Taylor Series are Limits of Legendre Expansions - Gvsu - Grand ...

Legendre Expansions on [−h, h]
If f is analytic in a disk of radius R > 0 about the origin,
then for h sufficiently small and positive, one may write
(
∞∑
∫ )
2n + 1 t=h
f(z) =
f(t)P n (t/h)dt P n (z/h).
2h
n=0
t=−h
y
(–h,0) (h,0)
x
Moreover,
2n + 1
lim
n→∞ ∣ 2h
≤
∫ t=h
R
h
+
f(t)P n (t/h)dt
∣
1
√ (Rh )
.
2
− 1
t=−h
1
n
Taylor Series areLimits of Legendre Expansions – p.6/15