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 Cosh

 http://functions.wolfram.com/01.20.21.4006.01

 Input Form

 Integrate[z^n E^(p z^2) Cos[b z]^m Cosh[c z]^v, z] == 2^(-1 - m - v) ((-z^(1 + n)) ((-p) z^2)^((1/2) (-1 - n)) Binomial[m, m/2] Binomial[v, v/2] Gamma[(1 + n)/2, (-p) z^2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2]) + p^(-1 - n) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[E^((b^2 (m - 2 s)^2)/(4 p)) Binomial[m, s] (Sum[(I b (-(m/2) + s))^(n - q) ((b (m - 2 s) - 2 I p z)^2/p)^ ((1/2) (-1 - q)) (I b (m - 2 s) + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) - 2 I p z)^2/(4 p)], {q, 0, n}] + Sum[2^(-n + q) (I b (m - 2 s))^(n - q) ((b (m - 2 s) + 2 I p z)^2/p)^ ((1/2) (-1 - q)) ((-I) b (m - 2 s) + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b (m - 2 s) + 2 I p z)^2/(4 p)], {q, 0, n}]), {s, 0, Floor[(1/2) (-1 + m)]}] + p^(-1 - n) Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(Binomial[v, s] (Sum[2^(-n + q) (c (-2 s + v))^(n - q) (2 c s - c v + 2 p z)^(1 + q) (-((2 c s - c v + 2 p z)^2/p))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((2 c s - c v + 2 p z)^2/(4 p))], {q, 0, n}] + Sum[(c (s - v/2))^(n - q) (c (-2 s + v) + 2 p z)^(1 + q) (-((c (-2 s + v) + 2 p z)^2/p))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((c (-2 s + v) + 2 p z)^2/(4 p))], {q, 0, n}]))/ E^((c^2 (-2 s + v)^2)/(4 p)), {s, 0, Floor[(1/2) (-1 + v)]}] - p^(-1 - n) Sum[Binomial[v, s] Sum[E^((b^2 (-2 k + m)^2 - c^2 (-2 s + v)^2)/(2 p)) Binomial[m, k] (E^((-2 I b k + I b m - 2 c s + c v)^2/(4 p)) Sum[2^(-n + q) (I b (2 k - m) + c (-2 s + v))^(n - q) (-((I b (2 k - m) - 2 c s + c v - 2 p z)^2/p))^((1/2) (-1 - q)) (-2 I b k + I b m + 2 c s - c v + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, -((I b (2 k - m) - 2 c s + c v - 2 p z)^2/ (4 p))], {q, 0, n}] + E^((2 I b k - I b m - 2 c s + c v)^2/ (4 p)) Sum[2^(-n + q) (I b (-2 k + m) + c (-2 s + v))^(n - q) (-((I b (-2 k + m) - 2 c s + c v - 2 p z)^2/p))^((1/2) (-1 - q)) (2 I b k - I b m + 2 c s - c v + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, -((I b (-2 k + m) - 2 c s + c v - 2 p z)^2/ (4 p))], {q, 0, n}] + E^((-2 I b k + I b m - 2 c s + c v)^2/ (4 p)) Sum[((-I) b k + (I b m)/2 + c s - (c v)/2)^(n - q) (2 I b k - I b m - 2 c s + c v + 2 p z)^(1 + q) (-((2 I b k - I b m - 2 c s + c v + 2 p z)^2/p))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((2 I b k - I b m - 2 c s + c v + 2 p z)^2/(4 p))], {q, 0, n}] + E^((2 I b k - I b m - 2 c s + c v)^2/(4 p)) Sum[(I b (k - m/2) + c (s - v/2))^(n - q) (-2 I b k + I b m - 2 c s + c v + 2 p z)^(1 + q) (-((-2 I b k + I b m - 2 c s + c v + 2 p z)^2/p))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((-2 I b k + I b m - 2 c s + c v + 2 p z)^2/ (4 p))], {q, 0, n}]), {k, 0, Floor[(1/2) (-1 + m)]}], {s, 0, Floor[(1/2) (-1 + v)]}]) /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

