User:Egm6321.f11.team1/report6.bensplaypen.R*7.8

= R*7.8 - Plot Legendre Functions for $$n=0 \text{ to } 3$$ =

Given
Legendre polynomials

$$\begin{align} & {{P}_{0}}(x)=1 \\ & {{P}_{1}}(x)=x \\ & {{P}_{2}}(x)=\frac{1}{2}(3{{x}^{2}}-1) \\ & {{P}_{3}}(x)=\frac{1}{2}(5{{x}^{3}}-3x) \\ \end{align}$$

Legendre functions

$$\begin{align} & {{Q}_{0}}(x)=\frac{1}{2}\log \left( \frac{1+x}{1-x} \right) \\ & {{Q}_{1}}(x)=\frac{1}{2}x\log \left( \frac{1+x}{1-x} \right)-1 \\ & {{Q}_{2}}(x)=\frac{1}{4}(3{{x}^{2}}-1)\log \left( \frac{1+x}{1-x} \right)-\frac{3}{2}x \\ & {{Q}_{3}}(x)=\frac{1}{4}(5{{x}^{3}}-3x)\log \left( \frac{1+x}{1-x} \right)-\frac{5}{2}{{x}^{2}}+\frac{2}{3} \\ \end{align}$$

Find
The plot of the Legendre polynomials and functions together:

Show that for $$ n = 0 $$: $$\scriptstyle \mu \to \pm 1 \Rightarrow \left|Q_0 (\mu) \right| \to + \infty$$

Observe the limits of $$P_n(\mu)$$ and $$Q_n(\mu)$$ as $$\scriptstyle \mu \rightarrow \pm 1$$.

Solution
See the requested solutions plotted below.



What has been shown

Clearly with $$ n = 0 $$ for $$\scriptstyle \mu \to \pm 1 $$ the value of $$ \left|Q_0 (\mu) \right|$$ goes to $$ + \infty$$ as can be seem in the plot for $$Q_0$$ above

The limits of $$P_n(\mu)$$ and $$Q_n(\mu)$$ as $$\scriptstyle \mu \rightarrow \pm 1$$ are as follows:

$$P_0(\mu)$$ as $$\scriptstyle \mu \rightarrow + 1$$ = $$+1$$

$$P_0(\mu)$$ as $$\scriptstyle \mu \rightarrow - 1$$ = $$+1$$

$$P_1(\mu)$$ as $$\scriptstyle \mu \rightarrow + 1$$ = $$+1$$

$$P_1(\mu)$$ as $$\scriptstyle \mu \rightarrow - 1$$ = $$-1$$

$$P_2(\mu)$$ as $$\scriptstyle \mu \rightarrow + 1$$ = $$+1$$

$$P_2(\mu)$$ as $$\scriptstyle \mu \rightarrow - 1$$ = $$+1$$

$$P_3(\mu)$$ as $$\scriptstyle \mu \rightarrow + 1$$ = $$+1$$

$$P_3(\mu)$$ as $$\scriptstyle \mu \rightarrow - 1$$ = $$-1$$

$$Q_0(\mu)$$ as $$\scriptstyle \mu \rightarrow + 1$$ = $$+ \infty$$

$$Q_0(\mu)$$ as $$\scriptstyle \mu \rightarrow - 1$$ = $$- \infty$$

$$Q_1(\mu)$$ as $$\scriptstyle \mu \rightarrow + 1$$ = $$+ \infty$$

$$Q_1(\mu)$$ as $$\scriptstyle \mu \rightarrow - 1$$ = $$+ \infty$$

$$Q_2(\mu)$$ as $$\scriptstyle \mu \rightarrow + 1$$ = $$+ \infty$$

$$Q_2(\mu)$$ as $$\scriptstyle \mu \rightarrow - 1$$ = $$- \infty$$

$$Q_3(\mu)$$ as $$\scriptstyle \mu \rightarrow + 1$$ = $$+ \infty$$

$$Q_3(\mu)$$ as $$\scriptstyle \mu \rightarrow - 1$$ = $$+ \infty$$

So the limits depend on whether $$ n $$ is odd or even