I'm trying to perform the following integral (numerically)

(1)

And their behaviour is

where PV denotes principal Value.

In principle I don't know the functions $\rm{Im}(\alpha)$ and $\rm{Re}(\alpha)$, but I know their behaviour between $s=0$ and $s=2$. I just need to know if the equation (1) from above is correct. For that purpose I made my own function of $\rm{Im}(\alpha)$ that behaves like the figure shown (I called it $\alpha$ first)

alpha[s_] = rho[s] (0.12 E^-(0.35 s)); 

where

rho[s_] = Sqrt[1 - 4 m^2/s];

and

m = 0.13957;

Then I made an interpolation to the function (just to practice numerical integration with known points of a function)

Tab1 = Table[{s, alpha[s]}, {s, 4 m^2, 2, 0.001}];
im = Interpolation[Tab1] ;

Supposing that at this point everything is all right, I have some points of my function and I want to deal with the numerical integration (1) shown above.

I divided the two first terms of the $\rm{Re}(\alpha)$:

step = (2 - 0.14^2)/20;
si = Table[4 m ^2 + (i - 1) step, {i, 1, 21}];
ai = Table[{si[[i]], {0.520 + 0.902 si[[i]]}}, {i, 1, 21}];

and the integration part goes as follows:

bi = NIntegrate[
Table[im[s]/(s (s - si[[i]])), {i, 1, 21}], {s, 4 m^2, 2}, 
Method -> PrincipalValue]

Obiously this integration doesn't give the result I wanted because I have to specify in which points the denominator goes to zero (so it can apply the PV).

Here comes my question. This $bi$ diverges everytime $s=si$. How can I specify in my function these points? I tried using the option

Method->{"PrincipalValue", "SingularPointsIntegrationRadius"-> }]

but it seems Mathematica 9 (the one I'm using) doesn't have this option. I also tried taking few points and specifying the singular points, but the function also diverges when $s$ tends to $si$ so it doesn't work either. Does someone have any idea how to do it?

PS: I don't want to specify what the variables mean because it's a much larger problem.

I'm trying to perform the following integral (numerically)

(1)

And their behaviour is

where PV denotes principal Value.

In principle I don't know the functions $\rm{Im}(\alpha)$ and $\rm{Re}(\alpha)$, but I know their behaviour between $s=0$ and $s=2$. I just need to know if the equation (1) from above is correct. For that purpose I made my own function of $\rm{Im}(\alpha)$ that behaves like the figure shown (I called it $\alpha$ first)

alpha[s_] = rho[s] (0.12 E^-(0.35 s)); 

where

rho[s_] = Sqrt[1 - 4 m^2/s];

and

m = 0.13957;

Then I made an interpolation to the function (just to practice numerical integration with known points of a function)

Tab1 = Table[{s, alpha[s]}, {s, 4 m^2, 2, 0.001}];
im = Interpolation[Tab1] ;

Supposing that at this point everything is all right, I have some points of my function and I want to deal with the numerical integration (1) shown above.

I divided the two first terms of the $\rm{Re}(\alpha)$:

step = (2 - 0.14^2)/20;
si = Table[4 m ^2 + (i - 1) step, {i, 1, 21}];
ai = Table[{si[[i]], {0.520 + 0.902 si[[i]]}}, {i, 1, 21}];

and the integration part goes as follows:

bi = 
  NIntegrate[Table[im[s]/(s (s - si[[i]])), {i, 1, 21}], {s, 4 m^2, 2}, 
    Method -> PrincipalValue]

Obiously this integration doesn't give the result I wanted because I have to specify in which points the denominator goes to zero (so it can apply the PV).

Here comes my question. This $bi$ diverges everytime $s=si$. How can I specify in my function these points? I tried using the option

Method -> {"PrincipalValue", "SingularPointsIntegrationRadius" -> }]

but it seems Mathematica 9 (the one I'm using) doesn't have this option. I also tried taking few points and specifying the singular points, but the function also diverges when $s$ tends to $si$ so it doesn't work either. Does someone have any idea how to do it?

P.S.: I don't want to specify what the variables mean because it's a much larger problem.

