Is there a way to integrate the product of two functions numerically in Python?

Question

I have two functions which take multiple arguments:

import numpy as np
from scipy.integrate import quad

gamma_s=0.1 #eV
gamma_d=0.1 #eV
T=298 #K
homo=-5.5 #eV
Ef=-5 #eV
mu=0 #eV just displaces the function

#Fermi-Dirac distribution
k=8.617333262e-5 #eV/K
def fermi (E:float, mu:float, T:float) -> float:
    return 1/(1+np.exp((E-mu)/(k*T)))

#Lorentzian density of states
gamma=gamma_d+gamma_s
def DoS (E:float, gamma:float, homo:float, Ef:float) -> float:
    epsilon=homo-Ef
    v=E-epsilon
    u=gamma/2
    return gamma/(np.pi*((v*v)+(u*u)))

I know that if I want to integrate just one of them, say fermi, then I would use

quad(fermi, -np.inf, np.inf, args=(mu,T))

But I need the integral of their product fermi*DoS with respect to their common variable E, and I can't imagine how to do it with quad, since there is no mention of it in the documentation.

I guess I could define another function integrand as their product and compute its integral, however that sounds somewhat messy and I would prefer a cleaner way of doing it.

Answer 1

You don't have to define a new, standalone function if something inline is more appealing to you:

quad(lambda e: fermi(e, mu, T) * DoS(e, gamma, homo, T), -np.inf, np.inf)

That is, we use partial application to turn the product of fermi and DoS into a new Python lambda.

Just to give some mathematical justification for the need to do something like this...

Mathematically speaking, one can only integrate (integrable) functions (or elements of function spaces derived from integrable functions). To integrate the product of two functions, we have to say which function is meant by their product. After a while, this may feel obvious, but here I think it's worth noting that humans defined

(fg)(x) := f(x)g(x).

In the same way one must give mathematical meaning to the product of functions, one must give meaning to the product of two Python functions. Especially because Python functions can return all sorts of things, many of which make no sense to multiply, so there couldn't be a general definition.