  It is well-known that direct analytic continuation of DGLAP evolution kernel
(splitting functions) from space-like to time-like kinematics breaks down at
three loops. We identify the origin of this breakdown as splitting functions
are not analytic function of external momenta. However, splitting functions can
be constructed from square of (generalized) splitting amplitudes. We establish
the rule of analytic continuation for splitting amplitudes, and use them to
determine the analytic continuation of certain holomorphic and anti-holomorphic
part of splitting functions and transverse-momentum dependent distributions. In
this way we derive the time-like splitting functions at three loops without
ambiguity. We also propose a reciprocity relation for singlet splitting
functions, and provide non-trivial evidence that it holds in QCD at least
through three loops.
