The Legendre Polynomials

Definition: \( P_n(x) = \sum\limits_{m = 0}^{M} (-1)^m \dfrac{(2n - 2m)!}{2^n m! (n-m)! (n-2m)!} x^{n-2m}\)
with \(M = \dfrac{n}{2}\) if \(n\) is even or \(\dfrac{n-1}{2} \) if \(n\) odd.

• Choose a Legendre Polynomial from the menu.
• Use button "Plot Legendre Polynomial" to plot the choosen polynomial.
• The program optimizes the display range for the last plotted polynomial to give a good representation of it.
• One can replot a polynomial to rescale the display for it.
• "Clear Plot Area" will delete all plotted polynomials.

The first nine Legendre Polynomials used here:

• \( P_0(x) = 1 \)
• \( P_1(x) = x \)
• \( P_2(x) = \frac{1}{ 2} ( 3x^2 - 1 ) \)
• \( P_3(x) = \frac{1}{ 2} ( 5x^3 - 3x ) \)
• \( P_4(x) = \frac{1}{ 8} ( 35x^4 - 30x^2 + 3 ) \)
• \( P_5(x) = \frac{1}{ 8} ( 63x^5 - 70x^3 + 15x ) \)
• \( P_6(x) = \frac{1}{ 16} ( 231x^6 - 315x^4 + 105x^2 - 5) \)
• \( P_7(x) = \frac{1}{ 16} ( 429x^7 - 693x^5 + 315x^3 -35x )\)
• \( P_8(x) = \frac{1}{128} ( 6435x^8 -12012x^6 +6930x^4 -1260x^2 +35)\)