pyhs3.normalization.gauss_legendre_integral¶

pyhs3.normalization.gauss_legendre_integral(expression, variable, lower, upper)[source]¶

Compute a definite integral symbolically via Gauss-Legendre quadrature.

Gauss-Legendre quadrature approximates the integral over \([a, b]\) by evaluating the integrand at specific nodes and summing with weights. This implementation uses 64-point quadrature, which integrates polynomials up to degree 127 exactly.

The standard Gauss-Legendre formula is defined on \([-1, 1]\):

\[\int_{-1}^{1} f(t)\,dt \approx \sum_{i=1}^{N} w_i\,f(t_i)\]

To integrate over an arbitrary interval \([a, b]\), we apply a linear change of variables:

\[x = \frac{b-a}{2}t + \frac{a+b}{2}, \quad t \in [-1, 1], \quad x \in [a, b]\]

The Jacobian \(dx = \frac{b-a}{2}dt\) (the “half width”) transforms the integral to:

\[\int_a^b f(x)\,dx = \frac{b-a}{2} \int_{-1}^{1} f\left(\frac{b-a}{2}t + \frac{a+b}{2}\right) dt\]

Applying Gauss-Legendre quadrature yields:

\[\int_a^b f(x)\,dx \approx \frac{b-a}{2} \sum_{i=1}^{N} w_i\,f\left(\frac{b-a}{2}t_i + \frac{a+b}{2}\right)\]

where \(t_i\) are the standard Legendre nodes and \(w_i\) are the standard weights on \([-1, 1]\).