Theory of potential step chronoamperometry under limiting current conditions and for purely diffusional transport at a microband electrode has been a subject of several studies. However, no complete and explicit expressions for the Faradaic current density and the Faradaic current have been reported thus far. In the present study such expressions are derived using a novel theoretical approach. The microband is considered as a limiting case of an elliptic cylinder, when the length of the smallest diameter of the elliptic cross-section tends to zero. Solution to the problem of heat conduction around an elliptic cylinder, due to Tranter [Quart. J. Mech. Appl. Math. 4 (1951) 461], is utilised. Following Tranter, the method of separation of variables in the Laplace space is used, resulting in two Mathieu differential equations. The concentration of the depolarizer, the Faradaic current density, and the Faradaic current, are then expressed as inverse Laplace transforms of certain infinite series involving appropriate Mathieu functions. The series are amenable to further analytical examinations. In particular, it is proven that a quasi-steady state develops at large time. It is also demonstrated how the popular idea, of an hemicylinder electrode "equivalent" to a microband, has to be understood to be correct. Numerical evaluation of the series provides unprecedentedly highly accurate solution values. Hence, the present solutions should be preferred over formerly used low-accurate formulae, for the purposes of experimental data analysis, and for the testing of modelling/simulation techniques. (C) 2015 Elsevier Ltd. All rights reserved.