  We revisit a classical problem of theoretical electrochemistry: the response
of an electric double layer capacitor (EDLC) subject to a small, suddenly
applied external potential. We solve the Debye-Falkenhagen equation to obtain
exact expressions for key EDLC quantities: the ionic charge density, the ionic
current density, and the electric field. In contrast to earlier works, our
results are not restricted to the long-time asymptotics of those quantities.
The solutions take the form of infinite sums whose successive terms all decay
exponentially with increasingly short relaxation times. Importantly, this set
of relaxation times is the same among all aforementioned EDLC quantities; this
property is demanded on physical grounds but not generally achieved within
approximation schemes. The scaling of the largest relaxation timescale
$\tau_{1}$, that determines the long-time decay, is in accordance with earlier
results: depending on the Debye length, $\lambda_{D}$, and the electrode
separation, $2L$, it amounts to $\tau_{1}\simeq\lambda_{D} L/D$ for
$L\gg\lambda_{D}$, and $\tau_{1} \simeq 4 L^2/(\pi^2 D)$ for $L\ll\lambda_{D}$,
respectively (with $D$ being the ionic diffusivity).
