Creating a Matrix from Integrals

Question

I have a table of integrals that I want to put in an nxn matrix. I tried doing it in the following way

phix[x_, n_] := Exp[-n \[Alpha] x^2/2]
phiy[y_, m_] := Exp[-m \[Beta] y^2/2]
const = {List[
    Integrate[
     x^2 y^2 phix[x, n1] phix[x, n2] phiy[y, m1] phiy[y, m2], {x, 0, 
      Infinity}, {y, 0, Infinity}], {n1, 1, 3}, {n2, 1, 3}, {m1, 1, 
     3}, {m2, 1, 3}]} // MatrixForm

but what I get as output is the following, instead of the matrix form.

I also tried to use Table instead of list, but still don't get the output in the matrix form. I need the output as a matrix because I would like to calculate the e-values and e-vectors.

Any help would be greatly appreciated.

Answer 1

You probably want to use the option Assumptions.

Some coding suggestions:

• consider making the matrix of integrals first keeping phix and phiy undefined, and

• then provide definitions through Block or ReplaceAll.

Here is an example:

Clear[const, phix, phiy]

const = Table[
   Integrate[
    x^2 y^2 phix[x, n1] phix[x, n2] phiy[y, m1] phiy[y, m2], {x, 0, 
     Infinity}, {y, 0, Infinity}, 
    Assumptions -> {\[Alpha] > 0, \[Beta] > 0}], {n1, 1, 3}, {n2, 1, 
    3}, {m1, 1, 3}, {m2, 1, 3}];
Magnify[MatrixForm[const], 0.6]

Block[{phix, phiy},
 phix[x_, n_] := Exp[-n \[Alpha] x^2/2];
 phiy[y_, m_] := Exp[-m \[Beta] y^2/2];
 MatrixForm[const]
]

Magnify[MatrixForm[const], 0.6] /. {phix[u_, v_] :> 1/(v + u)^3, 
  phiy[u_, v_] :> 1/(u + v)^3}