For a doublet,the  stream function is \[\psi=\frac{-k}{2\pi}\frac{\sin\theta}{r}\]

\[V_{r}=\frac{\partial \phi}{\partial r}=\frac{1}{r}\frac{\partial \psi}{\partial \theta}\]

Here \[\psi=\frac{-k}{2\pi}\frac{\sin\theta}{r}
\\\Rightarrow \frac{\partial \psi}{\partial \theta}=\frac{-k}{2\pi}\frac{\cos\theta}{r}
\\\frac{\partial \phi}{\partial r}=\frac{1}{r}\left ( \frac{-k}{2\pi}\frac{\cos\theta}{r} \right )=\frac{-k}{2\pi}\frac{cos\theta}{r^{2}}\]

On integrating with respect to ‘r’

\[\phi=\left ( \frac{-k}{2\pi}cos\theta \right )\left ( -\frac{1}{r} \right )
\\\Rightarrow \phi=\frac{k}{2\pi}\frac{\cos\theta}{r}\]