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 Cos

 http://functions.wolfram.com/01.07.21.2747.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18