Pregunta: Find all f(z)=u(x,y)+iv(x,y),u=Re(f), v=Im(f), if u=x^(2)-y^(2)+x and hat (del )f=0

We will use Milne Thomsan Method

When u is given then : $\mathrm{f}\left(\mathrm{z}\right)=\int \{{\varphi }_1(\mathrm{z},0)-\mathrm{i}{\varphi }_2(\mathrm{z},0)\}\mathrm{d}\mathrm{z}+\mathrm{c}$

where ${\varphi }_1(\mathrm{x},\mathrm{y})=\frac{\partial \mathrm{u}}{\partial \mathrm{x}}\text{and}{\varphi }_2(\mathrm{x},\mathrm{y})=\frac{\partial \mathrm{u}}{\partial \mathrm{y}}$