# Quadrature for numerical integration over infinite intervals

I am looking for book recommendations or hints on numerical integration over infinite intervals. I am particularly interested in integrals of the form

$$intlimits_{-infty}^{+infty} g(x) exp(p_d(x)) mathrm{d} x$$,

where $$g(x)$$ is an arbitrary continuous function (but not necessarily continuously differentiable) and $$p_d(x)$$ is some polynomial of even degree $$d > 2$$ with negative leading coefficient.

Moreover, I know the first few weights and abscissas of the corresponding Gaussian quadrature but have no rule to compute more for higher accuracy as there is no known family of orthogonal polynomials with respect to $$exp(p_d(x)), ;d>2$$, that I can make use of.

I’d be grateful for any hints or literature recommendations because I haven’t found a nice summary of suitable methods yet. Thank you in advance.

MathOverflow Asked by BernieD on December 31, 2020

I would convert the integration range to a finite interval, $$int_{-infty}^infty f(x)dx=int_0^1left[f(1/t-1)+f(-1/t+1)right]t^{-2}dt,$$ and then use an adaptive Gauss-Kronrod routine. Many computational libraries have code for that, for example, Matlab or Mathematica.

Answered by Carlo Beenakker on December 31, 2020

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