  In mathematical physics, time-dependent Green's functions (GFs) are the
solutions of differential equations of the first and second time derivatives.
Habitually, the time-dependent GFs are Fourier transformed into the frequency
space. Then, analytical continuation of the frequency is extended to below or
above the real axis. After inverse Fourier transformation, retarded and
advanced GFs can be obtained, and there may be arbitrariness in such analytical
continuation. In the present work, we establish the differential equations from
which the retarded and advanced GFs are rigorously solved. The key point is
that the derivative of the time step function is the Dirac delta function plus
an infinitely small quantity, where the latter is not negligible because it
embodies the meaning of time delay or time advance. The retarded and advanced
GFs defined in this paper are the same as the one-body GFs defined with the
help of the creation and destruction operators in many-body theory. There is no
way to define the causal GF in mathematical physics, and the reason is given.
This work puts the initial conditions into differential equations, thereby
paving a way for solving the problem of why there are motions that are
irreversible in time.
