6.8
6.8 Is it possible to find a real function \(v(x,y)\) so that \(x^3+y^3+iv(x,y)\) is holomorphic?
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Consider \(f(x,y)=u(x,y)+iv(x,y)\) where \(u(x,y)=x^3+y^3\). Then \(u_{xx}+v_{yy}=6x+6y\) Note that \(u(x,y)\) is not harmonic because \(6x+6y \neq 0\) for some \(x,y \in \mathbb{R}\). Using Theorem 6.4 by the contrapositive: if \(u\) is not harmonic in \(G\) or \(v\) is not harmonic in \(G\), then \(f\) is not holomorphic in \(G\). Hence, because \(u(x,y)\) is not harmonic, by Proposition 6.4, we cannot find a real valued function \(v(x,y)\) so that \(x^3+y^3+iv(x,y)\) is holomorphic.