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Mapping integration points and weights

If we have an integral on the form \begin{equation*} \int_0^{\infty}f(t)dt, \end{equation*} we can choose new mesh points and weights by using the mapping \begin{equation*} \tilde{x}_i=tan\left\{\frac{\pi}{4}(1+x_i)\right\}, \end{equation*} and \begin{equation*} \tilde{\omega}_i= \frac{\pi}{4}\frac{\omega_i}{cos^2\left(\frac{\pi}{4}(1+x_i)\right)}, \end{equation*} where x_i and \omega_i are the original mesh points and weights in the interval [-1,1] , while \tilde{x}_i and \tilde{\omega}_i are the new mesh points and weights for the interval [0,\infty) .