tailieunhanh - Handbook of mathematics for engineers and scienteists part 142

Handbook of mathematics for engineers and scienteists part 142. 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. | . Airy Functions 955 . Asymptotic expansions as x to. x - Lf 1 V -1m 4v2 - D v2 - 32 4v2 m 1 21 V2nX m 8x m m 1 K x . nw i f 4v 2 - i 4v 2 - 32 4 2 - 2m -1 211 V 2x 1 m 8x m m 1 The terms of the order of O x M-1 are omitted in the braces. . Integrals with modified Bessel functions. XL x dx x n F v 1 V 3 v 1 Re A v -1 Jo 2V A v 1 1 v 1 2 2 4 J where F a b c x is the hypergeometric series see Subsection r XK dx p2V - 1F A . . 1 - vF a Jo A - v 1 2 2 4 r x F 1 . Re A Re v - 1. A v 1 2 2 4 . Airy Functions . Definition and Basic Formulas . Airy functions of the first and the second kinds. The Airy junction of the first kind Ai x and the Airy function of the second kind Bi x are solutions of the Airy equation y xx- xy 0 and are defined by the formulas 1 f Ai x cos 3t3 xt dt n Jo 1 f Ipwif_f 3 i zp t i cinfl- 3 i zp 1 nt Bi x exp 3 t i xt j i siny 3 t 1 xtj j c t. n Jo Wronskian W Ai x Bi x 1 n. . Relation to the Bessel functions and the modified Bessel functions. Ai x 3 x l-1 3 z - I1 3 z n y lx K1 3 z z 3x3 2 Ai -x 3 x J-1 3 z J1 3 z Bi x yjx l-1 3 z I1 3 z Bi -x yjx J-1 3 z - J1 3 z . 956 Special Functions and Their Properties . Power Series and Asymptotic Expansions . Power series expansions as x 0. Ai x cif x - C2g x Bi x V3 ci f x C2g x X 1 3 1X4 6 1x4x7 9 x3k f x 1 3 x 6 x x g3 V 2 4 2x5 7 2x5x8 10 2 x3k 1 x _x x x . v 3 _ k__ where C1 3 2 3 f 2 3 - and C2 3-1 3 r 1 3 - . . Asymptotic expansions as x œ. For large values of x the leading terms of asymptotic expansions of the Airy functions are Ai x 2n 1 2x 1 4 exp -z z x3 2 Ai -x n 1 2x 1 4 sin z j Bi x n-1 2x 1 4 exp z Bi -x n 1 2x 1 4 cos z 4 . Degenerate Hypergeometric Functions Kummer Functions . Definitionsand Basic Formulas . Degenerate hypergeometric functions a b x and F a b x . The degenerate hypergeometric functions Kummer functions a b x and F a b x are solutions of the degenerate hypergeometric