Why are the results different when I integrate an expression in different ways?

Question

When I integrate the expression like,

A1=(0.310314 ((0.00564116 - 4.25452 I) + (0.149772 - 55.6062 I) \[Beta] + \[Beta]^2 - (0. + 0.0765116 I) \[Omega] - I \[Beta] \[Omega]))/(236.578 + 3092.07 \[Beta] + 0.223032 \[Beta]^2 + \[Beta]^3 + 8.50904 \[Omega] + 111.212 \[Beta] \[Omega] +  0.0765116 \[Omega]^2 + \[Beta] \[Omega]^2) - (0.0239123 ((0.000620872 - 0.471257 I) + 0.0084749 \[Beta] - (0. + 0.0084749 I) \[Omega]))/((236.578 + 3092.07 \[Beta] + 0.223032 \[Beta]^2 + \[Beta]^3 + 8.50904 \[Omega] + 111.212 \[Beta] \[Omega] + 0.0765116 \[Omega]^2 + \[Beta] \[Omega]^2) (0.0732601 -I (55.7334 + \[Omega])))

Three ways

((Integrate[ A1 \[Rho][\[Omega]], \[Omega]] /. {\[Omega] -> \[Omega]1}) -
(Integrate[ A1 \[Rho][\[Omega]], \[Omega]] /. {\[Omega] -> \[Omega]2})) {\[Beta] -> 0.6295615718735122` + 0.07944775685111569` I}

((Integrate[A1 \[Rho][\[Omega]] /. {\[Beta] -> 0.6295615718735122` + 0.07944775685111569` I}, \[Omega]] /. {\[Omega] -> \[Omega]1}) - (Integrate[A1 \[Rho][\[Omega]] /. {\[Beta] -> 0.6295615718735122` + 0.07944775685111569` I}, \[Omega]] /. {\[Omega] -> \[Omega]2}))

Integrate[A1 \[Rho][\[Omega]] /. {\[Beta] ->0.6295615718735122` + 0.07944775685111569` I}, {\[Omega], \[Omega]2, \[Omega]1}]

give three different results

  0.356811 + 5.38461*10^-6 I

-0.407398 + 0.228095 I

0.355888 + 0.0000238939 I

where \[Rho][\[Omega]]=0.758333 + 1/3 (-55.6062 - \[Omega]) and \[Omega]1=-54.9468,\[Omega]2=-56.2655.

What is the problem here? And how to determine which one is right?

Answer 1

Clear["Global`*"]

The order of operations affects the precision. Use exact values until the final result and then use arbitrary precision rather than machine precision.

ρ[ω_] = 0.758333 + 1/3 (-55.6062 - ω) //
    Rationalize[#, 0] & // Simplify;
ω1 = -54.9468 // Rationalize;
ω2 = -56.2655 // Rationalize;

βrule = {β -> Rationalize[
     0.6295615718735122` + 0.07944775685111569` I, 0]};

A1 = (0.310314 ((0.00564116 - 4.25452 I) +
          (0.149772 - 55.6062 I) β + β^2 -
          (0. + 0.0765116 I) ω - I β ω))/
      (236.578 + 3092.07 β + 0.223032 β^2 +
        β^3 + 8.50904 ω + 111.212 β ω +
        0.0765116 ω^2 + β ω^2) -
     (0.0239123 
        ((0.000620872 - 0.471257 I) + 0.0084749 β -
          (0. + 0.0084749 I) ω))/
      ((236.578 + 3092.07 β + 0.223032 β^2 +
          β^3 + 8.50904 ω + 111.212 β ω +
          0.0765116 ω^2 + β ω^2) 
        (0.0732601 - I (55.7334 + ω))) //
    Rationalize[#, 0] & // Simplify;

Method 1

N[(
   (Integrate[A1 ρ[ω], ω] /. {ω -> ω1}) -
    (Integrate[A1 ρ[ω], ω] /. {ω -> ω2})) /.
  βrule, 15]

(* 0.342963552432318 + 0.002442574599151 I *)

Method 2

N[(
  (Integrate[
      A1 ρ[ω] /. βrule, ω] /. {ω -> ω1}) -
   (Integrate[
      A1 ρ[ω] /. βrule, ω] /. {ω -> ω2})),
 15]

(* 0.342963552432318 + 0.002442574599151 I *)

Method 3

N[Integrate[A1 ρ[ω] /. βrule, {ω, ω2, ω1}], 15]

(* 0.342963552432318 + 0.002442574599151 I *)

Comparing results,

% == %% == %%%

(* True *)