Legendre Polynomial/Examples/P0

Example of Legendre Polynomial

The zeroth Legendre polynomial is:

$\map {P_0} x = 1$

Proof

From Generating Function for Legendre Polynomials, $\map {P_0} x$ is the coefficient of the zeroth term of the generating function:

$\map G t = \dfrac 1 {\sqrt {1 - 2 x t + t^2} }$

Setting $t = 0$ in $\map G t$:

Thus:

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

\(=\)

\(\ds \valueat {\dfrac 1 {\sqrt {1 - 2 x t + t^2} } } {t \mathop = 0}\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\sqrt {1 - 2 x \times 0 + 0^2} }\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\sqrt 1}\)

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$

Sources

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