  The neglected Russian mathematician, N.~S.~Koshliakov, derived beautiful
generalizations of the classical Abel--Plana summation formula through a
setting arising from a boundary value problem in heat conduction. When we let
the parameter $p$ in this setting tend to infinity, his formulas reduce to the
classical Abel--Plana summation formula. Rigorous formulations and proofs of
these summation formulas are given. In his notebooks, Ramanujan derived
different analogues of the Abel--Plana summation formula. One particular
example provides a vast new generalization of the classical transformation
formula for Eisenstein series, which we generalize in Koshliakov's setting.
