Question

For purposes of this problem ignore the list of Legendre polynomials given on page 271 and the graphs given in Figure 6.4.6. Use Rodrigues’ formula (36) to generate the $P_1\left(x\right),P_2\left(x\right),...,P_7\left(x\right)$Legendre polynomials. Use a CAS tocarry out the differentiations and simplifications.

Short answer

The Legendre polynomials using Mathematica are,

$\begin{array}{cc}{\mathrm{P}}_1\left(\mathrm{x}\right)=& \frac{1}{2}(-1+3{\mathrm{x}}^2)\\ {\mathrm{P}}_2\left(\mathrm{x}\right)=& \frac{1}{2}\mathrm{x}(-3+5{\mathrm{x}}^2)\\ {\mathrm{P}}_3\left(\mathrm{x}\right)=& \frac{1}{8}(3-30{\mathrm{x}}^2+35{\mathrm{x}}^4)\\ {\mathrm{P}}_4\left(\mathrm{x}\right)=& \frac{1}{8}\mathrm{x}(15-70{\mathrm{x}}^2+63{\mathrm{x}}^4)\\ {\mathrm{P}}_5\left(\mathrm{x}\right)=& \frac{1}{16}(-5+105{\mathrm{x}}^2-315{\mathrm{x}}^4+231{\mathrm{x}}^6)\\ {\mathrm{P}}_6\left(\mathrm{x}\right)=& \frac{1}{16}\mathrm{x}(-35+315{\mathrm{x}}^2-693{\mathrm{x}}^4+429{\mathrm{x}}^6)\end{array}$

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

${\mathrm{y}}_1={\mathrm{c}}_0\left[1-\frac{\mathrm{n}(\mathrm{n}+1)}{2!}{\mathrm{x}}^2+\frac{(\mathrm{n}-2)\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+3)}{4!}{\mathrm{x}}^4-\frac{(\mathrm{n}-4)(\mathrm{n}-2)\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+3)(\mathrm{n}+5)}{6!}{\mathrm{x}}^6+\dots \right]$

and

${\mathrm{y}}_2={\mathrm{c}}_1\left[\mathrm{x}-\frac{(\mathrm{n}-1)(\mathrm{n}+2)}{3!}{\mathrm{x}}^3+\frac{(\mathrm{n}-3)(\mathrm{n}-1)(\mathrm{n}+2)(\mathrm{n}+4)}{5!}{\mathrm{x}}^5-\frac{(\mathrm{n}-5)(\mathrm{n}-3)(\mathrm{n}-1)(\mathrm{n}+2)(\mathrm{n}+4)(\mathrm{n}+6)}{7!}{\mathrm{x}}^7+\dots \right]$

, where the arbitrary constants are ${\mathrm{c}}_0=\left\{\begin{array}{ll}1& \mathrm{if}\mathrm{n}=0\\ {(-1)}^{\mathrm{n}/2}\frac{{\displaystyle 1\times 3\dots (\mathrm{n}-1)}}{{\displaystyle 2\times 4\dots \mathrm{n}}}& \mathrm{if}\mathrm{n}=2,4,6,\dots \end{array}\right.$ and

$c_1=\left\{\begin{array}{ll}1& \mathrm{if}\mathrm{n}=1\\ {(-1)}^{(\mathrm{n}-1)/2}\frac{{\displaystyle 1\times 3\dots \mathrm{n}}}{{\displaystyle 2\times 4\dots (\mathrm{n}-1)}}& \mathrm{if}\mathrm{n}=3,5,7,\dots \end{array}\right.$.

Derive the Legendre polynomials using CAS.

The commandis used to do the derivative for a specific value of n, where f(X,n) is a function. The symbol / and $\to$are used to make the substitution for a specific value of n . That yields,where m=0,1,2,3.....$D\left[f\right(x,n),\{x,n\left\}\right]·/n\to m$.

To use Rodrigues’s formula, the following command is used.

$Times\left[1/\left(2^n*n!\right),D\left[{\left(x^2-1\right)}^n,x,n\right]\right]/.n\to m$

For n=1 , the polynomial is P₁(X)