tailieunhanh - Handbook of mathematics for engineers and scienteists part 143

Handbook of mathematics for engineers and scienteists part 143. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 962 Special Functions and Their Properties . Gauss s linear relations for contiguous functions. ß - a F a ß y x aF a 1 ß y x - ßF a ß 1 y x 0 Y-a- 1 F a ß y x aF a 1 ß y x - y - 1 F a ß y - 1 x 0 Y - ß - 1 F a ß y x ßF a ß 1 y x - y - 1 F a ß y - 1 x 0 Y - a- ß F a ß y x a 1 -x F a 1 ß y x - y - ß F a ß - 1 y x 0 Y - a - ß F a ß y x - y - a F a - 1 ß y x ß 1 - x F a ß 1 y x 0. . Differentiation formulas. rf aB F a B y x F a 1 B 1 y 1 x ax y d ry O x a n B n p zi x F a B y x ----F a n B n y n x rfx y n rfn -Y-xlx 1F a B Y x Y-n nx7 1F a B Y-n x dxn rfn x n 1F a B y x a n xa 1F a n B Y x axn L J where a n a a 1 . a n - 1 . See Abramowitz and Stegun 1964 and Bateman and Erdelyi 1953 Vol. 1 for more detailed information about hypergeometric functions. . Legendre Polynomials Legendre Functions and Associated Legendre Functions . Legendre Polynomials and Legendre Functions . Implicit and recurrence formulas for Legendre polynomials and functions. The Legendre polynomials Pn x and the Legendre functions Qn x are solutions of the second-order linear ordinary differential equation 1 - x2 yXx - 2xy x n n 1 y 0. The Legendre polynomials Pn x and the Legendre functions Qn x are defined by the formulas 1 dn . P .2- x x2-1 n Qn x . Pn x ln . V Pm-1 x Pn-m x . 2 1 -x _1 m The polynomials Pn Pn x can be calculated using the formulas P0 x 1 P1 x x P2 x 2 3x2 - 1 Pß x 1 5x3 - 3x P4 x 1 35x4 - 30x2 3 2 8 zx2n 1 x . . Pn 1 x 1 xPn x 1 Pn-1 x . . Legendre Polynomials Legendre Functions and Associated Legendre Functions 963 The first five functions Qn Qn x have the form Qo x 1 ln 1 x Qi x X In1 x - 1 2 1 - x 2 1 - x Qi x 1 3x2 - 1 ln1 X - 3x Qs x 1 5x3 - 3x ln1 X - 5x1 2 4 1 - x 2 4 1 - x 23 Q4 x 35x4 - 30x2 3 ln X - x3 x. 16 1 - x 8 24 The polynomials Pn x have the explicit representation n 2 Pn x 2 V -1 mCmCnn-2mxn-2m m 0 where A stands for the integer part of a number A. . Integral representation. Useful formulas. Integral .