In numerical analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind:

$\int _{0}^{+\infty }e^{-x}f(x)\,dx.$ In this case

$\int _{0}^{+\infty }e^{-x}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})$ where xi is the i-th root of Laguerre polynomial Ln(x) and the weight wi is given by 

$w_{i}={\frac {x_{i}}{\left(n+1\right)^{2}\left[L_{n+1}\left(x_{i}\right)\right]^{2}}}.$ ## For more general functions

To integrate the function $f$ we apply the following transformation

$\int _{0}^{\infty }f(x)\,dx=\int _{0}^{\infty }f(x)e^{x}e^{-x}\,dx=\int _{0}^{\infty }g(x)e^{-x}\,dx$ where $g\left(x\right):=e^{x}f\left(x\right)$ . For the last integral one then uses Gauss-Laguerre quadrature. Note, that while this approach works from an analytical perspective, it is not always numerically stable.

More generally, one can also consider integrands that have a known $x^{\alpha }$ power-law singularity at x=0, for some real number $\alpha >-1$ , leading to integrals of the form:
$\int _{0}^{+\infty }x^{\alpha }e^{-x}f(x)\,dx.$ 