  We present convergence theory for corrected quadrature rules on uniform
Cartesian grids for functions with a point singularity. We begin by deriving an
error estimate for the punctured trapezoidal rule, and then derive error
expansions. We define the corrected trapezoidal rules, based on the punctured
trapezoidal rule, where the weights for the nodes close to the singularity are
judiciously corrected based on these expansions. Then we define the composite
corrected trapezoidal rules for a larger family of functions using series
expansions around the point singularity and applying corrected trapezoidal
rules appropriately. We prove that we can achieve high order accuracy by using
a sufficient number of correction nodes around the point singularity and of
expansion terms.
