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Is there any way to use the NIntegrate function with constants?

f = Exp[-Sqrt[x^2 + y^2 + z^2]/λ]*Cos[π*x/a]*Cos[π*y/b]*Cos[π*z/c]
d = Laplacian[f, {x, y, z}, "Cartesian"] // Simplify
ren = -2/Sqrt[x^2 + y^2 + z^2]
s = d + ren

The above ' s ' is the function I would like to integrate.

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  • $\begingroup$ If you don't specify a value for λ, NIntegrate[] won't work. $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Nov 4, 2017 at 19:44
  • $\begingroup$ You also need bounds for the integral. Maybe you want a function of the constants that does the integral? Then try something like sFcn[x_, y_, z_, a_, b_, c_, λ_] = s; and J[a_, b_, c_, λ_] := NIntegrate[ sFcn[x, y, z, a, b, c, λ], {x, 0, 1}, {y, 0, 1}, {z, 0, 1}] $\endgroup$
    – Will.Mo
    Commented Nov 4, 2017 at 20:32
  • $\begingroup$ How could ıI do this? $\endgroup$
    – umos
    Commented Nov 4, 2017 at 20:55
  • $\begingroup$ @umos Use the 'at' symbol to identify who you are speaking to. Try executing the code I posted above, and see if it does what you want. If you have further questions, try to be more specific. $\endgroup$
    – Will.Mo
    Commented Nov 5, 2017 at 7:02
  • $\begingroup$ The codes you have given gives me the error as I try to evaluate function J NIntegrate::inumr: The integrand -(2/Sqrt[x^2+y^2+z^2])+(E^(-(Sqrt[x^2+y^2+z^2]/[Lambda]_)) (2 [Pi] x Cos[([Pi] y)/Pattern[<<2>>]] Cos[([Pi] z)/Pattern[<<2>>]] a_ b_^2 c_^2 [Lambda]_ Sin[([Pi] x)/Pattern[<<2>>]]+Cos[([Pi] x)/Pattern[<<2>>]] (<<1>>)))/(Sqrt[x^2+y^2+z^2] a_^2 b_^2 c_^2 [Lambda]_^2) has evaluated to non-numerical values for all sampling points in the region with boundaries {{0,1},{0,1},{0,1}}. $\endgroup$
    – user53311
    Commented Nov 5, 2017 at 11:45

1 Answer 1

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As @J.M. said, NIntegrate doesn't work when the integrand is non-numerical. You also must specify numerical bounds. You can build a function that returns a result for arbitrarily specified numerical constants as follows

Your code:

f = Exp[-Sqrt[x^2 + y^2 + z^2]/λ]*Cos[π*x/a]*Cos[π*y/b]*Cos[π*z/c];
d = Laplacian[f, {x, y, z}, "Cartesian"] // Simplify;
ren = -2/Sqrt[x^2 + y^2 + z^2];
s = d + ren;

My code:

sFcn[x_, y_, z_, a_, b_, c_, λ_] = s;
J[a_, b_, c_, λ_] := 
  NIntegrate[
   sFcn[x, y, z, a, b, c, λ], {x, 0, 1}, {y, 0, 1}, {z, 0, 
    1}];

Example execution:

J[1, 1, 1, 1]

-2.36029

The reason it works is that the constants are all numerical when NIntegrate is finally called. If you insist on non-numerical input, e.g. J[a, 1, 1, 1], you will get the error again (NIntegrate::inumr).

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