Integration by Numerical Methods

Question

I have a big expression,

    (E^(-((Ep^2 + Eq^2 + p^2 + q^2)/(
2 σ^2))) (E^(-I (θp + θq))
   p q Sin[ϕp] Sin[ϕq] Sinh[ζ/
   2]^2 + ((Ep + m) Cosh[ζ/2] + 
    p Cos[ϕp] Sinh[ζ/2]) ((Eq + m) Cosh[ζ/2] + 
    q Cos[ϕq] Sinh[ζ/2])) (1/
  2 (Ep + m) (Eq + m) (1 + Cosh[ζ]) + 
 p q (Cos[ϕp] Cos[ϕq] + 
    E^(I (θp + θq))
      Sin[ϕp] Sin[ϕq]) Sinh[ζ/2]^2 + 
 1/2 ((Eq + m) p Cos[ϕp] + (Ep + 
       m) q Cos[ϕq]) Sinh[ζ]))/(2 (Ep + m) (Eq + 
 m) n n1 (m + Ep Cosh[ζ] + 
 p Cos[ϕp] Sinh[ζ]) (m + Eq Cosh[ζ] + 
 q Cos[ϕq] Sinh[ζ]))

   

I tried analytically integrating it over p,q,$\theta p$,$\theta q$,$\phi p$,$\phi p$ with the limits $(-\infty,\infty)$,$(-\infty,\infty)$,$(0,\pi)$,$(0,\pi)$,$(0,2\pi)$,$(0,2\pi)$, it does not work, i.e., I ran it for 24 hours before aborting. I have 16 such expressions so that is not feasible for me

NIntegrate tells me to give all values numerically. But I want an expression with just the above expression integrated out. Is there any way to do that?

Answer 1

Expand your expression into a sum of simple fractions and integrate step by step over the different variables

$$\text{df}=\frac{e^{-\frac{\text{Ep}^2+\text{Eq}^2+p^2+q^2}{2 \sigma ^2}} \left(\left((\text{Ep}+m) \cosh \left(\frac{\zeta }{2}\right)+p \sinh \left(\frac{\zeta }{2}\right) \cos (\text{$\phi $p})\right) \left((\text{Eq}+m) \cosh \left(\frac{\zeta }{2}\right)+q \sinh \left(\frac{\zeta }{2}\right) \cos (\text{$\phi $q})\right)+p q \sinh ^2\left(\frac{\zeta }{2}\right) e^{-i (\text{$\theta $p}+\text{$\theta $q})} \sin (\text{$\phi $p}) \sin (\text{$\phi $q})\right) \left(\frac{1}{2} (\text{Ep}+m) (\text{Eq}+m) (\cosh (\zeta )+1)+\frac{1}{2} \sinh (\zeta ) (q (\text{Ep}+m) \cos (\text{$\phi $q})+p (\text{Eq}+m) \cos (\text{$\phi $p}))+p q \sinh ^2\left(\frac{\zeta }{2}\right) \left(\cos (\text{$\phi $p}) \cos (\text{$\phi $q})+e^{i (\text{$\theta $p}+\text{$\theta $q})} \sin (\text{$\phi $p}) \sin (\text{$\phi $q})\right)\right)}{2 n \text{n1} (\text{Ep}+m) (\text{Eq}+m) (\text{Ep} \cosh (\zeta )+m+p \sinh (\zeta ) \cos (\text{$\phi $p})) (\text{Eq} \cosh (\zeta )+m+q \sinh (\zeta ) \cos (\text{$\phi $q}))}$$

    (  dfn = List @@ (df // Apart // Expand))[[1]] // TeXForm

    

$$\frac{\text{Ep} \ \text{Eq} \ \cosh ^2 \ \left(\frac{\zeta }{2}\right) \ \exp \left(-\frac{\text{Ep}^2}{2 \sigma ^2}-\frac{\text{Eq}^2}{2 \sigma ^2}-\frac{p^2}{2 \sigma ^2}-\frac{q^2}{2 \sigma ^2}\right)}{4\ n \ \text{n1}\ (\text{Ep} \ \cosh (\zeta )\ +\ m \ +\ p \ \sinh (\zeta ) \ \cos (\text{$\phi $p}))\ (\text{Eq} \ \cosh (\zeta )\ + \ m \ + \ q \ \sinh (\zeta ) \ \cos (\text{$\phi $q}))}$$

