first-kind Gauss–Chebyshev numerical integration
Calculate the integral over the finite interval (a, b) using the first-kind Gauss-Chebyshev integration method.
| \(\normalsize Gauss-Chebyshev\ 1st\ quadrature\\{\large\int_{\small -1}^{\small 1}\frac{1}{\sqrt{1-x^2}}}f(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i)\\ {\large\int_a^{b}}g(x)dx\simeq{\large\frac{b-a}{2}\displaystyle \sum_{\small i=1}^{n}}w_{i}\sqrt{1-x_i^2}g({\large\frac{b-a}{2}}x_i+{\large\frac{b+a}{2}})\\\) |
The integrand f(x) is assumed to be analytic and not a periodic function.
| \(\normalsize Gaussian\ quadrature\\ \hspace{20px} {\large\int_{\small a}^{\small b}}w(x)f(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i), \hspace{20px} {\large\int_{\small a}^{\small b}}g(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}{\large\frac{w_{i}}{w(x_i)}}g(x_i)\\ Gauss-Chebyshev\ 1st\ quadrature\\ \hspace{30px} interval(a,b):\hspace{20px} (-1,\ 1)\\ \hspace{30px} w(x):\hspace{80px} \frac{1}{\sqrt{1-x^2}}\\ \hspace{30px} polynomialsl:\hspace{10px} T_n (x)\\ \) |
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