Integrating interpolated function with singularities

Question

I have the following equations and List of values:

a0 = 0.520;
a1 = 0.902;
b0 = 0.520;
m = 0.13957;
rho[s_] = Sqrt[1 - 4 m^2/s];
step = (2)/20;
si = Table[(i) step, {i, 1, 21}]
ai = Table[{si[[i]], {a0 + a1 si[[i]]}}, {i, 1, 21}];
im = Table[0, {i, 1, 21}];

Now I have to interpolate imto any list of values in my region of interpolation, that is, si.

imi = Interpolation[Table[{si[[i]], im[[i]]}, {i, 1, 21}]];

And now I have to perform the following integral:

Rea = 
Table[ai[[i, 2]], {i, 1, 21}] + (1/Pi) Table[
 si[[i]] NIntegrate[imi[sp]/(sp (sp - si[[i]])), {sp, 4 m^2, 2}, 
   Method -> PrincipalValue, Exclusions -> Thread[si == sp], 
   AccuracyGoal -> 8], {i, 1, 21}];

Surprisingly, it says that:

NIntegrate::pvrng: Singular points must be specified in the integration range in order to use PrincipalValue

I don't understand this because I'm already specifying the Exclusions with Thread. Does someone know why does it say so?

Answer 1

The error message means you have to put the singularities si in the "integration range," {sp, 4m^2, 2}, which you can do like this:

Rea = Table[
    ai[[i, 2]], {i, 1, 21}] + (1/Pi) Table[
     si[[i]] NIntegrate[imi[sp]/(sp (sp - si[[i]])), 
       Evaluate@Flatten@{sp, 4 m^2, si, 2},           (* insert  si  here *)
       Method -> PrincipalValue, AccuracyGoal -> 8], {i, 1, 21}];

Rea
(*
  {{0.6102}, {0.7004}, {0.7906}, {0.8808}, {0.971},  {1.0612}, {1.1514}, 
   {1.2416}, {1.3318}, {1.422},  {1.5122}, {1.6024}, {1.6926}, {1.7828}, 
   {1.873},  {1.9632}, {2.0534}, {2.1436}, {2.2338}, {2.324},  {2.4142}}
*)