How to Exclude Certain Intervals from Numerical Integration in Mathematica?

[Derivator]

hi,

i want to integrad the follwing 'integrand' funtion from -50 to 50

k := 1
\[CapitalDelta]1 := 1
\[CapitalDelta]2 := 2*\[CapitalDelta]1
T := (\[CapitalDelta]2/k)/10
f[E_, T_] := (Exp[E/(k*T)] + 1)^(-1)
Integrand[E_, eV_, T_] := 
 Abs[E]/Sqrt[E^2 - \[CapitalDelta]1^2]*Abs[E + eV]/
  Sqrt[(E + eV)^2 - \[CapitalDelta]2^2]*(f[E, T] - f[E + eV, T])

(You can copy and past this stuff to mathematica)

Now, I want to EXCLUDE all intervalls from numerical integration, where

E^2 > \[CapitalDelta]1
and
(E+eV)^2 > \[CapitalDelta]2

I.e. I want to do something like

NIntegrate[Integrand[Energy, eV, T], {Energy, INTERVALLBORDERS FULLFILLING THE CONDITIONS ABOVE}]

eV is then plugged into it by the plot function:

Plot[NIntegrate[Integrand[Energy, eV, T], {Energy, INTERVALLBORDERS FULLFILLING THE CONDITIONS ABOVE}], {eV, 0, 20}]

How is it possible to tell mathematica to use the required integration borders??

--
derivator

 

[mmwave]

Dear derivator,

Thank you for your inquiry. To exclude certain intervals from the numerical integration, you can use the "Exclusions" option in the NIntegrate function. This option allows you to specify any points or intervals to be excluded from the integration.

In your case, you can use the following code:

NIntegrate[Integrand[Energy, eV, T], {Energy, -50, 50},
Exclusions -> {E^2 > \[CapitalDelta]1 && (E + eV)^2 > \[CapitalDelta]2}]

This will exclude any points or intervals where E^2 > \[CapitalDelta]1 and (E + eV)^2 > \[CapitalDelta]2 from the integration. You can also specify multiple conditions in the "Exclusions" option by using the logical operators && (AND) and || (OR).

I hope this helps. Let me know if you have any further questions.