  This article describes a complete theory of spatial coherence for undulator
radiation sources. Current estimations of coherence properties often assume
that undulator sources are quasi-homogeneous, like thermal sources, and rely on
the application of the van Cittert-Zernike theorem for calculating the degree
of transverse coherence. Such assumption is not adequate when treating third
generation light sources, because the vertical(geometrical) emittance of the
electron beam is comparable or even much smaller than the radiation wavelength
in a very wide spectral interval that spans over four orders of magnitude (from
0.1 Angstrom up to 10^3 Angstrom). Sometimes, the so-called Gaussian-Schell
model, that is widely used in statistical optics in the description of
partially-coherent sources, is applied as an alternative to the
quasi-homogeneous model. However, as we will demonstrate, this model fails to
properly describe coherent properties of X-ray beams from non-homogeneous
undulator sources. As a result, a more rigorous analysis is required. We
propose a technique, based on statistical optics and Fourier optics, to
explicitly calculate the cross-spectral density of an undulator source in the
most general case, at any position after the undulator. Our theory, that makes
consistent use of dimensionless analysis, allows relatively easy treatment and
physical understanding of many asymptotes of the parameter space, together with
their region of applicability. Particular emphasis is given to the asymptotic
situation when the horizontal emittance is much larger than the radiation
wavelength, and the vertical emittance is arbitrary. This case is practically
relevant for third generation synchrotron radiation sources.
