Legendre Polynomial/Examples/P4

Example of Legendre Polynomial

The $4$th Legendre polynomial is:

$\map {P_4} x = \dfrac 1 8 \paren {35 x^4 - 30 x^2 + 3}$

Proof

\(\ds \forall n \in \N: \, \)

\(\ds \paren {n + 1} \map {P_{n + 1} } x\)

\(=\)

\(\ds \paren {2 n + 1} x \map {P_n} x - n \map {P_{n - 1} } x\)

Bonnet's Recursion Formula

\(\ds \leadsto \ \ \)

\(\ds 4 \map {P_4} x\)

\(=\)

\(\ds 7 x \map {P_3} x - 3 \map {P_2} x\)

setting $n = 3$

\(\ds \)

\(=\)

\(\ds 7 x \paren {\dfrac 1 2 \paren {5 x^3 - 3 x} } - 3 \paren {\dfrac 1 2 \paren {3 x^2 - 1} }\)

Legendre Polynomial $P_3$ and Legendre Polynomial $P_2$

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \paren {35 x^4 - 21 x^2 - 9 x^2 + 3}\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds \map {P_4} x\)

\(=\)

\(\ds \dfrac 1 8 \paren {35 x^4 - 30 x^2 + 3}\)

simplifying

$\blacksquare$