  We investigate scalar and spinor field theories in a constant magnetic field
at finite temperature and chemical potential. In an external constant magnetic
field the exact solution of the two-point Green functions are obtained by using
the Fock-Schwinger proper-time formalism. We extend it to the thermal field
theory and find the expressions of the Green functions exactly for the
temperature, the chemical potential and the magnetic field. For practical
calculations the contour of the proper-time integral is carefully selected. The
physical contour is discussed in a constant magnetic field at finite
temperature and chemical potential. As an example, behavior of the vacuum
self-energy is numerically evaluated for the free scalar and spinor fields.
