In electrical circuit theory, the zero state response (ZSR), is the behaviour or response of a circuit with initial state of zero. The ZSR results only from the external inputs or driving functions of the circuit and not from the initial state. The total response of the circuit is the superposition of the ZSR and the ZIR, or Zero Input Response. The ZIR results only from the initial state of the circuit and not from any external drive. The ZIR is also called the natural response, and the resonant frequencies of the ZIR are called the natural frequencies. Given a description of a system in the s-domain, the zero-state response can be described as Y(s)=Init(s)/a(s) where a(s) and Init(s) are system-specific. Zero state response and zero input response in integrator and differentiator circuits One example of zero state response being used is in integrator and differentiator circuits. By examining a simple integrator circuit it can be demonstrated that when a function is put into a linear time-invariant (LTI) system, an output can be characterized by a superposition or sum of the Zero Input Response and the zero state response. A system can be represented as with the input on the left and the output on the right. The output can be separated into a zero input and a zero state solution with The contributions of and to output are additive and each contribution and vanishes with vanishing and This behavior constitutes a linear system. A linear system has an output that is a sum of distinct zero-input and zero-state components, each varying linearly, with the initial state of the system and the input of the system respectively. The zero input response and zero state response are independent of each other and therefore each component can be computed independently of the other. Zero state response in integrator and differentiator circuits The Zero State Response represents the system output when When there is no influence from internal voltages or currents due to previously charged components Zero state response varies with the system input and under zero-state conditions we could say that two independent inputs results in two independent outputs: and Because of linearity we can then apply the principles of superposition to achieve Verifications of zero state response in integrator and differentiator circuits To arrive at general equation The circuit to the right acts as a simple integrator circuit and will be