In numerical analysis Gauss–Laguerre quadrature is an extension of Gaussian quadrature method for approximating the value of integrals of the following kind:

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

In this case

${\displaystyle \int _{0}^{+\infty }e^{-x}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})}$

where xi is the i-th root of Laguerre polynomial Ln(x) and the weight wi is given by [1]

${\displaystyle w_{i}={\frac {x_{i}}{(n+1)^{2}[L_{n+1}(x_{i})]^{2}}}.}$

## For more general functions

To integrate the function ${\displaystyle f}$ we apply the following transformation

${\displaystyle \int _{0}^{\infty }f\left(x\right)dx=\int _{0}^{\infty }f\left(x\right)e^{x}e^{-x}dx=\int _{0}^{\infty }g\left(x\right)e^{-x}dx}$

where ${\displaystyle g\left(x\right):=e^{x}f\left(x\right)}$. 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 ${\displaystyle x^{\alpha }}$ power-law singularity at x=0, for some real number ${\displaystyle \alpha >-1}$, leading to integrals of the form:

${\displaystyle \int _{0}^{+\infty }x^{\alpha }e^{-x}f(x)\,dx.}$

This allows one to efficiently evaluate such integrals for polynomial or smooth f(x) even when α is not an integer.[2]

## References

1. Equation 25.4.45 in {{#invoke:citation/CS1|citation |CitationClass=book }} 10th reprint with corrections.
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