  In every point of a K\"ahler manifold there exist special holomorphic
coordinates well adapted to the underlying geometry. Comparing these K\"ahler
normal coordinates with the Riemannian normal coordinates defined via the
exponential map we prove that their difference is a universal power series in
the curvature tensor and its iterated covariant derivatives and devise an
algorithm to calculate this power series to arbitrary order. As a byproduct we
generalize K\"ahler normal coordinates to the class of complex affine manifolds
with (1,1)-curvature tensor. Moreover we describe the Spencer connection on the
infinite order Taylor series of the K\"ahler normal potential and obtain
explicit formulas for the Taylor series of all relevant geometric objects on
symmetric spaces.
