1. Show that the function h(x, y) = y - 3y + 2y is harmonic. 2....

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1. Show that the function h(x, y) = y³ - 3y² + 2y is harmonic. 2. Find an analytic function f(z) = f(x+iy) so that f(2)=h(x, y) = Re(f(z)). That is, find v(x, y) so that y³ - 3yr²+2y+iv(x, y) is analytic. The function u(x, y) is called the harmonic conjugate of h. (Hint: Since f(x, y) = h(x, y) +iv(x, y) is analytic, you can determine what u, and u, must be. Then integrating with v, with respect to y then gives you an expression that differs from v(x, y) by a function depending on r alone. Similarly, you can integrate u, with respect to r to get v(x, y) + k(y): equating the two gives you u(x, y).) Or maybe you can just guess the answer, and check that it works.) 1. Show that the function h(x, y) = y³ - 3y² + 2y is harmonic. 2. Find an analytic function f(z) = f(x+iy) so that f(2)=h(x, y) = Re(f(z)). That is, find v(x, y) so that y³ - 3yr²+2y+iv(x, y) is analytic. The function u(x, y) is called the harmonic conjugate of h. (Hint: Since f(x, y) = h(x, y) +iv(x, y) is analytic, you can determine what u, and u, must be. Then integrating with v, with respect to y then gives you an expression that differs from v(x, y) by a function depending on r alone. Similarly, you can integrate u, with respect to r to get v(x, y) + k(y): equating the two gives you u(x, y).) Or maybe you can just guess the answer, and check that it works.)

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