For each of the following functions, (a) v(x,y) = 2y 3x?y + y (b) u(x,...

 

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For each of the following functions, (a) v(x,y) = 2y – 3x?y + y³ (b) u(x, y) = y x²+y² [Hint: polar] (c) u(x, y) = xy – x + y (d) v(x,y) = x² – y² + 2y (i) Determine whether the function is harmonic. (ii) If it is harmonic, find its harmonic conjugate function. (iii) Form an analytic function f(z) = u(x,y) + iv(x,y) from the harmonic function and its harmonic conjugate. %3D For each of the following functions, (a) v(x,y) = 2y – 3x?y + y³ (b) u(x, y) = y x²+y² [Hint: polar] (c) u(x, y) = xy – x + y (d) v(x,y) = x² – y² + 2y (i) Determine whether the function is harmonic. (ii) If it is harmonic, find its harmonic conjugate function. (iii) Form an analytic function f(z) = u(x,y) + iv(x,y) from the harmonic function and its harmonic conjugate. %3D