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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 3

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 2 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 9.4 Exercises Exercise 9.1 Consider two functions f x y and g x y . They are said to be functionally dependent if there is a an h g such that f x y h g x y . f and g will be functionally dependent if and only if their Jacobian vanishes. If f and g are functionally dependent then the derivatives of f are fx h g gx fy h g gy. Thus we have d f g fx fy d x y gx gy If the Jacobian of f and g vanishes then fxgy - fygx h g gxgy - h g gygx 0. fxgy fygx 0. This is a first order partial differential equation for f that has the general solution f x y h g x y . Prove that an analytic function u x y iv x y can be written in terms of a function of a complex variable f z u x y zv x y . Exercise 9.2 Which of the following functions are the real part of an analytic function For those that are find the harmonic conjugate v x y and find the analytic function f z u x y iv x y as a function of z. 1. x3 3xy2 2xy y 2. ex sinh y 454 3. ex sin x cos y cosh y cos x sin y sinh y Exercise 9.3 For an analytic function f z u r 6 iv r 6 prove that under suitable restrictions f z 2u z1 2 1 log z const. .