Problem 6.8

Is it possible to find a real function v(x,y) so that \(x^{3}+y^{3}+iv(x,y)\) is holomorphic

Consider the contrapositive of Proposition 6.4 which says If \(u\) is not harmonic or \(v\) is not harmonic, then \(f=u+iv\) is not holomorphic in the region G.
Let \(u(x,y) = x^{3}+y^{3}\). Then \(u_{xx}= (3x^{2})_x=6x\) and \(u_{yy}=(3y^{2})_y=6y\). So \(u_{xx}+u_{yy}=6x+6y\) and we can choose values for \(x\) and \(y\) such that \(u_{xx}+u_{yy}\neq0\). Thus, \(x^{3}+y^{3}\) does not satisfy the Laplace equation \(u_{xx}+u_{yy}=0\) for all \((x,y)\) and is therefore not harmonic. Since \(u(x,y)\) is not harmonic, \(f=u+iv\) is not holomorphic by the contrapositive of Proposition 6.4. So no matter what \(v(x,y)\) we choose, \(f\) will never be harmonic because we already know that \(u(x,y)\) is not harmonic.

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