For a continuous-time LTI system, the transfer function is the ratio of the Laplace-domain output to input under zero initial conditions:

$$H(s)=\frac{Y(s)}{X(s)}.$$

If the system is described by a linear constant-coefficient differential equation, applying the Laplace transform turns derivatives into powers of $s$ and produces an algebraic relation.

For example,

$$a_1\dot y+a_0y=b_0x$$

with zero initial conditions gives

$$H(s)=\frac{b_0}{a_1s+a_0}.$$

The transfer function is also the Laplace transform of the impulse response when the relevant transforms exist:

$$H(s)=\mathcal L{h(t)}.$$

Evaluating $H(s)$ on the imaginary axis, $s=i\omega$, gives the frequency response where that evaluation is valid.

A transfer function describes zero-state input-output behavior. It does not by itself encode a particular nonzero initial condition.