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 Cosh

 http://functions.wolfram.com/01.20.21.3688.01

 Input Form

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

 Standard Form

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", "z"]]]], ")"]], "2"], RowBox[List["4", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18