  In this paper, we compute holographic Euclidean thermal correlators of the
stress tensor and $U(1)$ current from the AdS planar black hole. To this end,
we set up perturbative boundary value problems for Einstein's gravity and
Maxwell theory in the spirit of Gubser-Klebanov-Polyakov-Witten, with
appropriate gauge fixing and regularity boundary conditions at the horizon of
the black hole. The linearized Einstein equation and Maxwell equation in the
black hole background are related to the Heun equation of degenerate local
monodromy. Leveraging the connection relation of local solutions of the Heun
equation, we partly solve the boundary value problem and obtain exact two-point
thermal correlators for $U(1)$ current and stress tensor in the scalar and
shear channels.
