  We develop a linear response theory by computing the asymptotic value of the
order parameter from the linearized equation of continuity around the
nonsynchronized reference state using the Laplace transform in time. The
proposed theory is applicable to a wide class of coupled phase oscillator
systems and allows for any coupling functions, any natural frequency
distributions, any phase-lag parameters, and any values for the time-delay
parameter. This generality is in contrast to the limitation of the previous
methods of the Ott--Antonsen ansatz and the self-consistent equation for an
order parameter, which are restricted to a model family whose coupling function
consists of only a single sinusoidal function. The theory is verified by
numerical simulations.

Adaptation and Self-Organizing Systems;Statistical Mechanics