The transfer function is a mathematical representation that describes the system's output for each possible input in the frequency domain.

Consider a general n^th-order, linear, time-invariant differential equation. This equation characterizes the system where one variable represents the input, and another represents the output.

Applying the Laplace transform to both sides of this equation results in an algebraic expression.

Assuming that all initial conditions are zero, this equation is further simplified.

The ratio of the output's Laplace transform to the input's Laplace transform is called the transfer function.

The transfer function is represented as a block diagram, with the input on the left, the output on the right, and the system transfer function inside the block.

The transfer function's denominator is identical to the characteristic polynomial of the differential equation.

Consider a first-order differential equation. The transfer function for this equation is calculated by taking the Laplace transform on both sides, assuming zero initial conditions.

Upon simplification, the result is a transfer function representing the system's response to an input in the frequency domain.