{title2=$G(i\omega)$}

The <continuous-time frequency response> is the <frequency response> of a continuous-time <linear time-invariant system>. From zero initial state, a <BIBO stable> system driven by a sustained <sinusoid> $u(t)=A\cos(\omega t)$ approaches the steady response
$$
y_{\mathrm{ss}}(t)=A|G(i\omega)|\cos\bigl(\omega t+\arg G(i\omega)\bigr).
$$
Any decaying initial or switching transient is separate. The <modulus> $|G(i\omega)|$ is the amplitude gain; the <complex argument> is the shift in <oscillation phase>. Thus evaluating a <transfer function> at $s=i\omega$ measures both how much of an oscillation passes through and how far it lags.

For the <first-order low-pass filter>, with $K>0$, direct <complex number> arithmetic gives
$$
G(i\omega)=\frac{K}{1+i\omega\tau},\qquad
|G(i\omega)|=\frac{K}{\sqrt{1+(\omega\tau)^2}},\qquad
\arg G(i\omega)=-\arctan(\omega\tau).
$$
The phase uses the <inverse tangent> branch between $-\pi/2$ and $\pi/2$. Low <angular frequencies> have gain close to $K$ and little lag. High <angular frequencies> have gain approximately $K/(\omega\tau)$ and lag approaching $\pi/2$. This is why the model is a <first-order low-pass filter>: it suppresses fast variation compared with slow variation. Rowell relates <frequency response> gain and phase to <poles> and then introduces <Bode plots> in https://www.ocw.mit.edu/courses/2-004-dynamics-and-control-ii-spring-2008/2a0a97261561177bf1a91b07202bb09a_lecture_32.pdf[MIT 2.004, lecture 32, sections 1 and 2].

At the corner <angular frequency>,
$$
\omega_c=\frac1\tau,\qquad |G(i\omega_c)|=\frac K{\sqrt2},\qquad
\arg G(i\omega_c)=-\frac\pi4.
$$
For $K=2$ and $\tau=3\,\mathrm{s}$, $\omega_c=1/3\,\mathrm{rad}\,\mathrm{s}^{-1}$, the gain is $\sqrt2\approx1.4142$, and the lag is $45$ degrees. The <angular frequency> $\omega$ is measured in <radians> per second; ordinary <frequency> $f$ is measured in hertz (cycles per second), with $\omega=2\pi f$. Consequently the same corner is $f_c=1/(2\pi\tau)\approx0.0531\,\mathrm{Hz}$, not $1/3\,\mathrm{Hz}$.

\Image[/first-order-transfer-frequency-response.png]
{title=<Bode plot> of the <first-order low-pass filter> with $K=2$ and $\tau=3\,\mathrm{s}$}
{description=Gain and <oscillation phase> are plotted against <angular frequency> on a logarithmic horizontal axis. The vertical line marks $\omega_c=1/3\,\mathrm{rad}\,\mathrm{s}^{-1}$, where the gain is $\sqrt2$ and the phase is $-45$ degrees. The gain panel shows $20\log_{10}|G(i\omega)|$; its corner is $3.01$ decibels below the low-frequency gain.}


 Continuous-time frequency response (source code)