]) Gamma[4/3] HypergeometricPFQ[ {1/6, (2 \[Alpha])/3}, {1/3, 1 + (2 \[Alpha])/3}, (-(4/3)) (a z)^(3/2)] + 3 (1 + \[Alpha]) Gamma[4/3] HypergeometricPFQ[{1/6, (2 \[Alpha])/3}, {1/3, 1 + (2 \[Alpha])/3}, (4/3) (a z)^(3/2)] + 3^(1/3) a z \[Alpha] Gamma[2/3] (HypergeometricPFQ[{5/6, 2/3 + (2 \[Alpha])/3}, {5/3, 5/3 + (2 \[Alpha])/3}, (-(4/3)) (a z)^(3/2)] - HypergeometricPFQ[{5/6, 2/3 + (2 \[Alpha])/3}, {5/3, 5/3 + (2 \[Alpha])/3}, (4/3) (a z)^(3/2)])))/ (6 3^(2/3) \[Alpha] (1 + \[Alpha]) Gamma[2/3] Gamma[4/3])

 Standard Form

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

 MathML Form

 z α - 1 sinh ( 2 3 ( a z ) 3 / 2 ) Ai ( a z ) z 1 6 3 2 / 3 α ( α + 1 ) Γ ( 2 3 ) Γ ( 4 3 ) ( z α ( 3 ( α + 1 ) Γ ( 4 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[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[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] - 3 ( α + 1 ) Γ ( 4 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[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[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] + 3 3 a z α Γ ( 2 3 ) ( 2 F 2 ( 5 6 , 2 α 3 + 2 3 ; 5 3 , 2 α 3 + 5 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["5", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], "+", FractionBox["2", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], "+", FractionBox["5", "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] - 2 F 2 ( 5 6 , 2 α 3 + 2 3 ; 5 3 , 2 α 3 + 5 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["5", "6"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], "+", FractionBox["2", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"], "+", FractionBox["5", "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] ) ) ) z z α -1 2 3 a z 3 2 AiryAi a z 1 6 3 2 3 α α 1 Gamma 2 3 Gamma 4 3 -1 z α 3 α 1 Gamma 4 3 HypergeometricPFQ 1 6 2 α 3 -1 1 3 2 α 3 -1 1 4 3 a z 3 2 -1 3 α 1 Gamma 4 3 HypergeometricPFQ 1 6 2 α 3 -1 1 3 2 α 3 -1 1 1 3 -4 a z 3 2 3 1 3 a z α Gamma 2 3 HypergeometricPFQ 5 6 2 α 3 -1 2 3 5 3 2 α 3 -1 5 3 1 3 -4 a z 3 2 -1 HypergeometricPFQ 5 6 2 α 3 -1 2 3 5 3 2 α 3 -1 5 3 4 3 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["Sinh", "[", RowBox[List[FractionBox["2", "3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", " ", "z_"]], ")"]], RowBox[List["3", "/", "2"]]]]], "]"]], " ", RowBox[List["AiryAi", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]], "}"]], ",", 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["3", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]], "}"]], ",", 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["1", "/", "3"]]], " ", "a", " ", "z", " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "6"], ",", RowBox[List[FractionBox["2", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "3"], ",", RowBox[List[FractionBox["5", "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["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["5", "6"], ",", RowBox[List[FractionBox["2", "3"], "+", FractionBox[RowBox[List["2", " ", "\[Alpha]"]], "3"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "3"], ",", RowBox[List[FractionBox["5", "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["2", "/", "3"]]], " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]], " ", RowBox[List["Gamma", "[", FractionBox["4", "3"], "]"]]]]]]]]]

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

 2001-10-29