A particular case of dipole is the so-called doublet, in which the quantity a tends to zero so that the source and sink both move towards the origin.

The complex potential of a doublet

is obtained making the limit of the dipole potential for vanishing a with the constraint that the intensity of the source and the sink must correspondingly tend to infinity as a approaches zero, the quantity

being constant (if we just superimpose a source and sink at the origin the resulting potential would be W=0)

Hint: Develop and in a Taylor series in the neighborhood of the origin, assuming small a.