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 Cos

 http://functions.wolfram.com/01.07.21.2503.01

 Input Form

 Integrate[z^n Sin[c z + d]^\[Mu] Cos[a z + b]^v, z] == (Binomial[v, v/2] (1 - Mod[v, 2]) n! Sin[c z + d]^\[Mu] Sum[(((-1)^j z^(n - j))/((n - j)! ((-I) c \[Mu])^(j + 1))) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, j + 1], -\[Mu]}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 1]}, E^(2 I (c z + d))], {j, 0, n}])/ (2^v (1 - E^(2 I (c z + d)))^\[Mu]) + (n! Sin[d + c z]^\[Mu] Sum[Binomial[v, k] (E^((1/2) I (4 b k - 2 b v + 4 a k z - 2 a v z)) Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) ((-I) a (-2 k + v) - I c \[Mu])^ (-1 - j) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, 1 + j], -\[Mu]}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, 1 + j]}, E^(2 I (d + c z))]), {j, 0, n}] + Sum[(1/(-j + n)!) ((-1)^j z^(-j + n) (I a (-2 k + v) - I c \[Mu])^(-1 - j) HypergeometricPFQ[ {Subscript[c, 1], \[Ellipsis], Subscript[c, 1 + j], -\[Mu]}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, 1 + j]}, E^(2 I (d + c z))]), {j, 0, n}]/ E^((1/2) I (4 b k - 2 b v + 4 a k z - 2 a v z))), {k, 0, Floor[(1/2) (-1 + v)]}])/(2^v (1 - E^(2 I (d + c z)))^\[Mu]) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == -(\[Mu]/2) && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == -((a (-2 k + v) + c \[Mu])/(2 c)) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == -(((-a) (-2 k + v) + c \[Mu])/(2 c)) && Element[n, Integers] && n >= 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sin μ ( c z + d ) cos v ( a z + b ) z 2 - v ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] n ! ( 1 - v mod 2 \$CellContext`v 2 ) sin μ ( d + c z ) ( 1 - 2 ( d + c z ) ) - μ j = 0 n ( ( - 1 ) j z n - j ) ( n - j ) ! ( - c μ ) j + 1 j + 2 F j + 1 ( - μ 2 , , - μ 2 , - μ ; 1 - μ 2 , , 1 - μ 2 ; 2 ( d + c 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["\[Mu]", "2"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox["\[Mu]", "2"]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Mu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Mu]", "2"]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + 2 - v ( 1 - 2 ( d + c z ) ) - μ n ! sin μ ( d + c z ) k = 0 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - 1 2 ( 4 b k + 4 a z k - 2 b v - 2 a v z ) j = 0 n ( - 1 ) j z n - j ( a ( v - 2 k ) - c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - c μ - a ( v - 2 k ) 2 c , , - c μ - a ( v - 2 k ) 2 c , - μ ; 1 - c μ - a ( v - 2 k ) 2 c , , 1 - c μ - a ( v - 2 k ) 2 c ; 2 ( d + c 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["c", " ", "\[Mu]"]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Mu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["c", " ", "\[Mu]"]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + 1 2 ( 4 b k + 4 a z k - 2 b v - 2 a v z ) j = 0 n ( - 1 ) j z n - j ( - a ( v - 2 k ) - c μ ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( - a ( v - 2 k ) + c μ 2 c , , - a ( v - 2 k ) + c μ 2 c , - μ ; 1 - a ( v - 2 k ) + c μ 2 c , , 1 - a ( v - 2 k ) + c μ 2 c ; 2 ( d + c 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", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox[RowBox[List["-", "\[Mu]"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["v", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["c", " ", "\[Mu]"]]]], RowBox[List["2", " ", "c"]]]]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) /; n v + Condition z z n c z d μ a z b v 2 -1 v Binomial v v 2 -1 n 1 -1 \$CellContext`v 2 d c z μ 1 -1 2 d c z -1 μ j 0 n -1 j z n -1 j n -1 j -1 c μ j 1 -1 HypergeometricPFQ -1 μ 2 -1 -1 μ 2 -1 -1 μ 1 -1 μ 2 -1 1 -1 μ 2 -1 2 d c z 2 -1 v 1 -1 2 d c z -1 μ n d c z μ k 0 v -1 2 -1 Binomial v k -1 1 2 4 b k 4 a z k -1 2 b v -1 2 a v z j 0 n -1 j z n -1 j a v -1 2 k -1 c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 c μ -1 a v -1 2 k 2 c -1 -1 c μ -1 a v -1 2 k 2 c -1 -1 μ 1 -1 c μ -1 a v -1 2 k 2 c -1 1 -1 c μ -1 a v -1 2 k 2 c -1 2 d c z 1 2 4 b k 4 a z k -1 2 b v -1 2 a v z j 0 n -1 j z n -1 j -1 a v -1 2 k -1 c μ -1 j -1 n -1 j -1 HypergeometricPFQ -1 a v -1 2 k c μ 2 c -1 -1 a v -1 2 k c μ 2 c -1 -1 μ 1 -1 a v -1 2 k c μ 2 c -1 1 -1 a v -1 2 k c μ 2 c -1 2 d c z n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18