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

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Legendre Polynomials P 0 (x) = 1, P 1 (x) = x P 2 (x) = 1 2 (3x2 − 1), P 3 (x) = 1 2 (5x3 − 3x) In general, P n (x) = 2n − 1 xP n−1 (x) − n − 1 n n P n−2(x). Legendre polynomials satisfy the important orthogonality relation, ∫ x=1 x=−1 P k (x)P j (x)dx = { 2 2k+1 if k = j; 0 otherwise. Taylor Series areLimits of Legendre Expansions – p.4/15
Neumann’s Theorem (1862) If f is analytic in an ellipse containing the closed unit disk and having foci at (±1, 0), then for z contained in this ellipse, we may write f(z) = ∞∑ n=0 a n P n (z), where a n = 2n + 1 2 ∫ t=1 t=−1 f(t)P n (t)dt. Taylor Series areLimits of Legendre Expansions – p.5/15
Page 1 and 2: Taylor Series are Limits of Legendr
Page 3 and 4: A Common Question in Calculus Can w
Page 5: Taylor’s Theorem If f is analytic
Page 9 and 10: Legendre Expansions on [−h, h] If
Page 11 and 12: The Limit of the Legendre Expansion
Page 13 and 14: The Limit of the Legendre Expansion
Page 15 and 16: The Proof Assume first that we may
Page 17 and 18: The Proof • By Taylor’s Theorem
Page 19 and 20: The Proof • By Taylor’s Theorem
Page 21 and 22: The Proof Combining all these ideas
Page 23 and 24: Bounding the Series Terms The key i
Page 25 and 26: Bounding P n (z/h) To estimate |P n
Page 27 and 28: Bounding the Series Terms Applying
Page 29: Acknowledgements • Nathanial Burc

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

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