If W = ϕ + iΨ represents the complex potential for an electric field.

Given \(\Psi =x^2-y^2 + \dfrac{x}{x^2 +y^2}\), then the function ϕ is

Concept:

From CR equations,

\(\frac{{\partial \phi }}{{\partial {\rm{x}}}} = \frac{{\partial {\rm{\psi }}}}{{\partial {\rm{y}}}}\)

\(\frac{{\partial \phi }}{{\partial {\rm{y}}}} = \frac{{\partial {\rm{\psi }}}}{{\partial {\rm{x}}}}\)

Calculation:

\(\frac{{\partial \phi }}{{\partial {\rm{x}}}} = \frac{\partial }{{\partial {\rm{y}}}}\left[ {{{\rm{x}}^2} - {{\rm{y}}^2} + \frac{{\rm{x}}}{{{{\rm{x}}^2} + {{\rm{y}}^2}}}} \right] = - 2{\rm{y}} - \frac{{2{\rm{xy}}}}{{{{\left( {{{\rm{x}}^2} + {{\rm{y}}^2}} \right)}^2}}}\)

Integrating w.r.t ‘x’ by keeping y constant

\(\Rightarrow \phi = {\rm{\;}} - 2{\rm{xy}} + \frac{{\rm{y}}}{{{{\rm{x}}^2} + {{\rm{y}}^2}}} + {\rm{C\;}}\)

\(\frac{{\partial {\rm{\psi }}}}{{\partial {\rm{y}}}} = \frac{\partial }{{\partial {\rm{x}}}}\left[ {{{\rm{x}}^2} - {{\rm{y}}^2} + \frac{{\rm{x}}}{{{{\rm{x}}^2} + {{\rm{y}}^2}}}} \right] = - 2{\rm{x}} - \frac{{{{\rm{x}}^2} - {y^2}}}{{{{\left( {{{\rm{x}}^2} + {{\rm{y}}^2}} \right)}^2}}}\)

Integrating w.r.t ‘y’ by keeping x constant 

\( \Rightarrow \phi = {\rm{}} - 2{\rm{xy}} + \frac{{\rm{y}}}{{{{\rm{x}}^2} + {{\rm{y}}^2}}} + {\rm{C\;}}\)