tailieunhanh - Handbook of mathematics for engineers and scienteists part 207

Handbook of mathematics for engineers and scienteists part 207. 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. | 1410 Functional Equations TABLE Particular solutions of the nonhomogeneous functional equation y x 1 - y x f x No. Right-hand side of equation f x Particular solution y x 1 1 x 2 x 22 x x - 1 3 x2 6 x x - 1 2x - 1 4 xn n 0 1 2 . n 1 Bn x where the Bn x are Bernoulli polynomials. text tn The generating function X Bn x e -1 n n 0 5 x 1 1 - tx-1 Q x -C 1 dt is the logarithmic derivative of the gamma function C . is the Euler constant 6 1 x x 1 1 x 7 aXx a 1 A 0 1 Xx aX -1 a 8 sinh a 2bx b 0 cosh a - b 2bx 2 sinh b 9 cosh a 2bx b 0 sinh a - b 2bx 2 sinh b 10 xax a 1 ax x a -1 x a-1 11 Inx x 0 ln r x where r x tx ie t dt is the gamma function J0 12 sin 2ax a nn cos a 2x - 1 2 sin a 13 sin 2nnx x sin 2nnx 14 cos 2ax a nn sin a 2x - 1 2 sin a 15 cos 2nnx x cos 2nnx 16 sin2 ax a m x sin a 2x - 1 2 4 sin a 17 sin2 nnx x sin2 nnx 18 cos2 ax a m x sin a 2x - 1 2 4 sin a 19 cos2 nnx x cos2 nnx 20 x sin 2ax a nn sin 2ax cos a 2x - 1 4 sin2 a 2 sin a 21 x sin 2nnx 2 x x - 1 sin 2nnx 22 x cos 2ax a nn cos 2ax sin a 2x - 1 4 sin2 a 2 sin a 23 x cos 2nnx 2 x x - 1 cos 2nnx 24 ax sin bx a 0 a 1 x a sin b x - 1 - sin bx a2 -2a cos b 1 25 ax cos bx a 0 a 1 x a cos b x - 1 - cos bx a2 - 2a cos b 1 . Linear Functional Equations in One Independent Variable 1411 2 . Solution for a 0 y x 01 x u sin x 02 x u cos x where the 0k x 0k x 1 are arbitrary periodic functions with period 1 k 1 2 . Remark. See also Subsection Example 1. y x 4. y x 1 - ay x f x . A nonhomogeneous first-order constant-coefficient linear difference equation. 1 . Solution 0 x ax y x if a 0 01 x a x sin x 02 x a x cos x y x if a 0 where 0 x 01 x and 02 x are arbitrary periodic functions with period 1 and y x is any particular solution of the nonhomogeneous equation. n 2 . For f x 2 bkxn and a 1 the nonhomogeneous equation has a particular solution k 0 n of the form y x 2 Akxn the constants Bk are found by the method of undetermined . k 0 coefficients. n 3 . For f x bkeXkx the nonhomogeneous .