The error for the Gaussian quadrature formulae of order \( N \) is given by $$ \begin{equation*} \int_a^bW(x)f(x)dx-\sum_{k=1}^Nw_kf(x_k)=\frac{f^{2N}(\xi)}{(2N)!}\int_a^bW(x)[q_{N}(x)]^2dx \end{equation*} $$ where \( q_{N} \) is the chosen orthogonal polynomial and \( \xi \) is a number in the interval \( [a,b] \). We have assumed that \( f\in C^{2N}[a,b] \), viz. the space of all real or complex \( 2N \) times continuously differentiable functions.