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 Exp

 http://functions.wolfram.com/01.03.21.0210.01

 Input Form

 Integrate[z^n E^(a Sqrt[z] + b z), z] == (2^(-1 - 2 n) Sum[(-1)^(-h + k) 4^k a^(-h - k + 2 n) (a + 2 b Sqrt[z])^(h + k) (-((a + 2 b Sqrt[z])^2/b))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (a (a + 2 b Sqrt[z]) Gamma[(1/2) (1 + h + k), -((a + 2 b Sqrt[z])^2/(4 b))] + 2 b Sqrt[-((a + 2 b Sqrt[z])^2/b)] Gamma[(1/2) (2 + h + k), -((a + 2 b Sqrt[z])^2/(4 b))]), {k, 0, n}, {h, 0, k}])/(b^(2 (1 + n)) E^(a^2/(4 b))) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n z a + b z z 2 - 2 n - 1 b - 2 ( n + 1 ) - a 2 4 b k = 0 n h = 0 k ( - 1 ) k - h 4 k a - h - k + 2 n ( a + 2 b z ) h + k ( - ( a + 2 b z ) 2 b ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( a ( a + 2 b z ) Γ ( 1 2 ( h + k + 1 ) , - ( a + 2 b z ) 2 4 b ) + 2 - ( a + 2 b z ) 2 b b Γ ( 1 2 ( h + k + 2 ) , - ( a + 2 b z ) 2 4 b ) ) /; n Condition z z n z 1 2 a b z 2 -2 n -1 b -2 n 1 -1 a 2 4 b -1 h 0 k k 0 n -1 k -1 h 4 k a -1 h -1 k 2 n a 2 b z 1 2 h k -1 a 2 b z 1 2 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k a a 2 b z 1 2 Gamma 1 2 h k 1 -1 a 2 b z 1 2 2 4 b -1 2 -1 a 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h k 2 -1 a 2 b z 1 2 2 4 b -1 n [/itex]

 Rule Form

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

 2002-12-18