Question

Use a CAS to graph $P_1\left(x\right),P_2\left(x\right),...,P_7\left(x\right)$on the interval[1,-1]

Short answer

The graphs obtained using Mathematica are plotted.

Step-by-step solution

Step 1:Define Legendre polynomials.

Let the Legendre equation be $(1-{\mathrm{x}}^2)\mathrm{y}\text{'}\text{'}-2\mathrm{xy}\text{'}+\mathrm{n}(\mathrm{n}+1)\mathrm{y}=0$. The two particular solutions are $y_1=c_0\left[1-\frac{n(n+1)}{2!}{x}^2+\frac{(n-2)n(n+1)(n+3)}{4!}{x}^4-\frac{(n-4)(n-2)n(n+1)(n+3)(n+5)}{6!}{x}^6+\dots \right]$and

$y_2=c_1\left[x-\frac{(n-1)(n+2)}{3!}{x}^3+\frac{(n-3)(n-1)(n+2)(n+4)}{5!}{x}^5-\frac{(n-5)(n-3)(n-1)(n+2)(n+4)(n+6)}{7!}{x}^7+\dots \right]$, where the arbitrary constants are $c_0=\left\{\begin{array}{ll}1& \mathrm{if}\mathrm{n}=0\\ {(-1)}^{\mathrm{n}/2}\frac{{\displaystyle 1\times 3\dots (\mathrm{n}-1)}}{{\displaystyle 2\times 4\dots \mathrm{n}}}& \mathrm{if}\mathrm{n}=2,4,6,\dots \end{array}\right.$and ${\mathrm{c}}_1=\left\{\begin{array}{ll}1& \mathrm{if}\mathrm{n}=1\\ {(-1)}^{(\mathrm{n}-1)/2}\frac{{\displaystyle 1\times 3\dots \mathrm{n}}}{{\displaystyle 2\times 4\dots (\mathrm{n}-1)}}& \mathrm{if}\mathrm{n}=3,5,7,\dots \end{array}\right.$.

Derive the graph of theLegendre polynomials using CAS.

The Legendre polynomials obtained in Problem 50 are,

$$\begin{gathered} P_1\left(x\right)=x \\ {P}_2\left(x\right)=\frac{1}{2}\left(-1+3x^2\right) \\ {P}_3\left(x\right)=\frac{1}{2}x\left(-3+5x^2\right) \\ {P}_4\left(x\right)=\frac{1}{8}\left(3-30x^2+35x^4\right) \\ {P}_5\left(x\right)=\frac{1}{8}x\left(15-70x^2+63x^4\right) \\ {P}_6\left(x\right)=\frac{1}{16}\left(-5+105x^2-315x^4+231x^6\right) \\ {P}_7\left(x\right)=\frac{1}{16}x\left(-35+315x^2-693x^4+429x^6\right) \end{gathered}$$

For n=1, the graph of polynomial P₁(X) is,

For n=2, the graph of polynomial P₂(X) is,

For n=3, the graph of polynomial P₃(X) is,

For n=4, the graph of polynomial P4(X) is,

For n=5, the graph of polynomial P₅(X) is,

For n=6, the graph of polynomial P₆(X) is,

For n=7, the graph of polynomial P₇(X) is,