Gauss–Laguerre quadrature

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. ==Generalized Gauss–Laguerre quadrature==

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. In this case, the weights are given in terms of the generalized Laguerre polynomials: :w_i = \frac{\Gamma(n+\alpha+1) x_i}{n!(n+1)^2 [L_{n+1}^{(\alpha)}(x_i)]^2} \,, where x_i are the roots of L_n^{(\alpha)}. This allows one to efficiently evaluate such integrals for polynomial or smooth f(x) even when α is not an integer. {{cite journal |first1=P. |last1=Rabinowitz |authorlink1=Philip Rabinowitz (mathematician) |first2=G. |last2=Weiss |title=Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form \int_0^{\infty} \exp(-x) x^n f(x)\,dx |journal=Mathematical Tables and Other Aids to Computation |volume=13 |pages=285–294 |year=1959 |doi=10.1090/S0025-5718-1959-0107992-3 ==References==