Legendre Polynomial/Examples

Examples

The first five Legendre polynomials are:

\(\ds \map {P_0} x\)

\(=\)

\(\ds 1\)

\(\ds \map {P_1} x\)

\(=\)

\(\ds x\)

\(\ds \map {P_2} x\)

\(=\)

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

\(\ds \map {P_3} x\)

\(=\)

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

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

\(=\)

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

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): generating function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generating function