I'm trying to perform the following integral (numerically) [![][1]][1] (1)

And their behaviour is

[![Behaviour if Re and $Im\alpha$][2]][2] where PV denotes principal Value.

In principle I don't know the functions $Im\alpha$ and Re$\alpha$ but I know their behaviour between $s=0$ and $s=2$. I just need to know if the equation (1) from above is correct. For that purpose I made my own function of $Im\alpha$ that behaves like the figure shown (I called it $\alpha$ first)

alpha[s_] = rho[s] (0.12 E^-(0.35 s)); 

Being

rho[s_] = Sqrt[1 - 4 m^2/s];

and

m = 0.13957;

now I made an interpolation to the function (just to practice numerical integration with known points of a function)

Tab1 = Table[{s, alpha[s]}, {s, 4 m^2, 2, 0.001}];
im = Interpolation[Tab1] ;

Suppose that until this point everything is alright. So I have some points of my function and I want to deal with the numerical integration (1) shown above. I divided the 2 first terms of the $Re\alpha$:

step = (2 - 0.14^2)/20;
si = Table[4 m ^2 + (i - 1) step, {i, 1, 21}];
ai = Table[{si[[i]], {0.520 + 0.902 si[[i]]}}, {i, 1, 21}];

And the integration part comes as follows:

bi = NIntegrate[
Table[im[s]/(s (s - si[[i]])), {i, 1, 21}], {s, 4 m^2, 2}, 
Method -> PrincipalValue]

Obiously this integration doesn't give the result I wanted because I have to specify in which points the denominator goes to zero (so it can apply the PV). Here comes my question. This $bi$ diverges everytime $s=si$. How can I specify in my function these points? I tried using the option

Method->{"PrincipalValue", "SingularPointsIntegrationRadius"-> }]

but it seems Mathematica 9 (the one I'm using) doesn't have this option. I also tried taking few points and specifying the singular points, but the function also diverges when $s$ tends to $si$ so it doesn't work either. Does someone have any idea how to do it?

Thank you very much.

PS: I don't want to specify what the variables mean because it's a much larger problem. [1]: https://i.sstatic.net/dgDvD.jpg [2]: https://i.sstatic.net/6rIX1.jpg

I'm trying to perform the following integral (numerically)

(1)

And their behaviour is

where PV denotes principal Value.

In principle I don't know the functions $\rm{Im}(\alpha)$ and $\rm{Re}(\alpha)$, but I know their behaviour between $s=0$ and $s=2$. I just need to know if the equation (1) from above is correct. For that purpose I made my own function of $\rm{Im}(\alpha)$ that behaves like the figure shown (I called it $\alpha$ first)

alpha[s_] = rho[s] (0.12 E^-(0.35 s)); 

where

rho[s_] = Sqrt[1 - 4 m^2/s];

and

m = 0.13957;

Then I made an interpolation to the function (just to practice numerical integration with known points of a function)

Tab1 = Table[{s, alpha[s]}, {s, 4 m^2, 2, 0.001}];
im = Interpolation[Tab1] ;

Supposing that at this point everything is all right, I have some points of my function and I want to deal with the numerical integration (1) shown above.

I divided the two first terms of the $\rm{Re}(\alpha)$:

step = (2 - 0.14^2)/20;
si = Table[4 m ^2 + (i - 1) step, {i, 1, 21}];
ai = Table[{si[[i]], {0.520 + 0.902 si[[i]]}}, {i, 1, 21}];

and the integration part goes as follows:

bi = NIntegrate[
Table[im[s]/(s (s - si[[i]])), {i, 1, 21}], {s, 4 m^2, 2}, 
Method -> PrincipalValue]

Obiously this integration doesn't give the result I wanted because I have to specify in which points the denominator goes to zero (so it can apply the PV).

Here comes my question. This $bi$ diverges everytime $s=si$. How can I specify in my function these points? I tried using the option

Method->{"PrincipalValue", "SingularPointsIntegrationRadius"-> }]

but it seems Mathematica 9 (the one I'm using) doesn't have this option. I also tried taking few points and specifying the singular points, but the function also diverges when $s$ tends to $si$ so it doesn't work either. Does someone have any idea how to do it?

PS: I don't want to specify what the variables mean because it's a much larger problem.