### 8.4 Existence of expansions using Fuchsian methods, T^{3}-case

In [54], Kichenassamy and Rendall proved the existence of expansions in the real analytic setting. In
other words, they proved that, given real analytic with , there is a unique solution
to Equations (16) – (17) with expansions of the form of Equations (37) – (38). Note that the number of
functions that are freely specifiable in the expansions coincides with the number of functions that need to
be specified in order to obtain a unique solution to the initial value problem corresponding to
Equations (16) – (17). For reasons mentioned in Section 8.2, the expansions suffer from a potential
consistency problem in the case of and . However, in [54] it was proven that if is
constant, the condition on can be relaxed to . The regularity condition of [54] was relaxed to
smoothness in [68].