tailieunhanh - Handbook of mathematics for engineers and scienteists part 212

Handbook of mathematics for engineers and scienteists part 212. 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. | . Functional Equations in Several Independent Variables 1445 15. f s x2 y2 f x f y . Gauss s equation. Solution f x exp Cx2 where C is an arbitrary constant. In addition the function f x 0 is also a solution. 16 f 2 x f2 y y 2 f x2 y2 y 2 Solution f x ax2 ft 1 2 where a and b are arbitrary positive constants. 17. f xn yn 1 n af x f y n is any number. Solution f x - exp Cxn a where C is an arbitrary constant. In addition the function f x 0 is also a solution. 18. f n x f y G f n is any number. 2 i 2 r Solution f x axn b 1 n where a and b are arbitrary positive constants. 19. f x y f x f x - y f x 2f x f y . Solutions f x 0 f x 1 Cx2 where C is an arbitrary constant. 20. f g-1 g x g y af x f y . Generalized Gauss equation. Here g x is an arbitrary monotonic function and g-1 x is the inverse of g x . Solution f x 0. exp Cg x where C is an arbitrary constant. The function f x 0 is also a solution. 21. M f x f y f M x y . Here M x y -1 x 2 y is a quasiarithmetic mean for a continuous strictly monotonic function with -1 being the inverse of . Solution f x -1 a x b where a and b are arbitrary constants. 1446 Functional Equations . Equations involving several unknown functions of a single argument. 22. f x g y h x y . Here f x g y and h z are unknown functions. Solution f x Ci exp C3x g y C2 exp Csy h z C1C2 exp C3z where C1 C2 and C3 are arbitrary constants. 23. f x g y h y f x y . Here f x g y and h z are unknown functions. Solutions f x C1x C2 g x 1 h x C1x first solution f x C1eax C2 g x eax h x C2 1 - eax second solution where a C1 and C2 are arbitrary constants. 24. fi x gi y f2 x g2 y f3 x g3 y 0. Bilinear functional equation. Two solutions f1 x C1f3 x f2 x C2f3 x g3 y -C1g1 y - C2g2 y g1 y C1g3 y g2 y C2g3 y f3 x -C1f1 x - Cfx where C1 and C2 are arbitrary constants the functions on the right-hand sides of the solutions are prescribed arbitrarily. 25. fi x gi y f2 x g2 y f3 x g3 y f4 x g4 y 0. Bilinear functional equation. Equations of this type often