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http://functions.wolfram.com/06.38.21.0045.01
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Integrate[z^(\[Alpha] - 1) Log[b z] CosIntegral[a z], z] ==
(1/(2 \[Alpha]^3))
((z^\[Alpha] ((-(I a z)^\[Alpha]) \[Alpha] Gamma[\[Alpha], (-I) a z] -
((-I) a z)^\[Alpha] \[Alpha] Gamma[\[Alpha], I a z] +
(a^2 z^2)^\[Alpha] HypergeometricPFQ[{\[Alpha], \[Alpha]},
{1 + \[Alpha], 1 + \[Alpha]}, (-I) a z] + (a^2 z^2)^\[Alpha]
HypergeometricPFQ[{\[Alpha], \[Alpha]}, {1 + \[Alpha], 1 + \[Alpha]},
I a z] - ((-I) a z)^\[Alpha] \[Alpha] Gamma[1 + \[Alpha]] Log[z] -
(I a z)^\[Alpha] \[Alpha] Gamma[1 + \[Alpha]] Log[z] +
(I a z)^\[Alpha] \[Alpha]^2 Gamma[\[Alpha], (-I) a z] Log[b z] +
((-I) a z)^\[Alpha] \[Alpha]^2 Gamma[\[Alpha], I a z] Log[b z] +
2 (a^2 z^2)^\[Alpha] \[Alpha] CosIntegral[a z]
(-1 + \[Alpha] Log[b z])))/(a^2 z^2)^\[Alpha])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["CosIntegral", "[", 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[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], ")"]], "\[Alpha]"]]], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], ")"]], "\[Alpha]"], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List[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[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[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], ")"]], "\[Alpha]"], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Alpha]"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], ")"]], "\[Alpha]"], " ", "\[Alpha]", " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Alpha]"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox["\[Alpha]", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox["\[Alpha]", "2"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]], " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], "\[Alpha]"], " ", "\[Alpha]", " ", RowBox[List["CosIntegral", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Ci </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mi> α </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> 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</mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> α </mi> <mo> , </mo> <mi> α </mi> </mrow> <mo> ; </mo> <mrow> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> 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z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> α </ci> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <ci> α </ci> <ci> α </ci> </list> <list> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> α </ci> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <ci> α </ci> <ci> α </ci> </list> <list> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> 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2 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> α </ci> </apply> <ci> α </ci> <apply> <ci> CosIntegral </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> α </ci> <apply> <ln /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
