Let G =C { 0}. Note that u(x, y) = log(x + y2) is harmonic...

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Let G =C\ { ≤ 0}. Note that u(x, y) = log(x² + y2) is harmonic in G. Verify by direct use of the line integral given in class, i.e. v(x, y) = f(-uydx+u,dy), that to within an arbitrary real constant, the function v(x, y) = argz is the harmonic conjugate of u. Hint: Let (ro. yo)= (1,0) and use paths of integration consisting line segments parallel to the real and imaginary axes. Let G =C\ { ≤ 0}. Note that u(x, y) = log(x² + y2) is harmonic in G. Verify by direct use of the line integral given in class, i.e. v(x, y) = f(-uydx+u,dy), that to within an arbitrary real constant, the function v(x, y) = argz is the harmonic conjugate of u. Hint: Let (ro. yo)= (1,0) and use paths of integration consisting line segments parallel to the real and imaginary axes.