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 .
*