EXERCISES 1. Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y) = 2x (1- y); (c) u(x, y) = sinh x sin và = (b)u(x, y)=2x-x3+3xy²; (x, y) = y/(x²+ y²). Ans. (a) v(x, y) = x²- y²+2y: (b) v(x, y) = 2y-3x²y + y3; (c) v(x, y) = cosh x cos y; (d) v(x, y) = x/(x² + y²). 2. Show that if v and V are harmonic conjugates of u(x, y) in a domain D, then v(x, y) and V (x, y) can differ at most by an additive constant. 2xy + ; (x² y²) would be not trie (see Hwa). a analytic (EX: Show u(x,y) = y³-3x²y is a harmonic function in ; the entire xy-plane and find a harmonic conjugate ofu. + Uy = 3y²-3x² ux = -bxy uxx = - by so Uxx + Uyy= D Uyys by in the xy-plane- 4. -uy = - (3y²-3x²) 3x²-3y² To find V, solve Vy = 4x = -6xy, vx مانی Use this first V= 5(-6xy)dy = -6x fydy = - 6x y² + ((x) 3 2 -3x²+(x) Now -3y² + d'(x) = 2 Vx=3x²-3y² to Solve for & (x) 3x²-3y² So (x) = 3x², We have (x)-f3x²dx = 3 × where is a Constant 3 Thus v = -3xy² + x²+ C is a Conjugate of u i [(x²=3xy²) +i (3x'y- harmonic