transfer function
If you want to summarize an entire linear system in a single formula — how it amplifies, delays, filters, or resonates — the transfer function is that summary. It is the ratio, in the s-domain, of the transformed output to the transformed input, computed for a system that starts at rest. One algebraic expression captures the system's whole input-output behavior.
Formally, with all initial conditions zero, you transform the differential equation and find Y(s) = H(s) X(s), where X(s) is the transformed input, Y(s) the transformed output, and H(s) = Y(s)/X(s) the transfer function. For a constant-coefficient equation, H(s) is simply a ratio of polynomials in s, read straight off the coefficients: for y'' + 2 y' + 5 y = x, H(s) = 1/(s^2 + 2 s + 5). The roots of the denominator are the poles, and their location in the s-plane tells you at a glance whether the system rings, decays, or grows.
The transfer function is the s-domain twin of the impulse response — they are a Laplace transform pair, H(s) and h(t). Its great practical virtue is that combining systems becomes algebra: systems in series multiply their transfer functions, and feedback becomes a simple rational rearrangement. The poles govern stability (all in the left half-plane means decaying, stable behavior), which is why the transfer function is the foundational object of linear-system and control analysis — though the control theory built on top of it is its own subject.
An RC low-pass filter obeys R C v' + v = x, so its transfer function is H(s) = 1/(R C s + 1) — a single pole at s = -1/(R C) that rolls off high frequencies.
Reading the transfer function straight off the circuit equation reveals the single pole that sets the filter's cutoff.
The transfer function is defined only for zero initial conditions; with nonzero initial state the s-domain output has extra terms beyond H(s) X(s), so a transfer function describes the forced response, not the full solution.