  Accurate evaluation of layer potentials is crucial when boundary integral
equation methods are used to solve partial differential equations. Quadrature
by expansion (QBX) is a recently introduced method that can offer high accuracy
for singular and nearly singular integrals, using truncated expansions to
locally represent the potential. The QBX method is typically based on a
spherical harmonics expansion which when truncated at order $p$ has $O(p^2)$
terms. This expansion can equivalently be written with $p$ terms, however
paying the price that the expansion coefficients will depend on the
evaluation/target point. Based on this observation, we develop a target
specific QBX method, and apply it to Laplace's equation on multiply connected
domains. The method is local in that the QBX expansions only involve
information from a neighborhood of the target point. An analysis of the
truncation error in the QBX expansions is presented, practical parameter
choices are discussed and the method is validated and tested on various
problems.
