A dynamic analysis is performed for heat transfer through a fin with constant cross-sectional area and with a specified fin tip temperature. The process starts with a dynamic energy balance around the fin. Laplace transforms and the Bromwich Integral are used to solve analytically the resulting partial differential equation. The final purely analytical solution is compared to the well known steady state solution. The two match exactly as time approaches infinity. Furthermore it is shown that the steady-state and dynamic characteristics of the fin are directly tied to the Biot number.