tailieunhanh - Handbook of mathematics for engineers and scienteists part 141

Handbook of mathematics for engineers and scienteists part 141. 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. | 948 Special Functions and Their Properties . Some formulas. 2vZv x x Zv-i x Zv i x Z x - Zv-i x - Zv i x -Zv x - Zv 1 x dx 2 Lx J d _ x Zv x x Zv-i x dx i d V . x J v x x Jv-n x x dx d _ x Zv x -x Zv i x dx 1 d n . . - x J v x --1 n x J n x x dx J-n x i n Jn x Ynx i nYn x n 0 i 2 . . Bessel functions for v n 2 where n 0 1 2 . sin x cos x J3 2 x T i . sin x -cos x V nx x J-3 2 x cos x - sin x x W sin x - . - t n 2 2k 7 V nx V 2 J 2k n - 2k 2x 2k k 0 n-i 2 cos x - k 0 i n 2k i 2k i n - 2k - i 2x 2fc i J-n-I 2 x SZ. cos x T n Jk 2 7 V nx V 2 2k n - 2k 2x 2 L k 0 z nn k L- 21 -i n 2k i x 2 0 2k i n -2k - i 2x 2fc i Yi 2 x - p2x cos x Yn i 2 x i J-n-i 2 x nr Y-i 2 x Y sin x Y-n-i 2 x --i n Jn i 2 x where A is the integer part of the number A. . Bessel functions for v n where n 0 1 2 . Let v n be an arbitrary integer. The relations J-n X -1 n Jn x Y-n x -1 n Yn x are valid. The function Jn x is given by the first formula in with v n and Yn x can be obtained from the second formula in by proceeding to the limit . Bessel Functions Cylindrical Functions 949 v n. For nonnegative n Yn x can be represented in the form 2 x 1 n-k - 1 2 V-2 1 Æ tfx yn 2t k i k n k 1 Y -. Mi -- - --E .j . k 0 k 0 v 7 n-1 where 0 1 -C 0 n -C k 1 C . is the Euler constant and 0 x k i In r x is the logarithmic derivative of the gamma function also known as the digamma function. . Wronskians and similar formulas. W Jv J-v - sin nv W Jv Yv nx nx Jv x J-v 1 x J-v x Jv-1 x ------- Jv x Yv 1 x - Jv 1 x Yv x ---- nx nx Here the notation W f g fg x - fXg is used. . Integral Representations and Asymptotic Expansions . Integral representations. The functions Jv x and Yv x can be represented in the form of definite integrals for x 0 nJv x cos x sin 0 - vG dG -sin nv exp -x sinh t - vt dt Jo Jo nYv x sin x sin 0 - vG dG - evt e vt cos nv e x sinh t dt. Jo Jo For v - x 0 21 vx v fœ sin xt dt Jv x nV2r - - v .h t2 - 1 v 1 2 21 vx v fœ cos .