info prev up next book cdrom email home

Legendre Series

Because the Legendre Functions of the First Kind form a Complete Orthogonal Basis, any Function may be expanded in terms of them

f(x) = \sum_{n=0}^\infty a_nP_n(x).
\end{displaymath} (1)

Now, multiply both sides by $P_m(x)$ and integrate
\int^1_{-1} P_m(x)f(x)\,dx = \sum_{n=0}^\infty a_n \int^1_{-1} P_n(x)P_m(x)\,dx.
\end{displaymath} (2)

\int^1_{-1} P_n(x)P_m(x)\,dx = {2\over 2m+1} \delta_{mn},
\end{displaymath} (3)

where $\delta_{mn}$ is the Kronecker Delta, so
\int^1_{-1} P_m(x)f(x)\,dx = \sum_{n=0}^\infty a_n {2\over 2m+1} \delta_{mn} = {2\over 2m+1} a_m
\end{displaymath} (4)

a_m = {2m+1\over 2} \int^1_{-1} P_m(x)f(x)\,dx.
\end{displaymath} (5)

See also Fourier Series, Jackson's Theorem, Legendre Polynomial, Maclaurin Series, Picone's Theorem, Taylor Series

© 1996-9 Eric W. Weisstein