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 Cos

 http://functions.wolfram.com/01.07.21.2424.01

 Input Form

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

 Standard Form

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 MathML Form

 z n sin ( d + c z ) cos ν ( a z ) z 1 2 ( 1 + 2 a z ) - ν cos ν ( a z ) n ! ( - ( d + c z ) j = 0 n ( - 1 ) j z n - j ( - c - a ν ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - c + a ν 2 a , , - c + a ν 2 a , - ν ; 1 - c + a ν 2 a , , 1 - c + a ν 2 a ; - 2 a z ) TagBox[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["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List["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["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["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]]]]], HypergeometricPFQ] - ( d + c z ) j = 0 n ( - 1 ) j z n - j ( c - a ν ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - a ν - c 2 a , , - a ν - c 2 a , - ν ; 1 - a ν - c 2 a , , 1 - a ν - c 2 a ; - 2 a z ) TagBox[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["a", " ", "\[Nu]"]], "-", "c"]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["a", " ", "\[Nu]"]], "-", "c"]], 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["a", " ", "\[Nu]"]], "-", "c"]], RowBox[List["2", " ", "a"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["a", " ", "\[Nu]"]], "-", "c"]], 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]]]]], HypergeometricPFQ] ) /; n Condition z z n d c z a z ν 1 2 1 2 a z -1 ν a z ν n -1 d c z j 0 n -1 j z n -1 j -1 c -1 a ν -1 j -1 n -1 j -1 HypergeometricPFQ -1 c a ν 2 a -1 -1 c a ν 2 a -1 -1 ν 1 -1 c a ν 2 a -1 1 -1 c a ν 2 a -1 -1 2 a z -1 d c z j 0 n -1 j z n -1 j c -1 a ν -1 j -1 n -1 j -1 HypergeometricPFQ -1 a ν -1 c 2 a -1 -1 a ν -1 c 2 a -1 -1 ν 1 -1 a ν -1 c 2 a -1 1 -1 a ν -1 c 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