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"p"]]]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n p z 2 cos m ( b z ) cosh v ( c z ) z 2 - m - v - 1 ( ( 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) ( s = 0 m - 1 2 b 2 ( m - 2 s ) 2 4 p ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( q = 0 n 2 q - n ( b ( m - 2 s ) ) n - q ( ( b ( m - 2 s ) + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( 2 p z - b ( m - 2 s ) ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b ( m - 2 s ) + 2 p z ) 2 4 p ) + q = 0 n ( b ( s - m 2 ) ) n - q ( ( b ( m - 2 s ) - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( b ( m - 2 s ) + 2 p z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b ( m - 2 s ) - 2 p z ) 2 4 p ) ) ) p - n - 1 + ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( s = 0 v - 1 2 - c 2 ( v - 2 s ) 2 4 p ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( q = 0 n 2 q - n ( c ( v - 2 s ) ) n - q ( 2 c s - c v + 2 p z ) q + 1 ( - ( 2 c s - c v + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( 2 c s - c v + 2 p z ) 2 4 p ) + q = 0 n ( c ( s - v 2 ) ) n - q ( c ( v - 2 s ) + 2 p z ) q + 1 ( - ( c ( v - 2 s ) + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( c ( v - 2 s ) + 2 p z ) 2 4 p ) ) ) p - n - 1 - ( s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] k = 0 m - 1 2 b 2 ( m - 2 k ) 2 - c 2 ( v - 2 s ) 2 2 p ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( 2 b k - b m - 2 c s + c v ) 2 4 p q = 0 n ( ( s - v 2 ) c + ( k - m 2 ) b ) n - q ( - 2 b k + b m - 2 c s + c v + 2 p z ) q + 1 ( - ( - 2 b k + b m - 2 c s + c v + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( - 2 b k + b m - 2 c s + c v + 2 p z ) 2 4 p ) + ( - 2 b k + b m - 2 c s + c v ) 2 4 p q = 0 n ( - b k + b m 2 + c s - c v 2 ) n - q ( 2 b k - b m - 2 c s + c v + 2 p z ) q + 1 ( - ( 2 b k - b m - 2 c s + c v + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( 2 b k - b m - 2 c s + c v + 2 p z ) 2 4 p ) + ( - 2 b k + b m - 2 c s + c v ) 2 4 p q = 0 n 2 q - n ( b ( 2 k - m ) + c ( v - 2 s ) ) n - q ( - ( b ( 2 k - m ) - 2 c s + c v - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( - 2 b k + b m + 2 c s - c v + 2 p z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( b ( 2 k - m ) - 2 c s + c v - 2 p z ) 2 4 p ) + ( 2 b k - b m - 2 c s + c v ) 2 4 p q = 0 n 2 q - n ( b ( m - 2 k ) + c ( v - 2 s ) ) n - q ( - ( b ( m - 2 k ) - 2 c s + c v - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( 2 b k - b m + 2 c s - c v + 2 p z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( b ( m - 2 k ) - 2 c s + c v - 2 p z ) 2 4 p ) ) ) p - n - 1 - z n + 1 ( - p z 2 ) 1 2 ( - n - 1 ) ( 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]]]]] ( 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]]]]] Γ ( n + 1 2 , - p z 2 ) ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 n Condition z z n p z 2 b z m c z v 2 -1 m -1 v -1 Binomial v v 2 -1 \$CellContext`v 2 -1 s 0 m -1 2 -1 b 2 m -1 2 s 2 4 p -1 Binomial m s q 0 n 2 q -1 n b m -1 2 s n -1 q b m -1 2 s 2 p z 2 p -1 1 2 -1 q -1 2 p z -1 b m -1 2 s q 1 Binomial n q Gamma q 1 2 -1 b m -1 2 s 2 p z 2 4 p -1 q 0 n b s -1 m 2 -1 n -1 q b m -1 2 s -1 2 p z 2 p -1 1 2 -1 q -1 b m -1 2 s 2 p z q 1 Binomial n q Gamma q 1 2 -1 b m -1 2 s -1 2 p z 2 4 p -1 p -1 n -1 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 -1 c 2 v -1 2 s 2 4 p -1 Binomial v s q 0 n 2 q -1 n c v -1 2 s n -1 q 2 c s -1 c v 2 p z q 1 -1 2 c s -1 c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 c s -1 c v 2 p z 2 4 p -1 q 0 n c s -1 v 2 -1 n -1 q c v -1 2 s 2 p z q 1 -1 c v -1 2 s 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 c v -1 2 s 2 p z 2 4 p -1 p -1 n -1 -1 s 0 v -1 2 -1 Binomial v s k 0 m -1 2 -1 b 2 m -1 2 k 2 -1 c 2 v -1 2 s 2 2 p -1 Binomial m k 2 b k -1 b m -1 2 c s c v 2 4 p -1 q 0 n s -1 v 2 -1 c k -1 m 2 -1 b n -1 q -2 b k b m -1 2 c s c v 2 p z q 1 -1 -2 b k b m -1 2 c s c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -2 b k b m -1 2 c s c v 2 p z 2 4 p -1 -2 b k b m -1 2 c s c v 2 4 p -1 q 0 n -1 b k b m 2 -1 c s -1 c v 2 -1 n -1 q 2 b k -1 b m -1 2 c s c v 2 p z q 1 -1 2 b k -1 b m -1 2 c s c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 b k -1 b m -1 2 c s c v 2 p z 2 4 p -1 -2 b k b m -1 2 c s c v 2 4 p -1 q 0 n 2 q -1 n b 2 k -1 m c v -1 2 s n -1 q -1 b 2 k -1 m -1 2 c s c v -1 2 p z 2 p -1 1 2 -1 q -1 -2 b k b m 2 c s -1 c v 2 p z q 1 Binomial n q Gamma q 1 2 -1 -1 b 2 k -1 m -1 2 c s c v -1 2 p z 2 4 p -1 2 b k -1 b m -1 2 c s c v 2 4 p -1 q 0 n 2 q -1 n b m -1 2 k c v -1 2 s n -1 q -1 b m -1 2 k -1 2 c s c v -1 2 p z 2 p -1 1 2 -1 q -1 2 b k -1 b m 2 c s -1 c v 2 p z q 1 Binomial n q Gamma q 1 2 -1 -1 b m -1 2 k -1 2 c s c v -1 2 p z 2 4 p -1 p -1 n -1 -1 z n 1 -1 p z 2 1 2 -1 n -1 Binomial m m 2 -1 Binomial v v 2 -1 Gamma n 1 2 -1 -1 p z 2 \$CellContext`m 2 -1 \$CellContext`v 2 -1 m m 0 v v 0 n [/itex]

 Rule Form

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

 2002-12-18