Integration of Chebyshev Polynomials

Question

I have a task of expanding a Lorentzian representation of a Dirac - Delta with Chebyhsev polynomials, basically, the integral I need to perform is $$\frac{1}{\pi}\int_{-1}^{1}dx\frac{\eta}{(x-\mu)^2 + \eta ^2} \cos (n\cdot \arccos(x))$$ where $n$ is an integer and, $\mu$ is ]-1, 1[ and $\eta$ is small but $> 0$. If you could so kindly provide me with ideas, analytical or numerically i would appreciate. I can say that i already tried several ideas to numerically integrate but it starts failing with high order $n$, $n > 1000$, where there are heavy oscillations.

Answer 1

What have you tried so far? Is there any reason you can't brute force the integral like so

import numpy as np
from scipy import integrate

def integral_value(n,mu,eta):
    return integrate.quad(lambda x: eta/((x-mu)**2+eta**2)*np.cos(n*np.arccos(x))*1/(np.pi),-1,1,limit=1000)

print("I=",integral_value(1500,0.5,0.5)[0])
print("Error =",integral_value(1500,0.5,0.5)[1])

with $n=1500, \, \mu=\eta=0.5$.