  The normal forms up to the third order for a Hopf-steady state bifurcation of
a general system of partial functional differential equations (PFDEs) is
derived based on the center manifold and normal form theory of PFDEs. This is a
codimension-two degenerate bifurcation with the characteristic equation having
a pair of simple purely imaginary roots and a simple zero root, and the
corresponding eigenfunctions may be spatially inhomogeneous. The PFDEs are
reduced to a three-dimensional system of ordinary differential equations and
precise dynamics near bifurcation point can be revealed by two unfolding
parameters. The normal forms are explicitly written as functions of the
Fr\'echet derivatives up to the third orders and characteristic functions of
the original PFDEs, and they are presented in a concise matrix notation, which
greatly eases the applications to the original PFDEs and is convenient for
computer implementation. This provides a user-friendly approach of showing the
existence and stability of patterned stationary and time-periodic solutions
with spatial heterogeneity when the parameters are near a Turing-Hopf
bifurcation point, and it can also be applied to reaction-diffusion systems
without delay and the retarded functional differential equations without
diffusion.
