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 Exp

 http://functions.wolfram.com/01.03.21.0760.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18