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In numerical analysis Gauss–Laguerre quadrature is an extension of the 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.

## Generalized Gauss–Laguerre quadrature

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 Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions. Dover. ISBN 978-0-486-61272-0. 10th reprint with corrections.
2. ^ Rabinowitz, P.; Weiss, G. (1959). "Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form ${\displaystyle \int _{0}^{\infty }\exp(-x)x^{n}f(x)dx}$". Mathematical Tables and Other Aids to Computation 13: 285–294. doi:10.1090/S0025-5718-1959-0107992-3.