How to find Laplace Transform

Questions
1

How to find Laplace Transform of the following function with respect to x :

Product[2 - (Gamma[Subscript[m, i], Subscript[m, i]/Subscript[\[CapitalOmega], i] (x/\[Gamma])^(1/ Subscript[n, i])]/Gamma[Subscript[m, i]] + Gamma[Subscript[t, i], Subscript[t, i]/Subscript[\[Rho], i] (x/\[Gamma])^(1/Subscript[k, i])]/ Gamma[Subscript[t, i]]), {i, 1, M}]

Thanks for your support!

2

I’m an amateur to Laplace transforms, but it looks to me that you can express your eventual solution in terms of integrals of type

LaplaceTransform[Gamma[m, C x^(1/n)], x, z]

with various m, C, and n.

Assuming that your n’s are positive integers and looking at the first few such results, it looks to me as if there is pattern lurking, which I leave to you to determine:

 In[1]:= Table[
  n -> LaplaceTransform[Gamma[m, C x^(1/n)], x, z], {n, 1, 6}]
 
 Out[1]= {1 -> ((1 - ((C + z)/C)^-m) Gamma[m])/z,
  2 -> 1/2 z^(
    1/2 (-3 - m)) (2 z^((1 + m)/2) Gamma[m] +
      C^m (-Sqrt[z] Gamma[m/2] Hypergeometric1F1[m/2, 1/2, C^2/(4 z)] +
          C Gamma[(1 + m)/2] Hypergeometric1F1[(1 + m)/2, 3/2, C^2/(
           4 z)])),
 3 -> 1/6 z^(
   1/3 (-5 - m)) (6 z^((2 + m)/3) Gamma[m] +
     2 C^(1 + m) z^(1/3)
       Gamma[(1 + m)/
       3] HypergeometricPFQ[{1/3 + m/3}, {2/3, 4/3}, -(C^3/(27 z))] -
     C^(2 + m)
       Gamma[(2 + m)/
       3] HypergeometricPFQ[{2/3 + m/3}, {4/3, 5/3}, -(C^3/(27 z))] -
     2 C^m z^(2/3)
       Gamma[m/
       3] HypergeometricPFQ[{m/3}, {1/3, 2/3}, -(C^3/(27 z))]),
 4 -> 1/24 z^(
   1/4 (-7 - m)) (24 z^((3 + m)/4) Gamma[m] +
     6 C^(1 + m) Sqrt[z]
       Gamma[(1 + m)/
       4] HypergeometricPFQ[{1/4 + m/4}, {1/2, 3/4, 5/4}, C^4/(
       256 z)] -
     3 C^(2 + m) z^(1/4)
       Gamma[(2 + m)/
       4] HypergeometricPFQ[{1/2 + m/4}, {3/4, 5/4, 3/2}, C^4/(
       256 z)] +
     C^(3 + m)
       Gamma[(3 + m)/
       4] HypergeometricPFQ[{3/4 + m/4}, {5/4, 3/2, 7/4}, C^4/(
       256 z)] -
     6 C^m z^(3/4)
       Gamma[m/4] HypergeometricPFQ[{m/4}, {1/4, 1/2, 3/4}, C^4/(
       256 z)]),
 5 -> 1/120 z^(
   1/5 (-9 - m)) (120 z^((4 + m)/5) Gamma[m] +
     24 C^(1 + m) z^(3/5)
       Gamma[(1 + m)/
       5] HypergeometricPFQ[{1/5 + m/5}, {2/5, 3/5, 4/5, 6/5}, -(C^5/(
        3125 z))] -
     12 C^(2 + m) z^(2/5)
       Gamma[(2 + m)/
       5] HypergeometricPFQ[{2/5 + m/5}, {3/5, 4/5, 6/5, 7/5}, -(C^5/(
        3125 z))] +
     4 C^(3 + m) z^(1/5)
       Gamma[(3 + m)/
       5] HypergeometricPFQ[{3/5 + m/5}, {4/5, 6/5, 7/5, 8/5}, -(C^5/(
        3125 z))] -
     C^(4 + m)
       Gamma[(4 + m)/
       5] HypergeometricPFQ[{4/5 + m/5}, {6/5, 7/5, 8/5, 9/5}, -(C^5/(
        3125 z))] -
     24 C^m z^(4/5)
       Gamma[m/
       5] HypergeometricPFQ[{m/5}, {1/5, 2/5, 3/5, 4/5}, -(C^5/(
        3125 z))]),
 6 -> 1/720 z^(
   1/6 (-11 - m)) (720 z^((5 + m)/6) Gamma[m] +
     120 C^(1 + m) z^(2/3)
       Gamma[(1 + m)/
       6] HypergeometricPFQ[{1/6 + m/6}, {1/3, 1/2, 2/3, 5/6, 7/6},
       C^6/(46656 z)] -
     60 C^(2 + m) Sqrt[z]
       Gamma[(2 + m)/
       6] HypergeometricPFQ[{1/3 + m/6}, {1/2, 2/3, 5/6, 7/6, 4/3},
       C^6/(46656 z)] +
     20 C^(3 + m) z^(1/3)
       Gamma[(3 + m)/
       6] HypergeometricPFQ[{1/2 + m/6}, {2/3, 5/6, 7/6, 4/3, 3/2},
       C^6/(46656 z)] -
     5 C^(4 + m) z^(1/6)
       Gamma[(4 + m)/
       6] HypergeometricPFQ[{2/3 + m/6}, {5/6, 7/6, 4/3, 3/2, 5/3},
       C^6/(46656 z)] +
     C^(5 + m)
       Gamma[(5 + m)/
       6] HypergeometricPFQ[{5/6 + m/6}, {7/6, 4/3, 3/2, 5/3, 11/6},
       C^6/(46656 z)] -
     120 C^m z^(5/6)
       Gamma[m/6] HypergeometricPFQ[{m/6}, {1/6, 1/3, 1/2, 2/3, 5/6},
       C^6/(46656 z)])}
3

Dear Peter,

Thanks for your answer. In fact, I would like to know which theorem you used to build your formula as I could not see Product function for i (1 to M) which has direct efferct on the output..

Your cooperation is highly appreciated!!