Summation for a function having singularity

Question

I need to do summation of this function with Mathematica.

For[t=0,t<500,t++;Sum[-Q*Log[Q],{k,-Infinity,Infinity},{n,-Infinity,Infinity}]]

Q= Q1+Q2+Q3

where

Q1= 1/(Sqrt[(0.2+ 0.2 k)^2] Sqrt[
  k^2]) ((0.0220725+ 0.0220725 k) Sqrt[k^2]
     Erf[Sqrt[(0.2+ 0.2 k)^2]] - 
   0.110363 Sqrt[(0.2+ 0.2 k)^2]
     k Erf[0.2 Sqrt[k^2]]) (0.886227 Erf[0.8 E^(-0.0004 t) - 0.2 n] + 
   0.886227 Erf[0.2- 0.8 E^(-0.0004 t) + 0.2 n])

Q2=1/(Sqrt[(0.2+ 0.2 k)^2] Sqrt[
  k^2]) ((0.0220725+ 0.0220725 k) Sqrt[k^2]
     Erf[Sqrt[(0.2+ 0.2 k)^2]] - 
   0.110363 Sqrt[(0.2+ 0.2 k)^2]
     k Erf[0.2 Sqrt[k^2]]) (-0.886227 Erf[
     0.8 E^(-0.0004 t) + 0.2 n] + 
   0.886227 Erf[0.2+ 0.8 E^(-0.0004 t) + 0.2 n])

Q3=(Erf[0.2+ 0.2 n] - 
   1. Erf[0.2 n]) (0.0271938 Erf[(0.2+ 
        0. I) - (0.\[VeryThinSpace]+ 0.8 I) Sqrt[
       E^(-0.0004 t)] + (0.2+ 0. I) k] + 
   0.0271938 Erf[(0.2+ 
        0. I) + (0.+ 0.8 I) Sqrt[
       E^(-0.0004 t)] + (0.2+ 
         0. I) k] - (0.+ 0.0271938 I) Erfi[
     0.8 Sqrt[
       E^(-0.0004 t)] - (0.+ 
         0.2 I) k] + (0.+ 0.0271938 I) Erfi[
     0.8 Sqrt[E^(-0.0004 t)] + (0.+ 0.2 I) k])

The problem is because Q cannot be defined at certain k,Sum cannot be done. (For example, when I try to sum Sum[-Q*Log[Q],{n,-55,55},{k,-55,55}] at t=0, the error message is "Power::infy: "Infinite expression 1/Sqrt[0] encountered", "indet: Indeterminate expression 0.ComplexInfinity encountered."

Is there a way that i can sum except the points where -Q*Log[Q] cannot be defined?

Thanks.

Answer 1

Since the sum is from -Infinity to +Infinity it will have to be computed algebraically, so you cannot intercept the errors like so :-

Sum[If[NumberQ[#], #, 0] &@Chop[-Q*Log[Q]],
  {k, -Infinity, Infinity}, {n, -Infinity, Infinity}];

One source of error is, for instance, in Q1, when k = 1 ...

1/(Sqrt[(0.2 + 0.2 k)^2] Sqrt[k^2])

the denominator is zero.

You can avoid this problem by summing without k = 1 :-

a1 = Sum[-Q*Log[Q], {k, -3, 0}, {n, -3, 3}];
a2 = Sum[-Q*Log[Q], {k, 2, 3}, {n, -3, 3}];
ans = a1 + a2

However, k = 1 is not the only problematic input. You may be able to identify the others and exclude them from the summation.