  Using an exact integrodifferential equation we study the properties of the
gauge invariant quark Green's function, defined with a path-ordered gluon field
phase factor along a straight line, in two-dimensional QCD in the large-N_c
limit. The Green's function is found to be infrared finite with singularities
represented by an infinite number of threshold type branch points with a power
equal to -3/2, starting at positive mass squared values. The solution is
analytically determined.
