Connection formulas for a magnetic flux tube that describe the approximate behaviour of the perturbations across thin layers where dissipative processes (here electrical conductivity) act are derived for the Alfven singularity. The tube may be twisted or untwisted. In an appropriate limit these formulas reduce to jumps across a narrow region. Such jumps are described in terms of introduced functions $F$ and $G$ and their related functions. Jump relations are used to derive approximate dispersion relations, leading to the determination of resonant absorption decay rates. Decay rates are determined for two specific density profiles, the linear one and the sinusoidal profile. Jump conditions pertaining to the slow mode are also discussed. The equivalent jump relations holding for Cartesian geometry are obtained and illustrated for a single magnetic interface, obtaining decay rates.