Drop any factor not dependent on the integration variable and integrate stepwise

     dfn[[1]] /. {x__?(FreeQ[\[Phi]q])*y_ :> y}

$$\frac{1}{\text{Eq} \ \cosh (\zeta )\ + \ m \ + \ q \ \sinh (\zeta ) \ \cos (\text{$\phi $q})}$$

    Assuming[Sinh[\[Zeta]] > 0 && E^\[Zeta] > 1 && Cosh[\[Zeta]] > 1 &&m >0 &&  Eq > 0 , 
       Integrate[1/( m + Eq Cosh[\[Zeta]] + q Cos[\[Phi]q] Sinh[\[Zeta]]),\[Phi]q, 0, 2 \[Pi]}]]

              

$$\begin{array}{cc} \{ & \begin{array}{cc} -\frac{4 \pi e^{\zeta }}{\sqrt{2 \text{Eq}^2 e^{2 \zeta }+\text{Eq}^2 e^{4 \zeta }+\text{Eq}^2+4 \text{Eq} e^{\zeta } m+4 \text{Eq} e^{3 \zeta } m+4 e^{2 \zeta } m^2+2 e^{2 \zeta } q^2-e^{4 \zeta } q^2-q^2}} & \left| \frac{-2 e^{2 \zeta } \text{Eq}-2 \text{Eq}-4 e^{\zeta } m+\sqrt{\left(2 e^{2 \zeta } \text{Eq}+2 \text{Eq}+4 e^{\zeta } m\right)^2-4 \left(e^{2 \zeta } q-q\right)^2}}{e^{2 \zeta } q-q}\right| \geq 2\land \left| \frac{-2 e^{2 \zeta } \text{Eq}-2 \text{Eq}-4 e^{\zeta } m-\sqrt{\left(2 e^{2 \zeta } \text{Eq}+2 \text{Eq}+4 e^{\zeta } m\right)^2-4 \left(e^{2 \zeta } q-q\right)^2}}{e^{2 \zeta } q-q}\right| <2 \\ \frac{4 \pi e^{\zeta }}{\sqrt{2 \text{Eq}^2 e^{2 \zeta }+\text{Eq}^2 e^{4 \zeta }+\text{Eq}^2+4 \text{Eq} e^{\zeta } m+4 \text{Eq} e^{3 \zeta } m+4 e^{2 \zeta } m^2+2 e^{2 \zeta } q^2-e^{4 \zeta } q^2-q^2}} & \left| \frac{-2 e^{2 \zeta } \text{Eq}-2 \text{Eq}-4 e^{\zeta } m+\sqrt{\left(2 e^{2 \zeta } \text{Eq}+2 \text{Eq}+4 e^{\zeta } m\right)^2-4 \left(e^{2 \zeta } q-q\right)^2}}{e^{2 \zeta } q-q}\right| <2\land \left| \frac{-2 e^{2 \zeta } \text{Eq}-2 \text{Eq}-4 e^{\zeta } m-\sqrt{\left(2 e^{2 \zeta } \text{Eq}+2 \text{Eq}+4 e^{\zeta } m\right)^2-4 \left(e^{2 \zeta } q-q\right)^2}}{e^{2 \zeta } q-q}\right| \geq 2 \\ \end{array} \\ \end{array}$$

Since all members in the list have product form in the variables, its possible to integrate the union of all factors as 1-d integrals over their respective intervals. This is making sense only if the conditions can be evaluated beforehand by fixing the numerical factors in their respective physical ranges.

Store the intgral list of the integral factors obtained as a list of rules int the form

  expr_ dfk :> expr intdfk

and apply it to the list dfn

  Plus@@  ( dfn//.rules)

If there are integrals unevaluated, do the by Nintegrate with numerical coefficients.

This strategy breaks down, if the conditions stemming from residuum evaluations, are more complex than the n-fold integral.