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 Cosh

 http://functions.wolfram.com/01.20.21.4000.01

 Input Form

 Integrate[z^n E^(p z) Cos[c z]^m Cosh[b + a z]^\[Nu], z] == (E^(p z) Binomial[m, m/2] n! (1 - Mod[m, 2]) Cosh[b + a z]^\[Nu] Sum[(1/(-j + n)!) (-1)^j z^(-j + n) (p - 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 (b + a z))], {j, 0, n}])/ (2^m (1 + E^(2 (b + a z)))^\[Nu]) + (n! Cosh[b + a z]^\[Nu] Sum[Binomial[m, s] (E^((p - I c (m - 2 s)) z) Sum[(1/(-j + n)!) (-1)^j z^(-j + n) (p - I c (m - 2 s) - 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 (b + a z))], {j, 0, n}] + E^((p + I c (m - 2 s)) z) Sum[(1/(-j + n)!) (-1)^j z^(-j + n) (p + I c (m - 2 s) - 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 (b + a z))], {j, 0, n}]), {s, 0, Floor[(1/2) (-1 + m)]}])/(2^m (1 + E^(2 (b + a z)))^\[Nu]) /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == (p - a \[Nu])/(2 a) && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == (p - I c (m - 2 s) - a \[Nu])/ (2 a) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (p + I c (m - 2 s) - 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 cos m ( c z ) cosh 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]]]]] n ! ( 1 - m mod 2 \$CellContext`m 2 ) cosh ν ( b + a z ) ( 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 ; p - a ν 2 a + 1 , , p - a ν 2 a + 1 ; - 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[FractionBox[RowBox[List["p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["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[FractionBox[RowBox[List["p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) ( 1 + 2 ( b + a z ) ) - ν + 2 - m n ! cosh ν ( b + a z ) ( 1 + 2 ( b + a z ) ) - ν s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( p - c ( m - 2 s ) ) z j = 0 n ( - 1 ) j z n - j ( p - c ( m - 2 s ) - a ν ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( p - c ( m - 2 s ) - a ν 2 a , , p - c ( m - 2 s ) - a ν 2 a , - ν ; p - c ( m - 2 s ) - a ν 2 a + 1 , , p - c ( m - 2 s ) - a ν 2 a + 1 ; - 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[FractionBox[RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", 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[FractionBox[RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]], ")"]]]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + ( p + c ( m - 2 s ) ) z j = 0 n ( - 1 ) j z n - j ( p + c ( m - 2 s ) - a ν ) - j - 1 ( n - j ) ! j + 2 F j + 1 ( p + c ( m - 2 s ) - a ν 2 a , , p + c ( m - 2 s ) - a ν 2 a , - ν ; p + c ( m - 2 s ) - a ν 2 a + 1 , , p + c ( m - 2 s ) - a ν 2 a + 1 ; - 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[FractionBox[RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", 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[FractionBox[RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]], ")"]]]], "-", RowBox[List["a", " ", "\[Nu]"]]]], RowBox[List["2", " ", "a"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", 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 n 1 -1 \$CellContext`m 2 b a z ν j 0 n -1 j z n -1 j p -1 a ν -1 j -1 n -1 j -1 HypergeometricPFQ p -1 a ν 2 a -1 p -1 a ν 2 a -1 -1 ν p -1 a ν 2 a -1 1 p -1 a ν 2 a -1 1 -1 2 b a z 1 2 b a z -1 ν 2 -1 m n b a z ν 1 2 b a z -1 ν s 0 m -1 2 -1 Binomial m s p -1 c m -1 2 s z j 0 n -1 j z n -1 j p -1 c m -1 2 s -1 a ν -1 j -1 n -1 j -1 HypergeometricPFQ p -1 c m -1 2 s -1 a ν 2 a -1 p -1 c m -1 2 s -1 a ν 2 a -1 -1 ν p -1 c m -1 2 s -1 a ν 2 a -1 1 p -1 c m -1 2 s -1 a ν 2 a -1 1 -1 2 b a z p c m -1 2 s z j 0 n -1 j z n -1 j p c m -1 2 s -1 a ν -1 j -1 n -1 j -1 HypergeometricPFQ p c m -1 2 s -1 a ν 2 a -1 p c m -1 2 s -1 a ν 2 a -1 -1 ν p c m -1 2 s -1 a ν 2 a -1 1 p c m -1 2 s -1 a ν 2 a -1 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