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 Cos

 http://functions.wolfram.com/01.07.21.2714.01

 Input Form

 Integrate[z^n E^(p z) Sin[c z + d] Cos[a z]^\[Nu], z] == ((1/2) I Cos[a z]^\[Nu] n! ((-E^(I d + (p + I c) z)) Sum[(1/(n - j)!) ((-1)^j z^(n - j) (p + I c - I a \[Nu])^(-1 - j) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, 1 + j], -\[Nu]}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, 1 + j]}, -E^(2 I a z)]), {j, 0, n}] + E^((-I) d + (p - I c) z) Sum[(1/(n - j)!) ((-1)^j z^(n - j) (p - I c - I a \[Nu])^(-1 - j) HypergeometricPFQ[ {Subscript[d, 1], \[Ellipsis], Subscript[d, 1 + j], -\[Nu]}, {1 + Subscript[d, 1], \[Ellipsis], 1 + Subscript[d, 1 + j]}, -E^(2 I a z)]), {j, 0, n}]))/(1 + E^(2 I a z))^\[Nu] /; Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == ((-I) p + c - a \[Nu])/(2 a) && Subscript[d, 1] == Subscript[d, 2] == \[Ellipsis] == Subscript[d, n + 1] == -((I p + c + a \[Nu])/(2 a)) && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z sin ( c z + d ) cos ν ( a z ) z n ! 2 cos ν ( a z ) ( 1 + 2 a z ) - ν ( - d + ( p + c ) z j = 0 n ( - 1 ) j z n - j ( n - j ) ! ( p + c - a ν ) j + 1 j + 2 F j + 1 ( - p + c - a ν 2 a , , - p + c - a ν 2 a , - ν ; 1 + - p + c - a ν 2 a , , 1 + - p + c - a ν 2 a ; - 2 a z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "p"]], "+", "c", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "p"]], "+", "c", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "p"]], "+", "c", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "p"]], "+", "c", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]] + - d + ( p - c ) z j = 0 n ( - 1 ) j z n - j ( n - j ) ! ( p - c - a ν ) j + 1 j + 2 F j + 1 ( - p + c + a ν 2 a , , - p + c + a ν 2 a , - ν ; 1 - p + c + a ν 2 a , , 1 - p + c + a ν 2 a ; - 2 a z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", "c", "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", "c", "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Nu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", "c", "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "p"]], "+", "c", "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]] ) /; n Condition z z n p z c z d a z ν n 2 -1 a z ν 1 2 a z -1 ν -1 d p c z j 0 n -1 j z n -1 j n -1 j p c -1 a ν j 1 -1 HypergeometricPFQ -1 p c -1 a ν 2 a -1 -1 p c -1 a ν 2 a -1 -1 ν 1 -1 p c -1 a ν 2 a -1 1 -1 p c -1 a ν 2 a -1 -1 2 a z -1 d p -1 c z j 0 n -1 j z n -1 j n -1 j p -1 c -1 a ν j 1 -1 HypergeometricPFQ -1 p c a ν 2 a -1 -1 p c a ν 2 a -1 -1 ν 1 -1 p c a ν 2 a -1 1 -1 p c a ν 2 a -1 -1 2 a z n [/itex]

 Rule Form

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

 2002-12-18