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?

Suppose we have the function \(f(x,y)=u(x,y) + iv(x,y)\) such that \(u(x,y)= x^3 + y ^3.\) If we take the contrapositive of Proposition 6.4 it states: If \(u\) is not harmonic in G or \(v\) is not harmonic in G. Then \(f=u+iv\) is not holomorphic in the region G. Notice \(u_{xx}+ u_{yy}=6x + 6y\), which is not equal to 0 when \(y\neq -x\). So by Definition 6.1, \(u\) is not harmonic for some \(x,y\in \mathbb{C} \) But \(x, y\) are not holomorophic on that line since it is impossible to have an open disk centered at \(z_0\). Hence, \(u\) is not holomorphic, for all \(x, y \in \mathbb{C}\). Therefore, it is not possible to find a real function \(v(x,y)\) so that \(x^3 + y^3 + iv(x,y)\) is holomorphic.

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