This paper demonstrates the generalized Rolle theorem and discussed the zero of Legendre polynomials.
According to the desired accuracy, we determine the number of terms in the Fourier series and Legendre polynomials, then the Practical simplified formulae are obtained.
All ordinary polynomials have series expansion of orthogonal polynomials, while Legendre polynomials, Hermite polynomials and Laguerre polynomials are special orthogonal polynomials.
The angular coordinate used a DVR based on Legendre polynomials and the radial coordinates utilized a DVR based on sine basis functions.
