5.1.3 勒让德多项式的正交性及相应的广义傅里叶级数

合集 - 数学物理方程 姜颖版 笔记(11)

1.3.4施图姆-刘维尔本征值问题2024-11-142.4.1 线性齐次常微分方程解的相关性2024-11-133.4.2二阶线性齐次常微分方程在正常点附近邻域内的级数解法2024-11-144.4.3勒让德方程正常点附近邻域上的求解2024-11-145.4.4二阶齐次常微分方程在正则奇点附近邻域的级数解法2024-11-146.4.5贝塞尔方程求解2024-11-147.4.6二阶非齐次常微分方程的求解2024-11-148.5.1.1球坐标系下的拉普拉斯方程分离变数法求解2024-11-149.5.1.2勒让德多项式2024-11-14

10.5.1.3 勒让德多项式的正交性及相应的广义傅里叶级数2024-11-14

11.5.1.4具有极轴转动对称性的拉普拉斯问题求解2024-11-14

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勒让德多项式的正交性

对于不同项的勒让德多项式：

$$\{\begin{array}{l}(1-x^2)P_n^{″}(x)-2x{P}_n^{\prime }(x)+n(n+1)P_n(x)\equiv 0,(1)\\ (1-x^2)P_m^{″}(x)-2x{P}_m^{\prime }(x)+m(m+1)P_m(x)\equiv 0,(2)\end{array}$$

证明其正交性：
${\int }_{-1}^1\{(1)\times P_m(x)-(2)\times P_n(x)\}dx$

${\int }_{-1}^1\{P_m(x)\frac{d}{dx}[(1-x^2)P_n^{\prime }(x)]-P_n(x)\frac{d}{dx}[(1-x^2)P_m^{\prime }(x)]\}dx+[n(n+1)-m(m+1)]{\int }_{-1}^1{P}_n(x)P_m(x)dx\equiv 0$

$(1-x^2)[P_m(x)P_n^{\prime }(x)-P_n(x)P_m^{\prime }(x){]}_{-1}^1+{\int }_{-1}^1(1-x^2)[P_m^{\prime }(x)P_n^{\prime }(x)-P_n^{\prime }(x)P_m^{\prime }(x)]dx$
$+[n(n+1)-m(m+1)]{\int }_{-1}^1{P}_n(x)P_m(x)dx\equiv 0$

$${\int }_{-1}^1{P}_n(x)P_m(x)dx=0,m\ne n$$

不同阶勒让德多项式正交！

$\{P_l(x),l=0,1,2,\cdots \}$ — $[-1,1]$上连续函数空间的正交完备函数基底

$$f(x)=\sum _{l=0}^{\mathrm{\infty }}{A}_l{P}_l(x),A_l=\frac{1}{{\int }_{-1}^1[P_l{]}^{2}dx}{\int }_{-1}^{1}f(x)P_l(x)dx$$

广义傅里叶级数展开

$${\int }_0^{\pi }{P}_n(\cos \theta )P_m(\cos \theta )\sin \theta d\theta \equiv 0,m\ne n$$

$$f(\theta )=\sum _{l=0}^{\mathrm{\infty }}{A}_l{P}_l(\cos \theta ),A_l=\frac{1}{{\int }_0^{\pi }[P_l(\cos \theta ){]}^2\sin \theta d\theta }{\int }_0^{\pi }f(\theta )P_l(\cos \theta )\sin \theta d\theta$$

$\theta \in [0,\pi ]$