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 Exp

 http://functions.wolfram.com/01.03.21.0724.01

 Input Form

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

 Standard Form

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 MathML Form

 z n ( b z ) μ ( c z ) ν z 2 - 2 n - 1 - c 2 ν 2 4 b μ - c z ν - b z μ ( z c ) ν ( b z ) μ ( b μ ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ν ) - h - k + 2 n ( 2 b z μ + c ν ) h + k ( - ( 2 b z μ + c ν ) 2 b μ ) 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]]]]] ( c ν ( 2 b z μ + c ν ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 b z μ + c ν ) 2 4 b μ ) + 2 b μ - ( 2 b z μ + c ν ) 2 b μ Γ ( 1 2 ( h + k + 2 ) , - ( 2 b z μ + c ν ) 2 4 b μ ) ) /; n Condition z z n b z μ c z 1 2 ν 2 -2 n -1 -1 c 2 ν 2 4 b μ -1 -1 c z 1 2 ν -1 b z μ z 1 2 c ν b z μ b μ -2 n 1 h 0 k k 0 n -1 k -1 h 4 k c ν -1 h -1 k 2 n 2 b z 1 2 μ c ν h k -1 2 b z 1 2 μ c ν 2 b μ -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c ν 2 b z 1 2 μ c ν Gamma 1 2 h k 1 -1 2 b z 1 2 μ c ν 2 4 b μ -1 2 b μ -1 2 b z 1 2 μ c ν 2 b μ -1 1 2 Gamma 1 2 h k 2 -1 2 b z 1 2 μ c ν 2 4 b μ -1 n [/itex]

 Rule Form

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

 2002-12-18