Circuit designers are used to employ analytical formulas and numerically stable expressions for the input-output behaviour of electronic components in preliminary calculations intended to select the most suitable circuit topology to meet prescribed design specifications. Manufactured memristors are highly-nonlinear dynamical circuit elements for new future electronics. However the Differential Algebraic Equation sets, used to capture accurately their nonlinear dynamics, typically consist of involved mathematical expressions, which prevent their analytical integration and the derivation of input-output formulas for circuit design. Adopting certain mathematical techniques, we were recently able to derive for the first time, formulas for the DC behaviour of a real-world memristor exhibiting both non-volatility and fading memory. Particularly, on the basis of an accurate mathematical model, this paper presents a set of analytical expressions for the memory state response of a tantalum oxide resistance switching memory, fabricated at the Palo Alto facilities of Hewlett Packard Labs, to any DC stimulus and for all initial conditions.