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 Cosh

 http://functions.wolfram.com/01.20.21.3919.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18