Beck Knight

12/01/2024 · Senior High School

* stirling's Formula \( P_{n}(x)=y_{0}+\frac{\alpha\left(\Delta y_{0}+\Delta y_{-1}\right)}{2}+\frac{\alpha^{2}}{2!} \Delta^{2} y_{-1}+\frac{\alpha\left(\alpha^{2}-i\right)}{3} \) \( \cdot\left(\frac{\Delta^{3} y_{-1}+\Delta^{3} y_{-2}}{2}\right)+\frac{\alpha^{2}\left(\alpha^{2}-i\right)}{4!} \Delta^{4} y_{-2}+\cdots \) \( \alpha=\frac{x-x_{0}}{h}, h=x_{i+1}-x_{i} \) \( P_{n}(x)=\frac{1}{h}\left[\left(\frac{\Delta y_{0}+\Delta y_{-1}}{2}\right)-\frac{1}{6}\left(\frac{\Delta^{3} y_{-1}+\Delta^{3} y_{-2}}{2}\right)\right. \) \( +\frac{1}{30}\left[\frac{\Delta^{5} y_{-2}+\Delta^{5} y_{-3}}{2}\right]+\cdots \)