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Special Functions

Chapter
Part of the Springer Tracts in Modern Physics book series (STMP, volume 186)

Abstract

The Legendre polynomial of degree l (l = 0,1,2,..., ∞) is defined by the formula
$$ P_l (u) = \frac{1} {{2^l l!}}{\mathbf{ }}\frac{{d^l }} {{du^l }}(u^2 - 1)^l {\mathbf{ }}. $$
(7)
(C.1) It is a polynomial with l zeros in the range (−1, +1) and with parity (− 1)l. The first five Legendre polynomials are P 0 = 1, (C.2) P 1 = u, (C.3) P 2 = 1/2(3u 2-1), (C.4) P 3 = 1/2(5u 3−3u), (C.5) P 4 = 1/8 (35u 4 − 30u 2 + 3). (C.6)

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

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