Can a dipole with finite spacing between poles be represented by pure multipoles centered at the origin?

Say for example that I have a dipole with finite spacing $2\epsilon$ between the poles. I have read that this field can be expressed as the summation of pure multipoles that are centered at the origin. Pure multipoles imply a point singularity at the origin where the pole spacing goes to $0$. The general exterior multipole expansion in cylindrical coordinates is given by

\begin{align} \psi(r,\theta) = P_0 \ln r + \sum_{k = 1}^\infty \dfrac{Q_k \cos(k\theta)+R_k\sin(k\theta)}{r^k}. \end{align}

I do not understand how a summation of these terms can converge to the field of a dipole with finite spacing. At the origin, all of the pure dipole terms are singular while the field of the finite dipole field is finite. At the poles of the finite dipole, the field is singular while the field of the pure multipole terms is not. If this can be done, how can i find the multipole moments?


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