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 SinIntegral

 http://functions.wolfram.com/06.37.21.0046.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Log[b z] SinIntegral[a z], z] == (1/(2 \[Alpha]^3)) ((z^\[Alpha] (I (a^2 z^2)^\[Alpha] HypergeometricPFQ[{\[Alpha], \[Alpha]}, {1 + \[Alpha], 1 + \[Alpha]}, (-I) a z] - I (a^2 z^2)^\[Alpha] HypergeometricPFQ[{\[Alpha], \[Alpha]}, {1 + \[Alpha], 1 + \[Alpha]}, I a z] + \[Alpha] ((-I) ((((-I) a z)^\[Alpha] - (I a z)^\[Alpha]) Gamma[1 + \[Alpha]] Log[z] + (I a z)^\[Alpha] Gamma[\[Alpha], (-I) a z] (-1 + \[Alpha] Log[b z]) - ((-I) a z)^\[Alpha] Gamma[\[Alpha], I a z] (-1 + \[Alpha] Log[b z])) + 2 (a^2 z^2)^\[Alpha] (-1 + \[Alpha] Log[b z]) SinIntegral[a z])))/ (a^2 z^2)^\[Alpha])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox["\[Alpha]", "3"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["\[Alpha]", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Alpha]"]], ",", RowBox[List["1", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["\[Alpha]", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Alpha]"]], ",", RowBox[List["1", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], ")"]], "\[Alpha]"], "-", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], ")"]], "\[Alpha]"]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Alpha]"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 log ( b z ) Si ( a z ) z 1 2 α 3 ( z α ( a 2 z 2 ) - α ( 2 F 2 ( α , α ; α + 1 , α + 1 ; - a z ) 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["\[Alpha]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Alpha]", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Alpha]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Alpha]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( a 2 z 2 ) α - 2 F 2 ( α , α ; α + 1 , α + 1 ; a z ) 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["\[Alpha]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Alpha]", HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Alpha]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Alpha]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( a 2 z 2 ) α + α ( 2 ( a 2 z 2 ) α ( α log ( b z ) - 1 ) Si ( a z ) - ( - Γ ( α , a z ) ( α log ( b z ) - 1 ) ( - a z ) α + ( ( - a z ) α - ( a z ) α ) Γ ( α + 1 ) log ( z ) + ( a z ) α Γ ( α , - a z ) ( α log ( b z ) - 1 ) ) ) ) ) z z α -1 b z SinIntegral a z 1 2 α 3 -1 z α a 2 z 2 -1 α HypergeometricPFQ α α α 1 α 1 -1 a z a 2 z 2 α -1 HypergeometricPFQ α α α 1 α 1 a z a 2 z 2 α α 2 a 2 z 2 α α b z -1 SinIntegral a z -1 -1 Gamma α a z α b z -1 -1 a z α -1 a z α -1 a z α Gamma α 1 z a z α Gamma α -1 a z α b z -1 [/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["Log", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["\[Alpha]", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Alpha]"]], ",", RowBox[List["1", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["\[Alpha]", ",", "\[Alpha]"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "\[Alpha]"]], ",", RowBox[List["1", "+", "\[Alpha]"]]]], "}"]], ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], ")"]], "\[Alpha]"], "-", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], ")"]], "\[Alpha]"]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Alpha]"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["\[Alpha]", "3"]]]]]]]]

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

 2001-10-29