  We study minimal timelike surfaces in $\mathbb R^3_1$ using a special
Weierstrass-type formula in terms of holomorphic functions defined in the
algebra of the double (split-complex) numbers. We present a method of obtaining
an equation of a minimal timelike surface in terms of canonical parameters,
which play a role similar to the role of the natural parameters of curves in
$\mathbb R^3$. Having one holomorphic function that generates a minimal
timelike surface, we find all holomorphic functions that generate the same
surface. In this way we give a correspondence between a minimal timelike
surface and a class of holomorphic functions. As an application, we prove that
the Enneper surfaces are the only minimal timelike surfaces in $\mathbb R^3_1$
with polynomial parametrization of degree 3 in isothermal parameters.
