We find exact, analytic solutions of the holographic AC conductivity at arbitrary frequency $omega$ and temperature $T$, in contrast to previous works where the AC conductivity was analytically obtained usually at small $omega$ and $T$. These solutions enable us to study the analyticity properties of the current-current correlator $G(omega)$ in detail. The first system we study is the AdS$_5$ planar black hole with momentum dissipation, whose extremal limit has an AdS$_2$ factor. Then we study an AdS$_4$ Einstein-dilaton system whose special cases are maximal gauged supergravities. The solutions show how the poles move and how branch cuts emerge as the temperature varies. As a byproduct, we obtain an $R$-current correlator in $mathcal{N}=4$ super-Yang-Mills theory on a sphere at finite temperature in the large $N$ and strong coupling limit.
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